Momentary Equilibrium in Transient Kinetics and Its Application for

May 24, 2013 - Momentary Equilibrium in Transient Kinetics and Its Application for Estimating the Concentration of Catalytic Sites. Evgeniy A. Redekop...
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Momentary Equilibrium in Transient Kinetics and Its Application for Estimating the Concentration of Catalytic Sites Evgeniy A. Redekop,† Gregory S. Yablonsky,‡ Vladimir V. Galvita,† Denis Constales,*,¶ Rebecca Fushimi,‡,§ John T. Gleaves,∥ and Guy B. Marin† †

Laboratory for Chemical Technology (LCT), Ghent University, Krijgslaan 281, S5 B-9000 Gent, Belgium Parks College of Engineering, Aviation and Technology, Saint Louis University, 3450 Lindell Boulevard, St. Louis, Missouri, United States 63103 ¶ Department of Mathematical Analysis, Ghent University, Galglaan 2, Gent, Belgium 9000 § The Langmuir Research Institute, 396 North Euclid Avenue, Suite A, St. Louis, Missouri, United States 63108 ∥ Department of Energy, Environmental, and Chemical Engineering, Washington University in Saint Louis, 1 Brookings Drive, St. Louis, Missouri, United States 63130 ‡

S Supporting Information *

ABSTRACT: We describe the novel concept of momentary equilibrium (ME), a special event during a pulse-response transient experiment in which the non-steady-state rates of adsorption and desorption of a probe molecule are instantaneously balanced. In the absence of other reactions, any system with reversible adsorption will always pass through ME during a pulse-response experiment with effusion. We also suggest a new method for measuring the concentration of adsorption sites on heterogeneous catalysts and the corresponding equilibrium constant by observing momentary equilibrium in thin-zone temporal analysis of products (TZTAP) pulse-response experiments with modulated pulse intensity. The suggested method employs reversible adsorption of probe molecules, contrary to traditional methods of counting adsorption sites which utilize irreversible reactions.

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INTRODUCTION Chemical equilibrium has been one of the central concepts of chemical thermodynamics and kinetics since the times of Guldberg, Waage, and van’t Hoff.1,2 When a closed system is in equilibrium, the concentrations of all species within this system remain constant in time (dc/dt = 0), and in accordance with the principle of detailed equilibrium,3 the rate of each forward reaction is balanced by the rate of its reverse counterpart (r+ = r−). Importantly, the ratio between concentrations is governed by the equilibrium constant of the overall reaction Keq =

∏ cpn /∏ crn p

p

reversible reaction is instantaneously balanced by the reverse rate at some moment in time during a transient process initiated by a pulse input. We show that the ME state for a reversible reaction can be identified and characterized in temporal analysis of products (TAP) experiments,6 which provide high-fidelity non-steady-state kinetic information for heterogeneously catalyzed processes. Furthermore, we suggest a novel method for determining intrinsic adsorption parameters based on gas and surface concentration values at the ME state. The intrinsic parameters of adsorption, including the total concentration of adsorption sites and the equilibrium constant, are typically determined by the regression of the equilibrium or steady-state concentrations of probe molecules in the gas phase and on the catalyst using an appropriate isotherm model. These concentration values may be measured using one or several of the following methods: (1) Step-by-step titration of all available adsorption sites can be performed by pulsed chemisorption of probe molecules such as hydrogen or CO followed by a temperature programmed desorption (TPD) measurement; e.g., see Mekki et al.7 or Karakaya et al.8 (2) The concentrations of gas and surface species can be determined in an equilibrated closed system or an open system at steady state by very sensitive pressure and microbalance mass measurements; e.g.,

r

(1)

r

where subscripts r and p refer to a reactant and a product, respectively. Both closed and open systems are generally not in equilibrium. However, the ratios between concentrations of certain species within a nonequilibrated system can be approximated by the equilibrium constants of the relevant reaction steps which are in pseudoequilibrium. Pseudoequilibrium occurs only within specific periods of time and/or ranges of operating conditions (i.e., concentrations and temperature). Pseudoequilibrium approximation is widely used in classical theoretical models of catalytic kinetics to express the occupancy of active sites by reaction intermediates through the concentrations of observable reactants and products, e.g., the Michaelis−Menten equation4 and the Langmuir−Hinshelwood−Hougen−Watson equations.5 Here, we describe the concept of momentary equilibrium (ME), another state in which chemical systems exhibit equilibrium properties. ME occurs in open systems with effusion when the forward rate of a © 2013 American Chemical Society

Special Issue: NASCRE 3 Received: Revised: Accepted: Published: 15417

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Figure 1. Temporal analysis of products (TAP): (A) the principle of thin-zone (TZ) TAP experiments; (B) the kinetically “model-free” reconstruction of TZ reaction rates and gas concentrations via the Y-Procedure followed by the calculation of surface storages by rate integration.

