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On Surface Heterogeneity and Catalytic Kinetics Dmitry Yu. Murzin Laboratory of Industrial Chemistry, Process Chemistry Centre, A° bo Akademi University, Biskopsgatan 8, 20500 Turku/A° bo, Finland
The effect of surface heterogeneity on catalysis and chemisorption is discussed. Available recent data on calorimetry, as well as on adsorption isotherms and reaction kinetics, are carefully reviewed. Despite the fact that the current catalytic engineering practice is very much limited to Langmuir-Hinshelwood kinetics based on concepts of ideal surfaces, there is a number of cases when such an approach is not sufficient enough for the modeling of the kinetics of heterogeneous catalytic reactions. Introduction Numerous papers, books, and symposia have as the main objective the heterogeneity of the solid surfaces (apriori or induced) and the consequence of such heterogeneity on adsorption, surface science, and catalysis.1-30 The reason that the binding energies of the reactants are coverage dependent is either the fact that the surface sites are not identical and/or the presence of lateral interactions between adsorbed species. Surface heterogeneity reflects itself in adsorption kinetics, adsorption isotherms, temperature programmed desorption, heat effects, catalytic kinetics, and so forth. The concepts of lateral interactions and surface heterogeneity are so widely used in surface science, chemical physics, physical chemistry, and catalysis that they became standard textbook knowledge. However, for engineering purposes, kinetic models of catalytic reactions often do not involve a very complex nature of heterogeneous catalysts and lateral interactions in the adsorbed layer. At the same time, according to the very definition of catalysis, it is a kinetic process, and thus, reliable kinetic models, which describe the rate of catalytic reactions, are of vital importance for solving applied problems in mathematical modeling, design, and intensification of chemical processes. The necessity of kinetic investigations in heterogeneous catalysis is closely connected to the tasks, which a chemical engineer has to deal with, that is, proper reactor design, evaluation of the side reactions, and the impact of dynamic effects on reactor performance. Despite the successful stories in technical process development, often the underlying assumptions of the Langmuir theory of ideal surfaces result in wrong behavior in different parameter regimes. Note, that already in the 1950-60s there was an understanding that the formulation of the rate expressions based on the original theory of Langmuir adopted by Hinshelwood and widely applied in this form for technical process development is a crude approximation. With the advance of the modern physical tools for the characterization of surfaces and adsorbed species, new concepts emerged, and it is now well established that the mean field assumption breaks down and surface reconstruction frequently occurs; adsorbed molecules change the structure of the surface layer and catalytic properties, reaction rate is dependent on spatial arrangement, and so forth. Recently,31 an attempt was made to deny experimental evidences on surface heterogeneity collected over the years on the basis of (a) reaction kinetics, (b) adsorption
isotherms, and (c) calorimetric data and to consider the whole concept as an anachronism. A conclusion was made31 that “Our long-term calorimetric studies of chemisorption and analyses of the data available made us to be confident that the notions on the occurrence of a catalytically significant heterogeneity of metal surfaces do not represent the facts.” The author was citing data available in the literature with respect to the heats of adsorption of gases, claiming that constant or near constant dependence of heat of adsorption as a function of coverage is obtained31 and that “there is not a grain of evidence to suggest that the surface heterogeneity reveals itself in any manner in the molar heats of chemisorption”.31 Calorimetry Due to the fundamental importance of the conclusions presented in31 and their contradiction with the numerous reports,1-30 it seems reasonable to look carefully into the literature data, cited31 as evidences for the fact that the heats of adsorption do not depend on coverage. Let us consider first supported metals and cite a few publications mentioned in ref 31 as providing no evidence on surface heterogeneity. Adsorption of hydrogen and oxygen over iridium was discussed,32 and it was explicitly stated32 that the heat of adsorption decreased monotonically with coverage for hydrogen, while for oxygen the adsorption heat remained approximately constant up to nearly half the monolayer and then followed a convex curve and ended in a plateau (Figure 1). The shape of the differential calorimetric isotherms for oxygen was associated with a very strong O-Ir bond. Extensive work of the Dumesic group cited in ref 31 as supporting independence of adsorption heat on coverage in fact reports experimental data on the dependence of the heat of adsorption on coverage for hydrogen and CO on Pt and Rh,33-35 nitrogen on a commercial iron catalyst,35 isobutane on Pt,36 ethylene adsorption on Pt/SiO2 and PtSn/SiO2 catalysts,37 and H2, C2H4, and C2H2 on Pt powder,38 unequivocally demonstrating a very profound decrease of heats of adsorption with coverage. Just excerpts from the original papers are presented in the Supporting Information (Supporting Information Figures S1-S4). Figure 2 shows the microcalorimetric results for hydrogen adsorption on platinum powder. At 303 K, hydrogen adsorbs dissociatively on platinum with an initial heat of 90 kJ/mol. The differential heat decreases slowly as hydrogen adsorbs. At higher hydrogen cover-
10.1021/ie049044x CCC: $30.25 © 2005 American Chemical Society Published on Web 02/19/2005
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Figure 1. Differential calorimetric isotherms of hydrogen and oxygen adsorption on Ir/SiO2 (redrawn from ref 32 with permission from Elsevier). Figure 3. Measured differential heat of adsorption of benzene as a function of coverage on Pt(111) at 300 K (redrawn with permission from ref 42. Copyright 2004 American Chemical Society).
Figure 2. Differential heat of hydrogen adsorption on platinum powder at 303 K (redrawn from ref 38 with permission from Elsevier).
age, the differential heat decreases to ∼55 kJ/mol and remains constant at this value until the surface becomes saturated with hydrogen. Just a brief examination of the data clearly indicates that there is a wealth of data unequivocally confirming decrease of the heat of adsorption with coverage on supported metals. In the last years, adsorption energies not only for supported catalysts but also for single-crystal surfaces were reported39,40 and used to study high-vapor-pressure substances (e.g., O2, CO, NO, and ethylene).41 Later on, the same technique was extended to low-vapor-pressure substances (e.g., benzene).42 It was confirmed that also on single crystals the measured molar heats of adsorption are coverage dependent (Figure 3). This is in agreement with the statement of Ertl that the original view of Langmuir of the lattice of a single-crystal surface representing periodic arrangements of equivalent adsorption sites is correct only in exceptional cases.43 In ref 31, discussion on the data obtained for single crystals was avoided by mentioning that “It may appear that some data obtained under ultravacuum at different crystal faces do not correspond to our conclusions. Such data require a special discussion, which does not enter into the scope of the present paper”. The importance of taking into account the reactivity of different crystal faces became apparently clear in the recent years with the steadily increasing application of nanosized materials in heterogeneous catalysis. In conventional preparations of catalysts including copre-
cipitation or filling of the support with an aqueous solution of metal particle precursor, it is difficult to control the catalyst particle size and shape. Recently, novel synthesis methods for nanocatalysts and nanophase materials and novel nanostructural methods have been reported in the literature.44 For instance, Ru nanoparticles of ∼5 nm in size as demonstrated by transmission electron microscopy (TEM) have been supported on γ-alumina and have shown very high activity for ammonia synthesis.44 In the recent years, metal nanoparticles formed in various media and stabilized by different mechanisms are the objects of intense studies due to their unique properties, which are significantly influenced by the nanoparticle size, organization of the nanoparticle crystal lattice, nanoparticle surface, and chemical nature of the microenvironment surrounding the nanoparticle.45 It is well-known that the catalytic activity and selectivity of supported metal particle catalysts are strongly dependent on the size and shape of the particles. Some examples are from oxidation and hydrogenation reactions,46,47 where high catalytic activities as well as high enantio- or chemoselectivities have been achieved over nanosized metal supported catalysts. Scientifically interesting phenomena over metal nanoparticles, like the observed highest catalytic activity in oxidation at the transition of the electronic state from metallic to nonmetallic and the selectivity changes with increasing metal particle size in the hydrogenation of multifunctional compounds, need still more detailed fundamental understanding from the surface science viewpoint. At the same time, there is no doubt that there is a large potential for the development and the application of metal nanoparticles with tailored physical and chemical properties in both materials science and catalysis. Metal nanoparticles supported on inorganic and organic matrixes have shown promising features, like higher catalytic activity and/or selectivity than those of conventional catalysts in many catalytic reactions. The origin for these effects is the size quantization of most electronic properties. The metal nanoparticles exhibit unique properties that differ from those of the bulk substances, that is, different heat capacity, vapor pressure, and melting point. Moreover, as indicated above, when decreasing the metal particle size sufficiently enough, a transition of the electronic state from metallic to nonmetallic occurs. Additionally, metal nanoparticles exhibit a large surface-to-volume ratio
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and increased number of edges, corners, and faces, leading to altered catalytic activity and selectivity. It is thus not surprising that the reactivity of different faces should be taken into account in the kinetic modeling of catalytic reactions over nanosized materials.48 In a recent study on the influence of catalyst surface structure on the kinetics of ethene hydrogenation of hydrocarbons on Pt/silica,49 it was proposed that the apparent structure sensitivity (i.e., kinetic discontinuity) exhibited by this reaction can be attributed to the influence of particle size on the reaction kinetics, more specifically to the relative ease of forming a surfacereaction inhibiting layer of adsorbed hydrocarbon on the surface of small Pt particles. There are then essential grounds to expect that future kinetic modeling on catalytic reactions over nanometersized metal particles, which we can coin as nanokinetics, will include surface heterogeneity (e.g., apriori due to different activities of distinct catalyst sites and/or induced due to lateral interactions). On top of the metal properties per se, the local metal environment can affect both the metal particle size and shape.50,51 This environment is changed, for example, by varying the pretreatment of the catalyst.51 Moreover, the acidity of the support affects the stabilization of the metal particle.50 These phenomena as well as the wellknown impact of the catalyst preparation methods on catalytic activity speak in favor of surface heterogeneity. In light of the previous considerations, it is clear that catalytic activity does not depend only on chemical composition as advanced by Boreskov in the early 1950s to emphasize the chemical nature of catalysis and still advocated in ref 31 in its most simplistic form as contradicting with surface heterogeneity. At this stage, it can be concluded that the calorimetric data presented above demonstrate that the heats of adsorption of simple gases, like CO and hydrogen, as well as more complex organic molecules, depend on