see Ostrovskii9 and Zhu et al.10 (3) The equilibrium or steadystate concentrations of gas and surface species can be estimated from quantitatively calibrated spectroscopic data; e.g., see Sirita et al.11 (4) The total uptake of an isotopic label on a catalyst can be determined under state−state reaction conditions in steady-state isotopic transient kinetic analysis (SSITKA) experiments.12,13 Measurement of site concentrations using each of these methods poses a considerable challenge, especially under nonsteady-state conditions. For example, pulsed chemisorption must often be performed in a temperature range which is significantly lower than the temperature at which the reaction is typically carried out in order to slow down the immediate desorption of probe molecules. Very sensitive microbalance measurements and quantitative spectroscopic data are rarely available. Finally, SSITKA, which provides only the integral amount of an isotopic label on a catalyst during a steady-state process, is not suitable for fast non-steady-state measurements. These limitations motivate development of novel techniques for measuring the concentration of adsorption sites and other adsorption parameters, particularly for high surface-area multicomponent heterogeneous catalysts under non-steadystate conditions and realistic reaction temperatures. In order to address this challenge, we have employed the concept of ME to obtain reliable equilibrium concentration values from pulseintensity modulated (PIM) TAP experiments. A series of simulated TAP experiments were used to test the concept of ME and its application for quantifying the concentration of adsorption sites, and the ME methodology was applied to CO adsorption on platinum nanoparticles supported on hydrotalcite. The current limitations of the suggested methodology are discussed, followed by perspectives on its potential applications.

conversion of reactants takes place.17−20 The injected molecules of gas travel through the microreactor via Knudsen diffusion and eventually escape into an adjacent vacuum chamber containing a quadrupole mass spectrometer (QMS) detector. The QMS is tuned to an appropriate mass-to-charge ratio before each pulse period in order to detect various reactants and products exiting the microreactor. The primary QMS signals are recorded with millisecond resolution and translated into exit-flow rates for various species. A special algorithm called the Y-Procedure can then be used to further translate these exit-flow rate curves into the gas concentrations and reaction rates localized within the catalytic zone.21,22 Contrary to conventional data-analysis techniques23,24 which are based on the model regression, the YProcedure does not require a priori assumptions about the nature of an underlying reaction mechanism to extract rateconcentration transients. TAP experiments provide several opportunities for precise determination of the concentration of sites on a catalyst. Long sequences of small TAP pulses are particularly well suited for step-by-step titration of sites in those cases where probe molecules irreversibly change the catalyst. Two types of titration experiments may be performed: an irreversibly adsorbing substance can be used to gradually fill adsorption sites (e.g., oxygen uptake on metals22,25) or a probe gas can be used in a reverse titration, removing preadsorbed species from a catalyst until they are fully depleted. The latter procedure has been used to measure the total oxygen storage capacity (OSC) by pulsing CO over preoxidized catalysts26,27 and to quantify carbon deposits on catalysts by incrementally burning carbon off with oxygen pulses. If the number of probe molecules in each TAP pulse is much smaller than the quantity of catalytic species with which they react, then each pulse can be considered state defining.28 State-defining pulses do not significantly change the catalyst which they kinetically characterize. The evolution of kinetic parameters as a function of the changing catalyst state, when extracted from each pulse in a state-altering multipulse sequence, can provide valuable information about reaction mechanisms and may even reveal the existence of multiple types of active sites.29 Pulse-by-pulse titration of sites is typically performed with probe molecules which, at a given temperature, do not desorb from the catalyst over the course of the experiment. The

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BACKGROUND ON THE TEMPORAL ANALYSIS OF PRODUCTS (TAP) The principle of TAP pulse-response experiments14−16 is schematically depicted in Figure 1A. In brief, a pulse-response experiment is initiated by a small and narrow pulse of a gas mixture which is sent into an evacuated packed microreactor. Figure 1A shows a TAP microreactor packed in the thin-zone (TZ) configuration, which maintains the macroscopic spatial uniformity of catalytic samples even when high per-pulse 15418

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adsorption of such molecules within a certain temperature range can be considered as apparently irreversible. The necessity of apparently irreversible interactions between the catalyst and probe molecules for site titration presents a limitation for pulse-response methods such as TAP because the adsorption steps of many industrially relevant catalytic processes are apparently reversible under elevated operating temperatures. In such cases, the concentration of adsorption sites must be estimated as one of the regression parameters, extrapolated from a lower temperature where reactant desorption is negligible, or measured ex situ. The body of evidence which has been recently accumulated in support of temperature-dependent catalyst morphology30−33 motivates development of new methods for measuring the total concentration of catalytic sites at more realistic operating temperatures. Here, we suggest a novel approach for determining the concentration of adsorption sites in TAP experiments, which employs characterization of the gas and catalyst compositions at momentary equilibrium during a reversible adsorption. Contrary to conventional techniques of pulsed chemisorption, this novel approach is not limited to low temperatures and can be performed on short time scales (minutes).

Table 1. Simulation Parameters Used for Virtual TZ TAP Experiments with Reversible Adsorption parameter reactor length, L (m) TZ (catalyst) length, Lcat (m)

METHOD AND THEORY In order to introduce the concept of momentary equilibrium (ME) and a new method for estimating intrinsic adsorption parameters, we simulated virtual TAP experiments with reversible adsorption. The simulated data were then analyzed via the Y-Procedure to obtain rate-concentration transients and elucidate their kinetic behavior. Reversible adsorption of carbon monoxide was used as a model because it is involved in many industrially relevant catalytic processes and is frequently used as a probe reaction in catalyst characterization; e.g., see Freund et al.34 and Lear et al.35 CO adsorption on metal catalysts is rather complex; CO adsorbs in several distinct forms on energetically different sites, 36 forms surface islands due to lateral interactions,37 and may cause surface reconstructions.38 However, in simplified microkinetic models,1,39,40 CO adsorption is often assumed to occur on a single type of catalytic site Z