coverage for metal powders, supported metal catalysts, and single crystals. Such a dependence is not necessarily linear but can display more complex behavior. Similar considerations are valid not only for supported metals but also for solid acid catalysts, as an enormous amount of calorimetric data exists on heats of adsorption on such microporous materials as zeolites. In particular, the adsorption of ammonia in zeolites has been extensively studied over many years.52,53 Due to its small size and rather strong base character, ammonia is a preferred molecule to probe the number, type, and strength of acid sites in catalysts. In addition, in zeolite science, ammonia adsorption techniques have been widely used to study, inter alia, molecule/cation interactions, cation distributions, and energetic heterogeneity. Supporting Information Figure S5 displays differential heats of adsorption of ammonia over the two zeolites NaY and NaY(NaBr),52 showing clearly that there is a continuous decrease of the heat of adsorption from 80 to ∼40-45 kJ/mol as adsorption progresses. Adsorption Isotherms As correctly pointed out,31 the applicability of a particular isotherm per se cannot be proof that the surface is heterogeneous or that there are lateral interactions finally leading to nonequivalence of surface sites. A simple example is the adsorption of ammonia on a
Figure 4. Comparison of Temkin and Langmuir isotherms with experimental data on ammonia adsorption on titanium silicate.54
titanium silicate,54 showing that for this catalyst in the studied range of partial pressures the difference is in fact minor, although the Temkin55 isotherm is slightly better than the the Langmuir isotherm (Figure 4). However, for systems where nonuniformity (apriori or induced) is more pronounced, in the context of the considerations presented in the previous section, it is not surprising that the simple Langmuir adsorption isotherm based on the concept of ideal surfaces cannot always describe experimental observations, when there is a dependence of heat of adsorption on coverage. As a recent example,56 the adsorption of the hydrocarbon 2,2,4-trimethylpentane is presented in Supporting Information Figure S6 along with a comparison between the Freundlich and Langmuir isotherms. Note that while the Temkin isotherm corresponds to a linear decrease of adsorption heat with coverage, the isotherm of Freundlich is derived for cases of nonlinear decrease. Interestingly, the concept of surface nonideality started to be applied recently for the adsorption of proteins in immobilized metal ion affinity chromatography to understand protein-immobilized metal/ion interactions and to explain cooperativity and binding heterogeneity in quantitative terms.57,58 The specific advantages of the Temkin isotherm are summarized:57 for low to moderate coverages, adsorption is described by two physically meaningful parameters, which can be determined by equilibrium adsorption experiments. Moreover, it satisfies Henry’s law, a necessary requirement in employing local equilibrium theory to predict chromatographic behavior from equilibrium adsorption expressions. It is fair to mention that as the simple Langmuir isotherm cannot explain experimental observations for a variety of systems, attempts were done to make modifications of it, still keeping the assumption of ideal surfaces, via including the multicentered adsorption, sometimes leading to improved fitting and description similar to isotherms based on concepts of surface heterogeneity. Implicit agreement that conventional Langmuir isotherms cannot explain experimental data on nitrogen adsorption on Fe also follows from ref 31 where, as more simple models failed, along with a logarithmic adsorption isotherm, a rather complex model for multicentered chemisorption was tested
P ) Kθ2[1 + (z - 2)θ]2/[1 - θ]4
(1)
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Such a case with one reversible step and two equilibrium steps was considered.31 For ammonia synthesis, an equation
r ) k′PN2
Figure 5. Comparison between Temkin and multicenterd adsorption isotherms.
with z being the number of surface centers that are covered by a full rotation of a chemisorbed molecule without counting a rotation center. A comparison between the logarithmic and multicentered isotherms presented in Figure 5 shows that although the difference is minor, it is nevertheless visible and what is more important systematic. At the present stage, it can be tentatively concluded that besides calorimetric evidences also data on chemisorption equilibrium are available, which do not follow the conventional approach of ideal surfaces, but should include more complex treatment. At the same time, it is apparently clear that there are systems where statistical treatment of adsorption isotherms does not reveal essential differences between Langmuir isotherms and more sophisticated mathematic treatments, based on surface heterogeneity. Kinetics It was repeatedly stated in the literature13,59 that if a particular rate equation fits well to experimental data for the kinetics of the steady-state catalyzed reaction, it is not as such a proof of a postulated mechanism; moreover, alternative models could lead to rate equations, which equally well fit the data. However, it is worth it to consider kinetic evidences for surface nonideality. On the basis of the discussion presented above, it is clear that there are cases when surface heterogeneity is not manifested in steady-state conditions. Looking at calorimetric data of the heats of adsorption as a function of coverage, it can be predicted that sometimes there is a plateau in the dependence of adsorption heats on coverage at low coverages. Within the framework of the Temkin formalism of apriori nonuniform surfaces at low and large coverages, the expressions for ideal surfaces and nonideal surfaces coincide. It is not surprising, therefore, that quite often in three-phase (gas/liquid/solid) systems when the surface is totally covered by reactants/products/solvents and steady-state kinetics is obeying zero or first order, kinetic treatment does not require involvement of the surface heterogeneity.60,61 Thus, when the surface coverage is very low or alternatively is very high, models for nonuniform surfaces should degenerate to those for uniform surfaces.
( ) ( ) PH23
0.5
- k′′
PNH32
PNH32
0.5
(2)
PH23
was derived utilizing the concept of ideal surfaces and the assumption that the surface is almost completely covered by adsorbed nitrogen. Note that, based on experimental data, which cover the pressure range from below 1 atm up to 500 atm near the equilibrium, the reaction rate is described by the Temkin-Pyzhev equation,62 which is different from eq 2, namely,
r ) k+PN2
( ) ( ) PH23
m
PNH32
- k-
PNH32
1-m
(3)
PH23
where m is a constant ranging from 0 to 1. Under equilibrium, the reaction rate equals 0; therefore, k+/k- ) K, where K is the equilibrium constant. The value of m is not always equal to 0.5; for instance, it was reported to be 0.2 for Co and 0.3 for Ni,62 which do not follow from the treatment based on the threestep mechanism with one rate determining reversible step.31 The majority of statements on calorimetry, adsorption isotherms, and especially ammonia synthesis kinetics presented in ref 31 were already reported by the same author.63 Many statements, which are referred to in the present paper, were in fact critically considered by Temkin,62 although without the adequate response in ref 31. It is remarkable that in the Temkin-Pyzhev equation only two parameters are required to be calculated from experimental data; nevertheless, such a simple equation is capable of describing kinetics in such a wide range of temperatures and pressures for different catalytic materials. This probably explains the fact that it is still used for the design of ammonia catalytic converters. At the same time, there is no doubt that it is possible to find equations based on the concept of ideal surfaces and multiple step sequence, which equally well could describe some experimental data on ammonia synthesis, like the following one2
r ) k+2θN2θ/ - k-2θN2 ) k+2K1PN2θ/2 k-2
[
PNH3
]
2
K3K4K5K6(xK7PH2)
3
θ/2 (4)
with the coverage of vacant sites given by
θ/ ) 1/1 + K1PN2 + xK7PH2 + PNH3
(
K5K6xK7PH2
1+
1 P + K6 NH3
1 K4xK7PH2
(
1+
1
))
K3xK7PH2
(5)
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Figure 6. Comparison between models for ideal (I) and nonideal (NI) surfaces (eqs 8 and 9, respectively).
Equation 4 is based on the following sequence of steps2
(1) N2(g) + * ≡ N2* (2) N2* + * w 2N* (3) N* + H* ≡ NH* + * (4) NH* + H* ≡ NH2* + * (5) NH2* + H* ≡ NH3* + *
(6)
(6) NH3* ≡ NH3(g) + * (7) H2(g) + 2* ≡ 2H* N2 + 3H2 ) 2NH3 and requires quasi-equilibrium assumption for all steps, but the second one. Otherwise, due to the nonlinear character of these steps, an explicit expression could not be obtained in a closed form. However, if the mechanism (e.g., the sequence of steps) is considered to be essentially the same, one can find a range of conditions, when the models based on the surface heterogeneity differ from models on ideal surfaces. Let us consider a simple isomerization reaction which occurs through the following sequence of steps:
(1) A + Z f ZA (2) ZA f Z + B
(7)
Here, A and B are reactants, ZA is an adsorbed intermediate, and Z is the surface site. For ideal surfaces, the reaction rate is given by
k1PAk2 k1PA + k2
(8)
Applying the framework of biographically nonuniform surfaces,22 it holds that
r ) k′PAR
A1 + Z T ZI1 + A2
(10)
B1 + Z1 T ZI2 + B2
(11)
where A1, A2, B1, and B2 are reactants, Z is the surface site, and I1 and I2 are adsorbed species. One of the applications of this methodology was related to the oxidative dehydrogenation of 1-butene to 1,3-butadiene on iron and tin/antimony oxide catalysts.64c This development64a,b was largely unnoticed, as the original papers were not readily available; therefore, in the Supporting Information, several examples, specifically constructed from the general considerations,64a,b are presented. A generalized form of a reaction rate equation was obtained64
C r ) k′ m n D H
(12)
where C, D, and H are some functions of the reactant partial pressures and k′ depends on the degree of nonuniformity. For the two-step sequence
(1) A1 + Z T ZI + A2 (2) B1 + ZI T Z + B2
(13)
A1 + B1 T B2 + A2 the values of C, D, and H are correspondingly
A)B
r)
ideal and nonideal surfaces, are close to each other; however, the deviation is systematic. Moreover, these two models give qualitatively different behavior at boundary values of partial pressures. For instance, according to an ideal surface model, at high partial pressures, the reaction rate obeys zero-order kinetics and, at low pressures, the reaction order is equal to unity, in drastic contrast to the mechanism for a nonideal surface. The model of biographically nonuniform surfaces advanced by Temkin22 provides a useful, although limited, framework, as it was mainly used to treat a simple two-step sequence and further extended64 to special cases, like Christiansen sequences. Only linear steps were considered which occur either with or without changes of the number of adsorbed species
(9)
where R is the Polanyi parameter with a typical value of 0.5. Now let us compare the reaction rates as a function of the partial pressure of reagent A for some particular values of parameters (Figure 6). We can easily see that in a certain range of partial pressures the descriptions offered by two models, namely,
C ) ω2D - ω-2H, D ) ω1 + ω-2, H ) ω-1 + ω2
(14)