representative of a typical TAP experiment. The details of numerical simulations can be found elsewhere.23,43 One of the objectives of this study was to determine the degree to which realistic levels of high-frequency noise will affect the results of ME analysis. It has previously been shown21 that the YProcedure amplifies high-frequency oscillations which are universally present in TAP data, causing significant complications in data interpretation. We therefore introduced artificial noise into the simulated exit-flow rate data employing a noise model developed by Roelant et al.,44 the parameters of which are also listed in Table 1. Figure 1A schematically depicts one such virtual experiment. Here, a pulse of CO/Ar mixture is introduced into the evacuated TAP microreactor with an adsorbing catalyst packed in the thin-zone configuration. The exit-flow rates of both argon and CO, also shown in Figure 1A, were normalized by the number of corresponding molecules contained within the initial pulse. The CO curve initially lies below the argon curve due to adsorption. After the peak, the CO curve crosses the argon curve and decays toward zero, always lying above the argon curve due to desorption of previously adsorbed molecules. Two-step postprocessing of simulated CO exit-flow rate data is schematically shown in Figure 1B. In the first step, the transient gas concentration of carbon monoxide CCO and its consumption rate RCO within the uniform catalytic zone are obtained from exit-flow rate curves via the Y-Procedure. In the second step, the surface concentration of CO on the catalyst CZCO is reconstructed by integration of the CO consumption rate in time according to

k+

k−

(2)

According to the law of mass-actions and the mean-field approximation, the non-steady-state rate of CO consumption during reversible adsorption, RCO, is given by R CO(t ) = k+CZ(t )CCO(t ) − k −CZCO(t ) = k+[CZ,tot − CZCO(t )]CCO(t ) − k −CZCO(t )

0.036 0.8 × 10−3 (unless otherwise specified) 0.024 (unless otherwise specified) 1.8 × 10−5 0.53 15.0 × 10−6

position of the TZ middle from the entrance, LTZ (m) cross-sectional area, A (m2) bed voidage, ε catalyst mass, mcat (kg) Transport reference diffusion coefficient of argon at 2.0 × 10−3 298.15 K, D (m2/s) Kinetics intrinsic adsorption constant, k (m3/mol/s) 10.0 × 103 intrinsic desorption constant, k (1/s) 30.0 total concentration of adsorption sites, CZ,tot 2.5 × 10−4 (mol/kgcat) Other Simulation Conditions temperature, T (K) 473.15 total pulse intensity, Np (mol) 0.5 × 10−9−3.0 × 10−8 pulse composition, CO/Ar 0.4:0.6 number of grid points per meter, Ng 1.0 × 104 Noise and Filtering absolute error, εabs (V) 0.2 × 10−6 relative error, εrel 0.01 correlation time, τcorr (s) 0.789 × 10−3 main frequency, ω (Hz) 50.0 absolute amplitude, aabs (V) 0.2 × 10−6 relative amplitude, arel 0.01 Y-Procedure smoothing parameter, σ 2.0

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CO + Z ⇌ ZCO

value Geometry

(3)

3

where k+ (m /mol/s) and k− (1/s) are the forward and reverse kinetic constants, CZ,tot (mol/kg) is the total concentration of adsorption sites, CZ and CZCO (mol/kg) are, respectively, the concentrations of currently free and occupied sites, and CCO (mol/m3) is the gas concentration of CO. The same simplified model has been previously used in many TAP studies dealing with CO adsorption, e.g., in Dekker et al.41 and Nijhuis et al.42 In this study, it was assumed that the adsorption of CO is reversible and proceeds according to eqs 2 and 3. The virtual TAP experiments presented below were simulated using rate expression 3 and the geometric, transport, and kinetic parameters listed in Table 1 with values 15419

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∫0

tend

R CO(t ) dt

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indicate the level of spatial nonuniformity during an experiment. The upper bound of the gray area corresponds to the catalytic zone inlet; the lower bound corresponds to the catalytic zone outlet, and the dashed line inside the gray-shaded area indicates its spatial mean. Finally, the noisy solid line is the curve reconstructed from exit-flow rate data via the YProcedure. The spatial nonuniformity of all three intrapulse characteristics did not exceed 20% at any time, which according to the literature20 indicates that the thin-zone assumption required by the Y-Procedure was valid in this experiment. The reconstructions of the consumption rate and the surface CO concentration closely approximate their mean spatial values and always lie within the corresponding gray-shaded areas. The gas CO concentration, on the other hand, is clearly overestimated upon its reconstruction from the exit-flow rate data. The shape of the CCO curve from Figure 2A resembles the shape of the CO exit-flow rate curve from Figure 1A. The behavior of the gas-phase CO concentration in the catalytic zone is governed by two factors: the global concentration gradient imposed by external conditions at the domain boundaries and the local adsorption/desorption process on the catalyst. Driven by the global concentration gradient, CCO in the catalytic zone reaches its peak and then decays to zero as all molecules inevitably escape the microreactor. The surface concentration of CO behaves similarly, though it peaks later than the gas concentration of CO. This delay in peak surface concentration occurs because adsorption from the gas phase continues even after the majority of CO molecules have been driven out of the catalytic zone by the global concentration gradient. Within the simulation, the CO consumption rate shown in Figure 2A passes through an initial peak, in a similar manner to the gas and surface concentrations, but it then crosses zero, passes through a minimum, and slowly decays to zero from the negative side. Initially, the CO consumption rate is positive and rises rapidly due to adsorption initiated and sustained by the incoming peak of gas-phase CO. After the gas-phase CO leaves the catalytic zone, desorption can no longer be compensated by adsorption and begins dominating the overall reaction dynamics. This causes the consumption rate to decrease. At the moment when the consumption rate changes its sign, the surface concentration of CO attains its maximum and starts decreasing, at which point both the adsorption and desorption terms of eq 3 begin their final decay toward zero. One of the most interesting features of the transient consumption rate is the presence of a special point in time where adsorption is instantaneously balanced by desorption, at which the consumption rate is zero and the surface CO concentration is at its maximum. This special time point is marked in Figure 2A by red circles. This event, which we have called momentary equilibrium (ME), is unique to open systems with pulse input and has a physicochemical meaning similar to the true chemical equilibrium of closed systems. Figure 2B shows by comparison the behavior of RCO, CCO, and CZCO in a uniform closed system moving toward equilibrium. All three kinetic characteristics evolve monotonously toward their equilibrium values and stay there indefinitely. Contrary to this case of true equilibrium, the consumption rate and CO concentrations during a pulse-response experiment exhibit more complex features and inevitably pass through the ME point. In the absence of other reactions, the existence of at least one ME during a pulse-response experiment is self-evident from eq 3. The ME point of a pulse-response experiment is not

(4)

The details of the practical implementation of these postprocessing steps can be found in Redekop et al.22,45 The following subsections describe a set of intrapulse kinetic characteristics CCO(t), CZCO(t), and RCO(t) reconstructed from simulated exit-flow rate curves. Henceforward, the term “intrapulse” (i.e., within an individual pulse) is used to distinguish the instantaneous kinetic characteristics which are time-resolved on the order of milliseconds from the averaged “interpulse” (i.e., from one pulse to the next) characteristics which are time-resolved on the order of seconds. The latter include all quantities derived from the moments of exit-flow rate curves, such as per-pulse conversions14 and “primary kinetic characteristics” introduced by Yablonskii et al.46,47 Momentary Equilibrium (ME) Observed in Intrapulse Kinetic Behavior. The transient intrapulse kinetic characteristics of CO adsorption, i.e., CCO(t), RCO(t), and CZCO(t), are depicted in Figure 2A for the virtual experiment with initial pulse intensity of 0.4 × 10−8 (molCO). Several pieces of information are presented in this figure for each intrapulse characteristic. The gray-shaded area represents the spread of a given characteristic within the catalytic zone, which is plotted to

Figure 2. Nonsteady-state consumption rate, RCO(t), gas concentration, CCO(t), and surface concentration, CZCO(t), of CO during a single-site reversible adsorption: (A) Rate-concentration data during a TZ TAP pulse-response experiment: simulated values within the catalyst zone (gray-shaded area), mean simulated values (dashed line), values reconstructed by the Y-Procedure (solid line), momentary equilibrium (red circles), and the uncertainty range (between vertical dotted lines); (B) Rate-concentration data during a batch experiment in a closed system, i.e., the approach of true chemical equilibrium. 15420

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The Y-Procedure analysis of an averaged exit-flow rate followed by the calculation of CO surface concentration according to eq 4 results in a single set of rate-concentration curves passing through an ME point with coordinates CZCO,ME and CCO,ME. Then, the same measurement must be repeated for multiple pulse sizes to increase the peak surface concentration of adsorbing molecules. The variation of pulse sizes will elucidate the dependency of CZCO,ME on CCO,ME which can be regressed with an isotherm to yield Keq and CZ,tot estimates. Equilibrium data derived from the ME positions for virtual TAP experiments are shown in Figure 3. The same figure also shows the Langmuir isotherm and the total concentration of sites originally used for simulations.

stable, and a reacting system never remains in this state for any measurable time. Besides true chemical equilibrium, it is also interesting to compare the concept of ME to the concept of “TAP equilibrium” discussed by Yablonsky.46 The latter equilibrium-like state is hypothetically reached by a reacting system in the limit of an infinitely long TAP reactor or in the limit of a long pulse sequence in which the initial composition of each successive pulse matches the integral composition of the preceding exit-flow. The ratio of zeroth moments of reactants and products in this equilibrium-like state is governed by an effective equilibrium constant given by the true equilibrium constant multiplied by the ratio of effective diffusion coefficients. Attaining such a hypothetical “TAP equilibrium” in a real experiment presents a considerable challenge. Momentary equilibrium, on the other hand, is achieved in a finite time at a finite distance inside the microreactor. Next, we demonstrate how to employ the concept of ME for estimating the intrinsic adsorption parameters. Using ME for Measuring the Concentration of Adsorption Sites in Pulse-Intensity Modulation (PIM) Experiments. The momentary equilibrium exhibited by CO during a pulse-response experiment is characterized by the gas and surface CO concentrations made available via the YProcedure. These ME concentrations can be used to estimate catalyst adsorption parameters, including the intrinsic equilibrium constant and total concentration of sites. Setting the consumption rate in eq 3 to zero leads to CZCO,ME =

KeqCZ,totCCO,ME 1 + KeqCCO,ME

Figure 3. Langmuir adsorption isotherm: CZCO,ME vs CCO,ME (i.e., eq 5) used for simulations (solid line) with Keq = 333.3 and CZ,tot = 2.5 × 10−4 (molZ/kgcat) (dotted line) and reconstructed ME concentrations (circles).