where ωi are the frequencies of the steps, that is, ω1 ) k1PA1, ω-1 ) k-1PA2, ω2 ) k2PB2, and ω-2 ) k-2PB2. Despite the convenient (albeit restrictied to particular types of reaction mechanisms) mathematical framework, there are no physical grounds to assume even nonuniformity;62 therefore, models giving a similar description but taking into account lateral interactions in the adsorbed layer were put forward. The importance of lateral interactions between chemisorbed molecules has been demonstrated for many cases, that is, lateral interactions result in the ordering of adsorbed particles. With the development of the low-energy electron diffraction technique, this phenomenon has been observed in thousands of adsorbate/substrate systems. The analysis of LEED data obtained at different coverages and temperatures makes it possible to construct adsorbatesubstrate phase diagrams. Comparing measured and
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calculated phase diagrams, one can obtain values for lateral interactions. Alternatively, lateral interactions can be assessed from a judicious analysis of desorption kinetics. Information on lateral interactions can also be extracted from high-resolution scanning tunneling microscopy (STM) data by analyzing the distribution of adsorbed particles. The theory of interatomic interactions on the catalyst surface is far from being complete. Usually, the repulsive interactions between nearest neighbors and attractive interactions between next-nearest neighbors are considered in the modeling. The whole set of experimental data on adsorption, desorption, and thermal desorption, including the analysis of temperature programmed reaction (TPR) spectra, is used to assess the value of the interaction parameter. In the widely used lattice gas model,16,23 the relationships between the rate of an elementary reaction and coverage are complex and cannot be written in a closed form when this model is used. In the model, each adsorbate is assumed to be localized on a two-dimensional array of surface sites and the site is assumed to be either vacant or occupied by a single adsorbate. The regular lattice systems with nearest-neighbor interactions (Ising models) are among the simplest models, still being able to reflect many characteristics of real systems.65 As closed solutions cannot be obtained, physically reasonable approximations were proposed. In the Bragg-Williams approximation, configurational degeneracy and nearest-neighbor interaction energy are treated as though molecules were distributed randomly among the sites. Using this approximation, the adsorption isotherm was derived65 and can be presented in the following form:
θ ln(aP) ) (cw/kT)θ + ln 1-θ
(15)
where a is the adsorption coefficient, θ is the coverage, c is the total number of nearest neighbors, and w is the potential energy of interaction between a pair of nearest neighbors. From an isotherm of the type of eq 15 in the region of medium coverage, the logarithmic adsorption isotherm (the Temkin isotherm) can be easily obtained. The value of the interaction parameter w was discussed66 using the Lennard-Jones 6-12 potential, which stems in part from quantum mechanical calculations and in part from empirical considerations. Quasi-chemical approximation of the lattice gas model assumes that the adsorbates maintain an equilibrium distribution on the surface. The lattice gas model with this approximation was used for the description of TPD spectra as well as in some instances for the reaction of gases on metal surfaces. Among the studied reactions were the steady-state oxidation of CO over Ir and hydrogen over Pt and the kinetics of the CO-NO and CO-O2 reactions over Rh.16,27 The model of this type provided acceptable mathematical description for various experimental data on the reaction of NO and CO on the Pt(100) surface.67 This reaction is important from the viewpoint of environmental catalysis because the starting reactants are the toxic components of exhaust from internal combustion engines. The development of efficient afterburning agents for exhaust seems possible only if physicochemical processes at the gas-catalyst interface are well understood and if a kinetic model for the non-steady-state conditions is available. The reaction of NO and CO on the Pt(100) surface has complex
dynamic behavior. The multiplicity of steady states, the explosive nature of the reaction, and the reaction oscillations were observed.67 Another example is the reaction of NO + H2 on Pt(110) which also shows quite complex behavior17 with 11 reversible and irreversible elementary steps considered in the reaction mechanism. Theoretical investigations of this model included lateral interactions only for two steps in the forward direction and two steps in the reverse direction, leading to the following rate expressions
ri ) kiθpIa (16)
p
Ia ) [θ/ +
∑1 θp exp(ap/RT)]
m
where θ/ is the coverage of vacant sites, m is the number of nearest-neighbor sites, and is the energy of lateral interactions,17 which were determined by a fitting procedure to provide the best description of all experimental data. This model was able17 to reproduce many kinds of nonlinear behavior including kinetic oscillations and the transition to chaos. These considerations alone already give a feeling that the main advantage of models based on surface heterogeneity is in explaining dynamic catalysis under conditions far from the steady state. In the latter case, diverse nonlinear phenomena exist: the multiplicity of steady states (stable and unstable), hysteresis phenomena, the ignition and extinction of the process, critical phenomena, phase transitions, a high sensitivity of the process to changes in the parameters, oscillations and wave phenomena, chaotic regimes, the formation of dissipative structures, and self-organization phenomena. In fact, the nonlinearity of macroscopic rate laws arises partly due to the complex cooperative interaction of adsorbed atoms and molecules with each other and with the catalyst surface. Unfortunately, the cumbersome character of the lattice gas model reduces its utilization in applied catalytic kinetics, as the kinetic equations should be solved simultaneously with a reactor model, representing the flow and temperature patterns in catalytic reactors and also including heat and mass transfer in the catalyst particles. A feasible alternative, which can be used for practical engineering purposes is the surface electronic gas model originally proposed by Temkin.22 This model has a suitable form to address the kinetic problems of applied catalysis,18,68-73 more specifically to model the kinetics of complex reactions. In the literature, application of the Temkin formalism (e.g., evenly nonuniform surfaces corresponding to a logarithmic adsorption isotherm) was essentially limited to the two-step sequence.74 Kiperman et al.75 mentioned a series of processes such as the hydrogenation of organic compounds, configuration isomerization, oxidation of methane, ethylene, CO, and alcohols, oxidative ammoxidation, oxidative dehydrogenation, oxidative chlorination, methanol synthesis, and isotopic exchange, when the expressions corresponding to nonuniform surfaces more adequately described the experimental data than classical kinetic equations, based on the models of ideal adsorbed layers. In those examples, the reaction mechanisms were treated as if they occur through a two-step sequence on biographically nonuniform surfaces; however, in the region of medium cover-
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age, the kinetic equations for biographical and induced nonuniform (e.g., the case of lateral interactions) surfaces have the same form. The treatment based on the two-step sequence originated from the complexity of deriving the explicit form of rate equations for other reaction mechanisms on biographically nonuniform surfaces.22 However, the surface electronic gas model has no restrictions from this point of view, as the implicit form can be easily used for the data fitting.18 In some particular cases when the reaction occurs either at low or at high surface coverage, the kinetics is insensitive to surface nonuniformity.75 Thus, the description has to be exactly the same when applying the concept of either uniform or nonuniform surfaces, as generally speaking the uniform surface can be treated just as a special case of the more general model of nonuniform surfaces. Using the surface electronic gas model, one arrives at the description based on the assumption of uniform surface by simply setting the effective charges of adsorbed species equal to zero.18 Therefore, within the framework of the surface electronic gas model, if the parameter estimation is statistically correct, the kinetic models of uniform surfaces can never be superior to the models for nonuniform surfaces. This seems to be another advantage of utilizing the surface electronic gas model in distinction to the models of biographical nonuniformity with a certain distribution of adsorption energies. For example, Corma et al.76 made a comparison of models for ideal and nonideal surfaces using the Hougen-Watson approach (only one elementary step is the rate-controlling step) and applying the expressions for adsorption isotherms for nonuniform surfaces in the region of medium coverage. As several simplifications were applied, a result which looks rather puzzling from the first glance was obtained,76 namely, that in several cases the fits based on the models of nonuniform surfaces were extremely bad in comparison with those of uniform surfaces. The reason for these statistically unacceptable descriptions is that the assumptions of medium coverage did not hold. A comparison between models of induced and biographical surfaces was recently presented for the ammonia synthesis.70 Schematically in the simplest treatment, the ammonia synthesis mechanism near the equilibrium can be pictured in the following way:
(1) N2 + Z S ZN2 (2) ZN2 + 3H2 ≡ Z + 2NH3
(17)
N2 + 3H2 ) 2NH3 Here, ZN2 is an adsorbed intermediate in the form of dinitrogen, step 1 is a reversible one, and step 2 is an equilibrium one. It was demonstrated that the rate expression is70
(
) ( )
PH23 T ln U k+PN2 r) 1- 2 ηC PNH32
R
( )
2 T ln U k+ PNH3 - 2 η C K P H 23
1-R
(18) where K is the equilibrium constant (K ) k1/k-1*K2 ) PNH32/PH23PN2), R is the Polanyi parameter, that is, a bridge between kinetics and thermodynamics,77 and U ) (k1/k-1)PN2 ) (k1/k-1K) (PNH32/PH23). Equation 18 is