(5)

This expression relates compositions of gas and surface phases in the ME point to each other, and in this example, it is equivalent to the classical Langmuir isotherm.48 From eq 5, it follows that a shift of the ME position in the rate-composition space (i.e., changing CZCO,ME and CCO,ME) can be used to estimate Keq and CZ,tot. The ME position can be manipulated by controlling the composition and size of TAP pulses. We suggest the following pulse-intensity modulated (PIM) TZ TAP experiment employing the principle of shifted equilibria for characterization of reversible adsorption on catalytic surfaces. First, a sequence of pulses must be recorded at a given temperature and constant pulse intensity. If probe molecules which were adsorbed during a pulse fully desorb within the time scale of this pulse, that is before the next pulse is introduced, than exit-flow rate curves should not differ within a pulse sequence. These exit-flow rate curves can be averaged in order to improve the signal-to-noise ratio. However, if some of the adsorbed probe molecules remain on the surface longer than the time-delay between two successive pulses, than exitflow rate curves will change in size within the sequence as a result of cumulative increase in surface coverage. Consequently, the position of the instantaneous ME surface storage will drift toward lower values within this pulse sequence, and the averaging of such data will not be justified. We discuss the implications of incomplete desorption of probe molecules on the time scale of a single pulse for the applicability range of the ME method in the Conclusions and Perspectives section. The results presented hereafter were obtained for those cases in which probe molecules completely desorb by the end of each pulse.

The Langmuir isotherm in Figure 3 exhibits two types of asymptotic behavior: a linear dependency for low ME coverages and a saturation plateau for high ME coverages. This suggests that the quality of parameter estimation for an adsorption isotherm depends directly on the range of surface coverages sampled in PIM TZ TAP experiments. The ratio of pulseintensity to the total number of sites on the sample must be varied by at least 1 order of magnitude in order to capture the essential nonlinearity of an isotherm. Otherwise, CZCO,ME vs CCO,ME dependency will appear to be linear, and it may not be possible to estimate the site concentration separately from the equilibrium constant. The distinction between these two asymptotic cases is even more pronounced when one considers not only the zero-rate isotherm sampled by ME points but also the entire set of time points describing the evolution of adsorption kinetics toward and away from momentary equilibrium. Trajectories in the Rate-Composition Space for StateDefining and State-Altering Experiments. The estimation of intrinsic adsorption parameters in PIM TZ TAP experiments, as described in the previous subsection, utilizes the values of intrapulse kinetic characteristics at a single point of momentary equilibrium for each pulse-intensity, i.e., RCO, CCO, and CZCO at tME. Other time points within these experiments contain additional information about adsorption kinetics, which can be used to estimate the intrinsic adsorption and desorption constants individually, as opposed to their ratio Keq. In order to extract this additional information, intrapulse kinetic characteristics must be analyzed with respect to each other within each 15421

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pulse. In order to visualize the evolution of intrapulse characteristics during a pulse-response experiment, we have previously suggested22 a representation of time-dependent intrapulse data as kinetic trajectories in a rate-composition space. Each geometric point in this space represents a particular state of a reacting system with coordinates corresponding to rates and concentrations (gaseous and surface). Given an initial condition, all points accessible by the system belong to a surface defined by the underlying reaction network and its kinetic parameters, including concentrations of sites and all kinetic constants. The ME-derived isotherm discussed previously represents only one slice of this kinetic surface when the consumption rate is equal to zero. Figure 4 contains simulated and reconstructed intrapulse kinetic data in a rate-composition space for different pulse-

Figure 5. Estimation of adsorption parameters from reconstructed rate-composition data: (A) the state-defining trajectory corresponding to 0.5 nmolCO plotted in modified coordinates of eq 6 (first 700 points); (B) the state-altering trajectory corresponding to 12.0 nmolCO plotted in modified coordinates of eq 6 (first 1000 points). In both plots, simulated data are shown as a dashed line, while reconstructed data are shown as a solid line. The plots also depict the adsorption parameters which can be estimated from reconstructed trajectories.

Figure 4. 3D view of kinetic trajectories in the rate-composition space, RCO(t) vs CC(t) vs CZCO(t): simulated trajectories (dashed lines), trajectories reconstructed by the Y-Procedure (solid lines), ME points (circles) connected by a fitted Langmuir isotherm (solid line in zerorate plane), and projections of simulated trajectories onto the RCO − CCO plane (dotted lines). The direction of time is indicated by an arrow. Numbers adjacent to kinetic trajectories show the corresponding pulse intensity in molCO.

of the trajectory followed a straight line, after which spurious oscillations developed due to noise, rendering further analysis impossible. This result is counterintuitive because it is usually the tail of a desorption curve which is used for estimating the value of a desorption constant. Patches of the characteristic kinetic surface sampled by more intense pulses are large enough to capture surface curvature stemming from the nonlinear cross-term of eq 3. In this case, the cross-term is not negligible. Nevertheless, eq 3 may still be presented in linear form if the intrinsic desorption constant has previously been determined from a small planar trajectory. Rearranging terms and dividing eq 3 by CCO leads to

intensities of the virtual PIM experiments with reversible CO adsorption. Each trajectory emanates from and returns to the origin, where rate and concentrations are zero due to the complete reversibility of CO adsorption in this model and the fact that the microreactor is continuously evacuated. As pulseintensity increases, kinetic trajectories sample increasingly larger patches of the characteristic surface defined by the simulation parameters. The patch sampled by the smallest pulse is nearly flat because at the lower limit of CO coverage eq 3 can be approximated by R CO(t ) ≈ k+CZ,totCCO(t ) − k −CZCO(t )

(6)