very similar to eq 3, which was originally proposed by Temkin for ammonia synthesis. Practically in the region of medium coverage, the difference between them is so small that it can be neglected at least for engineering purposes.70 However, if the reverse reaction can be neglected, at high nitrogen pressures, the reaction rate according to the model of induced nonuniformity differs from that for the model, based on biographical nonuniformity. This could be an explanation for the experimentally observed deviations from the first order in nitrogen at high nitrogen pressures.70 In fact, it is now commonly accepted that adsorbed nitrogen atoms represent the most abundant surface species; therefore, the mechanism (eq 17) is presented here only for illustration purposes. However, it can be easily demonstrated that an equation similar to eq 18 can be obtained on the basis of the surface electronic gas model also for a mechanism which supposes dissociative adsorption of nitrogen as the rate-limiting step. Although it is not directly related to the topic of the present paper, it is noteworthy to discuss here the Polanyi parameter (R). The relationship between kinetics and thermodynamics exemplified by this parameter is a general feature of chemical reactions. Linear free energy relationships (LFER) are widely used in homogeneously catalyzed reactions. These relationships bind reaction constants (k) with equilibrium constants (K) in a series of analogous elementary reactions, k ) gKR, where g and R (0 < R < 1) are constants. One of the typical examples is the Brønsted or the Hammett-Taft equation. Evans and Polanyi introduced this relationship to organic reactions, Semenov extended the application to chain reactions, and Temkin applied the linear free energy relationship to heterogeneous catalysis.77,78 Following successful application of the Hammett equation in homogeneous catalysis, several publications appeared also about direct utilization of Hammett relationships to heterogeneous catalytic hydrogenation,79,80 as well as acid-catalyzed reactions, like hydration, hydrolysis, isomerization, and electrophilic aromatic substitution.81 The Brønsted-Evans-Polanyi relation assumes linear dependences of the Gibbs activation energy and the Gibbs energy of an elementary reaction. If this relationship holds for heterogeneous catalysis, it is valid also for ideal surfaces. Note that recently surface-catalyzed chemical processes, including chemisorption, ammonia synthesis, methanation, and so forth, were discussed in terms of the Brønsted-Evans-Polanyi relation.82 For nonuniform surfaces, it was somewhat intuitively suggested22 that ∆G* ) R(∆G0) + const, with R being independent of the surface site or surface coverage. A recent analysis83 of activation energies for N2, CO, NO, and O2 dissociation on a number of different metals plotted as a function of the calculated dissociative chemisorption potential energy for the dissociation products indicated that although the values of chemisorption energy for close-packed surfaces and special step sites, where five metal atoms can be used for dissociation, are different, the slope in the variations of the transition-state energy with the final-state energy is the same for these two different sites, with the Polanyi parameter close to 0.9. The results were interpreted by proposing that, for a given metal surface geometry, the transition-state structures are essentially independent of the molecule and the metal considered. Moreover, the bond length in the transition state is quite long, and the constituent
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atoms have largely lost their molecular identity. The relationship is adsorbate independent, since the transition-state geometries are so similar for different reactants.83 Consequences for Applied Catalytic Kinetics: Concluding Remarks The considerations presented above indicate that there is a wealth of literature demonstrating the limitations of a classical Langmuir approach in treating the kinetics of processes involving catalytic surfaces. However, for practical purposes of kinetic and reactor modeling, very often traditional kinetic descriptions are used. This situation was considered to be a paradox of heterogeneous catalysis.84 In fact, there are several reasons why classical kinetic approaches are preferred. First, keeping in mind that in catalytic reaction engineering not only kinetics but also reactor design should be incorporated, it is much easier to deal with simple, explicit models then to take into account such fine effects as surface heterogeneity. It also applies that rather rigorous models for heat and mass transfer as well as flow and temperate patterns are required to provide a balanced description. Second, many catalytic reactions occur in the domain of coverages and steady-state conditions, displaying first- or zero-order kinetics, when surface nonuniformity (apriori or induced) is not manifested. As an example, liquidphase hydrogenations were discussed above. Third, when the degree of nonuniformity is small (e.g., rate of surface reactions and adsorption/desorption are weakly depending on coverage) and the precision of kinetic measurements is low, the difference in kinetic description for nonuniform and uniform models could be minor. For the case of stepwise nonuniformity with distinct crystallographic faces showing different activities, only when these differences are neither small nor very high, it is possible to observe