[R CO(t ) + k −CZCO(t )] = k+CZ,tot − k+CZCO(t ) CCO(t )

which defines a plane in the rate-composition space. Expression 6 suggests that the trajectories of sufficiently small pulses presented in RCO/CCO vs CZCO/CCO coordinates will give a nearly linear plot with a slope equal to the intrinsic desorption constant and a y-intercept equal to the in adsorption constant multiplied by the total concentration of sites. The smallest trajectory from Figure 4 is replotted in these coordinates in Figure 5A as an example. Only the initial seven hundred points

(7)

Hence, if the intrinsic desorption constant is known, a large nonplanar trajectory should give a linear plot in coordinates (RCO − k−CZCO)/CCO vs CZCO. Figure 5B demonstrates the largest simulated trajectory plotted in the aforementioned coordinates. In order to calculate the left-hand side of eq 7 for this large state-altering trajectory, we have used the value of the 15422

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reactions are reversible. Therefore, in the case of reversible reactions, an experiment is state-defining not only if it preserves the initial catalyst state by the end of the pulse but also if the maximum surface coverage achieved during the pulse is much smaller than the total number of adsorption sites resulting in a plane kinetic trajectory.

intrinsic desorption constant estimated from a small statedefining trajectory. As time progresses within the experiment, the surface storage CZCO, which is used in Figure 5B as abscissa, increases until it reaches a maximum and then decreases. For this reason, the plot exhibits two branches which deviate slightly from a linear dependency. The slope of this line yields the intrinsic adsorption constant, while its x-intercept yields the total concentration of sites. It should be noted that only the first 1000 time points were used for this plot, beyond which the signal-to-noise ratio decreases considerably. Table 2 summarizes the adsorption parameters estimated from ME points and state-defining and state-altering

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EXPERIMENTAL EXAMPLE: CO ADSORPTION ON Pt/Mg(Al)O In order to illustrate the methodology for measuring the total concentration of adsorption sites outlined above, we applied PIM TZ TAP experiments to characterize CO adsorption on a hydrotalcite-supported platinum catalyst Pt/Mg(Al)O (1 wt % Pt). Catalysts of this type promoted with a second metal, such as Ga or In, have recently attracted considerable attention in the context of hydrocarbon dehydrogenation/hydrogenation processes.49,50 However, the nature of platinum−promoter interactions which improve the performance of platinum catalysts in alkane dehydrogenation reactions is not fully understood. Studying the adsorption of CO on these catalysts is important for elucidating the influence of promoting elements on their catalytic properties, including the concentration of catalytic sites and the electronic state of platinum nanoparticles. We therefore used CO adsorption to develop suitable experimental protocols for PIM TZ TAP measurements. A TAP-I setup (Autoclave Engineers) with electro-magnetically driven pulse-valves (pulse/delay generator from Berkeley Nucleonics Corporation, model 505) was used to demonstrate the developed protocol. The amount of adsorbing gas sent into the microreactor with each pulse can be controlled by two experimental procedures. One way to organize a PIM experiment is to use feed mixtures with variable content of adsorbing gas while keeping the pulse size constant. This procedure is time-consuming and is not conducive for fast measurements, which are necessary in complex experimental sequences. The second procedure for performing a PIM experiment is to alter pulse size while keeping a constant feed-mixture composition. The two most efficient and rapid methods for controlling the size of an injected pulse when using either TAP-I6 or TAP-II14 setups are: (1) to alter the voltage applied to the pulse-valves of the TAP setup or (2) to alter their opening time. The former method was used in this study. Prior to collecting data, an optimal voltage range must be identified for each specific pulse-valve which will be used in PIM experiments and for the current hardware settings of the valve (e.g., spring tension). The selected lowest value of the optimal voltage range must produce exit-flow rate curves with an acceptable signal-to-noise ratio for all fragments that will be monitored by a mass spectrometer during an experiment. The selected upper bound of the voltage range must produce exit-flow rate curves which are effectively described by the Knudsen diffusion model, since the contribution of convective flow will become significant as the pulse intensity increases above a certain value. After determining the optimal range of pulse intensities for our case study, PIM TZ TAP experiments were performed as follows. A sample containing 15 (mg) of Pt/Mg(Al)O (1 wt % Pt) catalyst prepared as described by Sun et al.49 was loaded into the TAP microreactor in a thin layer (∼1 mm) between two zones of inert washed quartz. The catalyst was initially preconditioned at 723 K with several oxidation/reduction cycles. CO adsorption data were recorded at 323, 353, 373, and 413 K by pulsing a 1:1 mixture of CO and argon (Alphagaz)

Table 2. Adsorption Parameters: True Values Used for Simulations and Values Estimated from ME Points and State-Defining and State-Altering Trajectories parameters k+ (m3/mol/s) k− (1/s) Keq CZ,tot (mol/ kgcat)

simulation

ME

statedefining

state-altering

10.0 × 103 30.0 333.3 2.5 × 10−4

N/A N/A 311.3 2.5 × 10−4

10.2 × 103 32.8 311.3a 2.5 × 10−4

9.4 × 103 32.8b 286.4 2.6 × 10−4

a Keq value for the estimation based on a state-defining trajectory is taken from the estimation based on ME points. bk− value for the estimation based on a state-altering trajectory is taken from the estimation based on a state-defining trajectory.

trajectories. Only the equilibrium constant Keq and the total concentration of sites CZ,tot can be estimated from ME points. The analysis of a state-defining trajectory originally yields the estimates of the desorption constant k− and the product of the total concentration of sites and the adsorption constant k+CZ,tot. However, the value of the equilibrium constant estimated from ME points earlier can be used to estimate the adsorption constant separately using k+ = k−Keq. In turn, the analysis of a state-altering trajectory required a k− value which is known from a state-defining trajectory to estimate CZ,tot and k+. Besides improving the conceptual understanding of adsorption kinetics and assisting in parameter estimation, this geometric interpretation of intrapulse characteristics clarifies the meaning of two concepts commonly used in TAP literature: state-defining and state-altering experiments.28 State-defining experiments are typically described as experiments in which the catalyst state does not change significantly. Experiments that do not satisfy this criterion are called state-altering. For irreversible processes, including catalytic reactions and catalyst deactivation, an indication of a state-defining experiment is the relative independence of exit-flow rate curves from the pulse number within a sequence. It is usually assumed that the quantity of reactants in such state-defining pulses is so small compared to the quantity of adsorption sites in the microreactor that the catalyst state remains approximately the same from one pulse to the next. In contrast, state-altering experiments change the catalyst state appreciably by using larger pulses or very long sequences of individually state-defining pulses. According to the methodology of interrogative kinetics (IK), only precisely controlled state-altering experiments consisting of many statedefining ones should be used for systematic catalyst characterization. However, this simplified understanding of TAP experiments is not suitable in cases of reversible adsorption. Even large pulses leading to nonplanar kinetic trajectories as shown in Figure 4 leave the catalyst state unchanged if all 15423

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over the completely reduced catalyst. The pulse size was varied between 1.0 × 10−9 and 1.45 × 10−8 molCO by changing the opening time of the pulse-valve from 85.0 to 115.0 ms. First, the effective diffusion coefficients were determined by regressing argon responses with the standard Knudsen model. An example of a regressed exit-flow rate curve is shown in Figure S1 of the Supporting Information. The estimated effective diffusion coefficients were then used to translate the original exit-flow rate curves into rate-concentration data by means of the Y-Procedure. The surface concentration of CO was obtained by integrating the reconstructed consumption rate in time according to eq 4. Figure 6 shows an example of reconstructed intrapulse kinetic characteristics, namely, the CO consumption rate, gas

Figure 7. 3D view of experimental kinetic trajectories in the ratecomposition space, RCO(t) vs CCO(t) vs CZCO(t), at 413 K: reconstructed trajectories (solid line with white triangles); ME points (circles) connected by a fitted Langmuir isotherm (solid line in zerorate plane). The direction of time is indicated by an arrow. Numbers adjacent to kinetic trajectories show the corresponding pulse intensity in molCO.

behave like simulated trajectories in Figure 4. In agreement with virtual experiments, the trajectories corresponding to small pulses are more flat in comparison to larger trajectories, and the positions of ME points demonstrate Langmuirian behavior. Substantial noise-related deviations from smooth curves are apparent in these data, which precluded the estimation of intrinsic adsorption parameters from individual state-altering and state-defining trajectories using eqs 6 and 7. The characteristic isotherms, i.e., the ME concentrations of CO on the catalyst as functions of the ME concentrations of CO in the gas, are plotted in Figure 8 for all experimental temperatures. The same plot also contains fitted Langmuir isotherms for each temperature. The estimated total concentration of CO adsorption sites lies within 0.3−0.55 (mmol/kgcat), while the temperature dependence of the equilibrium constant suggests that the heat of CO adsorption does not exceed 18 kJ/molCO.

Figure 6. Example of intrapulse kinetic characteristics reconstructed from experimental data via the Y-Procedure: RCO(t) (blue line), CCO(t) (green line), and CZCO(t) (red line). The approximate position of ME is indicated by red circles; the shaded area represents the uncertainty in the ME position, and the data-range suitable for further analysis is bound by two vertical dashed lines. The data corresponds to 1.45 × 10−8 molCO pulse at 413.0 K.

concentration of CO, and surface concentration of CO for a 1.45 × 10−8 molCO pulse at 413.0 K. In accordance with the virtual experiments described in the theoretical section of this paper, the gas concentration of CO in the catalytic zone assumed the familiar shape of an exit-flow rate curve with a peak around 10 ms. The behavior of the CO consumption rate also followed the predicted pattern; the rate first reached a peak, then passed through zero, and finally decayed to zero from the negative side. The rate curve in Figure 6 is very noisy, despite low-pass filtering applied as a part of the Y-Procedure. This noise amplification leads to a significant distortion of the surface CO concentration which was calculated by integrating the rate curve in time. More specifically, the surface CO concentration appears negative at the earliest time-points of the curve and at all points after 60 ms. Only a narrow time-interval between 10 and 50 ms bound in Figure 6 by two dashed vertical lines, where the noise-induced distortion is not yet significant, was used for downstream data analysis. The position of momentary equilibrium and the corresponding concentration values were determined only approximately from the reconstructed intrapulse characteristics as a result of these noise effects. The reconstructed rate-concentration curves for 373 K and for different pulse intensities are presented in Figure 7 as kinetic trajectories. Qualitatively, these experimental trajectories

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CONCLUSIONS AND PERSPECTIVES In this paper, we analyzed the behavior of non-steady-state consumption rates, gas, and surface concentrations during thinzone temporal analysis of products experiments with a singlesite reversible adsorption. This analysis revealed the occurrence of momentary equilibrium (ME), a moment in time when the adsorption rate is exactly balanced by the desorption rate. While the existence of at least one ME during a pulse-response experiment is quarantined for reversible adsorption, its existence is not guaranteed in the presence of additional reactions. Using the Y-Procedure to reconstruct rate-concentration data from TAP exit-flow rate curves, it is possible to determine when the ME event occurred and the concentrations of adsorbing species in the gas and on the catalyst at this moment. The knowledge of gas and surface compositions at the ME point provides a novel way to obtain adsorption isotherms from a series of transient pulse-response experiments with variable pulse-intensity (PIM experiments). These isotherms can subsequently be used to estimate the concentration of 15424