over a wide range of parameters a kinetic behavior typical for nonideal surfaces with fractional reaction orders. On top of that, as demonstrated,73 the multicentered nature of adsorbed species masks the influence of nonuniformity, which supports the utilization of models of ideal adsorbed layers to treat the adsorption behavior of large organic molecules. At the same time, even for practical purposes, the models should be as simple as possible but not simpler. In the case of non-steady-state operations with a substantial importance of dynamic behavior, simplified kinetic expressions fail to describe kinetic behavior and more sophisticated models should be applied. An example when the coverage dependence of activation energies based on surface science knowledge was included in the highly sophisticated reactor modeling was recently published.85 Recent advances in catalysis on nanosized catalytic particles will definitely lead to corresponding nanokinetics, when not only the activity of different crystallographic faces but also precise knowledge about adsorption modes86 and their dynamic changes as the reaction progresses will be taken into account. It can be thus concluded that surface heterogeneity manifests itself in catalysis and chemisorption. Despite the fact that the current catalytic engineering practice is very much limited to Langmuir-Hinshelwood kinetics based on concepts of ideal surfaces, there is a number of cases when such an approach is not sufficient enough
for the modeling of the kinetics of heterogeneous catalytic reactions. Acknowledgment The author is grateful to Dr. A. K. Avetisov and Dr. V. L. Kuchaev for discussions and valuable comments. Supporting Information Available: Figures showing the differential heats of adsorption of hydrogen, CO, N2, and ammonia; differential heat versus adsorbate coverage for the adsorption of H2 and CO; and adsorption isotherms of 2,2,4-trimethylpentane and kinetic equations for biographically nonuniform surfaces. This material is available free of charge via the Internet at http://pubs.acs.org. Literature Cited (1) Brown, W. A.; Kose, R.; King, D. A. Calorimetric measurements of the adsorption heat for ethene on Pt{211} and Pt{311}. Surf. Sci. 1999, 440, 271. (2) Chorkendorff, I.; Niemantsverdriet, J. W. Concepts of modern catalysis and kinetics; Wiley-VCH: Weinheim, Germany, 2003. (3) Dooling, D. J.; Rekoske, J. E.; Broadbelt, L. J. Microkinetic models of catalytic reactions on nonuniform surfaces: Application to model and real systems. Langmuir 1999, 15, 5846. (4) Dooling, D. J.; Broadbelt, L. J. Microkinetic models and dynamic Monte Carlo simulations of nonuniform catalytic systems. AIChE J. 2001, 47, 1193. (5) Duca, D.; Barone, G.; Varga, Z.; Manna, G. L. Hydrogenation of light hydrocarbons on palladium: theoretical study of the local surface arrangements. THEOCHEM 2001, 542, 207. (6) Ertl, G. Heterogeneous catalysis on atomic scale. J. Mol. Catal. A: Chem. 2002, 182-183, 5. (7) Ertl, G. Heterogeneous catalysis on the atomic scale. Chem. Rec. 2001, 1, 33. (8) Ge, Q.; Kose, R.; King, D. A. Adsorption energetics and bonding from femtomole calorimetry and from first principles theory. Adv. Catal. 2000, 45, 207. (9) Hammer, B.; Norskov, J. K. Theoretical surface science and catalysis - calculations and concepts. Adv. Catal. 2000, 45, 71. (10) Hansen, E.; Neurock, M. Predicting lateral surface interactions through density functional theory: application to oxygen on Rh(100). Surf. Sci. 1999, 441, 410. (11) Hansen, E. W.; Neurock, M. Modeling surface kinetics with first-principles-based molecular simulation. Chem. Eng. Sci. 1999, 54, 3411. (12) Hildebrand, M.; Mikhailov, A. S.; Ertl, G. Traveling nanoscale structures in reactive adsorbates with attractive lateral interactions. Phys. Rev. Lett. 1998, 81, 2602. (13) Johannessen, T.; Larsen, J. H.; Chorkendorff, I.; Livbjerg, H.; Topsøe, H. Catalyst dynamics. Consequences for classical kinetic descriptions of reactors. Chem. Eng. J. 2001, 82, 219. (14) Kose, R.; Brown, W. A.; King, D. A. Role of lateral interactions in adsorption kinetics: CO/Rh{100}. J. Phys. Chem. B 1999, 103, 8722. (15) Kose, R.; Brown, W. A.; King, D. A. Calorimetric heats of dissociative adsorption for O2 on Rh{100}. Surf. Sci. 1998, 402404, 856. (16) Lombardo, S. J.; Bell, A. T. A review of theoretical models for adsorption, diffusion, desorption and reaction of gases on metals. Surf. Sci. Rep. 1991, 13, 1. (17) Makeev, A. G.; Nieuwenhuys, B. E. Mathematical modeling of the NO+H2/Pt(100) reaction: “Surface explosion”, kinetic oscillations, and chaos. J. Chem. Phys. 1998, 108, 3740. (18) Murzin, D. Yu. Modeling of adsorption and kinetics in catalysis over induced nonuniform surfaces: surface electronic gas model. Ind. Eng. Chem. Res. 1995, 34, 1208. (19) Nieskens, D. L. S.; van Bavel, A. P.; Niemantsverdriet, J. W. The analysis of temperature programmed desorption experiments of systems with lateral interactions; implications of the compensation effect. Surf. Sci. 2003, 546, 159. (20) Park, Y. K.; Aghalayam, P.; Vlachos, D. G. A generalized approach for predicting coverage-dependent reaction parameters
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Received for review September 30, 2004 Revised manuscript received December 29, 2004 Accepted January 18, 2005 IE049044X