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The experimental case study of CO adsorption on the Pt/ Mg(Al)O catalyst suggests that high-frequency noise in TAP data may have a considerable impact on the estimation of adsorption parameters via ME analysis. Amplified by the YProcedure, the noise does not only lead to uncertainty in the ME position but also causes unphysical artifacts in the downstream analysis including the negative values of surface coverage in the tail of the curve (see Figure 6) and the deformed shape of kinetic trajectories (see Figure 7). These artifacts precluded the estimation of adsorption parameters via the differential analysis of rate-composition transients which is theoretically described in Figure 5. Further improvements of the TAP hardware and the Y-Procedure algorithm are needed to mitigate the effects of noise on the quality of data interpretation. The developed PIM protocol for measuring the concentration of catalytic sites has several limitations. For a given temperature window, the protocol is limited to certain ranges of adsorption and desorption activation energies within which both the adsorption and desorption of probe molecules proceed on the time scale of one pulse (0.1−1.0 s). “Too slow” adsorption can not be studied due to a relatively short residence time of a probe gas inside the catalytic zone. “Too slow” desorption, on the other hand, will not allow all desorbing probe molecules to be registered by the mass spectrometer within one recording period. In the latter case, the recoding period allocated for each pulse can be increased within reasonable limits (3 nm and the shape changes induced by temperature. J. Phys. Chem. B 2005, 109, 24465−24472. (31) Barnard, A. S.; Konishi, H.; Xu, H. F. Morphology mapping of platinum catalysts over the entire nanoscale. Catal. Sci. Tech. 2011, 1, 1440−1448. (32) Yoshida, H.; Matsuura, K.; Kuwauchi, Y.; Kohno, H.; Shimada, S.; Haruta, M.; Takeda, S. Temperature-dependent change in shape of platinum nanoparticles supported on CeO2 during catalytic reactions. Appl. Phys. Express 2011, 4, 065001. (33) Chang, S. L. Y.; Barnard, A. S.; Dwyer, C.; Hansen, T. W.; Wagner, J. B.; Dunin-Borkowski, R. E.; Weyland, M.; Konishi, H.; Xu, H. Stability of porous platinum nanoparticles: Combined in situ TEM and theoretical study. J. Phys. Chem. Lett. 2012, 3, 1106−1110. (34) Freund, H.-J.; Meijer, G.; Scheffler, M.; Schlogl, R.; Wolf, M. CO oxidation as a prototypical reaction for heterogeneous processes. Angew. Chem., Int. Ed. 2011, 50, 10064−10094. (35) Lear, T.; Marshall, R.; Antonio Lopez-Sanchez, J.; Jackson, S. D.; Klapotke, T. M.; Baumer, M.; Rupprechter, G.; Freund, H.-J.; Lennon, D. The application of infrared spectroscopy to probe the surface morphology of alumina-supported palladium catalysts. J. Chem. Phys. 2005, 123, 174706. (36) McCabe, R.; Schmidt, L. Binding states of CO on single crystal planes of Pt. Surf. Sci. 1977, 66, 101−124. (37) Hoebink, J.; Huinink, J.; Marin, G. A quantitative analysis of transient kinetic experiments: The oxidation of CO by O2 over Pt. Appl. Catal., A 1997, 160, 139−151. (38) Tao, F.; Dag, S.; Wang, L.-W.; Liu, Z.; Butcher, D. R.; Bluhm, H.; Salmeron, M.; Somorjai, G. A. Break-up of stepped platinum catalyst surfaces by high CO coverage. Science 2010, 327, 850−853. (39) Mhadeshwar, A. B.; Wang, H.; Vlachos, D. G. Thermodynamic consistency in microkinetic development of surface reaction mechanisms. J. Phys. Chem. B 2003, 107, 12721−12733. (40) Allian, A. D.; Takanabe, K.; Fujdala, K. L.; Hao, X.; Truex, T. J.; Cai, J.; Buda, C.; Neurock, M.; Iglesia, E. Chemisorption of CO and mechanism of CO oxidation on supported platinum nanoclusters. J. Am. Chem. Soc. 2011, 133, 4498−4517. (41) Dekker, F.; Nazloomian, J.; Bliek, A.; Kapteijn, F.; Moulijn, J.; Coulson, D.; Mills, P.; Lerou, J. Carbon monoxide oxidation over platinum powder: A comparison of TAP and step-response experiments. Appl. Catal., A 1997, 151, 247−266. (42) Nijhuis, T.; Makkee, M.; van Langeveld, A.; Moulijn, J. New insight in the platinum-catalyzed CO oxidation kinetic mechanism by using an advanced TAP reactor system. Appl. Catal., A 1997, 164, 237−249. (43) Roelant, R. Mathematical determination of reaction networks from transient kinetic experiments. PhD thesis, Universiteit Gent, 2010. (44) Roelant, R.; Constales, D.; Yablonsky, G. S.; Van Keer, R.; Rude, M. A.; Marin, G. B. Noise in temporal analysis of products (TAP) pulse responses. Catal. Today 2007, 121, 269−281. (45) Redekop, E. A. Non-steady-state catalyst characterization with thin-zone tap experiments. PhD thesis, Washington University in Saint Louis, 2012.

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