Paradox of Heterogeneous Catalysis - American Chemical Society

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Ind. Eng. Chem. Res. 2004, 43, 3113-3126

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“Paradox of Heterogeneous Catalysis”: Paradox or Regularity? Victor E. Ostrovskii† Karpov Institute of Physical Chemistry, ul. Vorontsovo Pole 10, Moscow 105064, Russia

“The paradox of heterogeneous catalysis” reflects an opinion that no data file on stationary rates of a heterogeneous catalytic reaction can clarify whether the surface heterogeneity is of principal importance for the catalysis and that the surface heterogeneity is an open question. On the other hand, followers of notions on the principal importance of surface heterogeneity consider the power-law kinetic equations for catalytic processes and also logarithmic isotherms and exponential rate-coverage dependences for middle-coverage chemisorption equilibriums and rates respectively as the manifestations and proofs of surface heterogeneity. In this paper, it is shown that the available power-law kinetic equations and the relations translated by the logarithmic isotherms and exponential rate-coverage dependences can be deduced on the basis of the notion on surface uniformity. In addition, about 40 available metal-gas systems demonstrating the constancy of the molar heats of chemisorption over wide ranges of coverages are listed, and the temporal tendency of the measured molar heats of chemisorption to approach to constant levels, as the techniques and procedures are improved, is demonstrated. It is concluded that no surface heterogeneity reveals itself in catalysis and chemisorption at metals and the paradox arose from algebraic peculiarities. I. Introduction I.1. Subject of the Investigation. “The paradox of heterogeneous catalysis” that is “the paradox of kinetics of catalytic reactions” (below, paradox) had been formulated by Boudart.1 Under these names, it was multiply discussed, for example, by Kiperman and Boudart2,3 and by Weller.4 The paradox lies in the fact that inadequate models of catalyzing surfaces can be successfully applied for deduction of kinetic equations capable of describing satisfactorily any set of experimental data on stationary rates of a heterogeneous catalytic reaction. This paradox is discussed mainly in the context of the utility of experimental data on the rates of stationary catalytic processes for a solid solution of the following fundamental problem of heterogeneous catalysis: whether the surface centers catalyzing any chemical reaction are homogeneous in their catalytic ability or whether a rather wide distribution of the centers by their catalytic activity exists; i.e., the surfaces are heterogeneous. Our experience and the results presented in refs 4 and 5 (Chapter 4) call into question the possibility of solving this problem by using nothing but kinetic methods. We intend to make an attempt to unlock the nature of this paradox, at least as applied to metal catalysts, and to show that there are no fundamental causes to assert that the surface heterogeneity reveals itself in stationary catalytic and chemisorption processes. The studies that led us to the ideas presented in this paper were started in the early 1960s, when we began to design microcalorimeters6,7 and other instrumentation and procedures for measurements of the coverage dependence of molar heats of chemisorption (MHCs). At that time, no calorimetric data on MHCs at elevated temperatures were available. For as long as we applied the adsorption technique and procedures traditional for that time, we obtained coverage-dependent chemisorp† Tel. and fax: 007 (095) 975-2450. E-mail: vostrov@ cc.nifhi.ac.ru.

tion heats at metals. However, once principal technical and procedural defects were revealed and rectified (this activity is detailed in ref 8), metal surfaces “ceased to be” heterogeneous. Calorimetric data came into conflict with the power-law kinetic equations for catalytic processes, the logarithmic isotherm, and the so-called Elovich rate-coverage dependence (in Russian literature, this dependence is referred to as the RoginskiiZeldovich equation because it was first proposed by these authors in ref 9), the applicability of which allegedly proves that surfaces are heterogeneous relative to the catalytic and adsorption ability of the surface centers. In this connection, we decided to analyze in detail all aspects of self-expression of surfaces in gassolid interactions. In this paper, the results of our 40year activity in this field are presented at the background of available notions and experimental results. Nipping on ahead, I can say that the well-known simple Langmuir-Hinshelwood design is quite enough to explain the power-law kinetic equations, logarithmic isotherm, and Elovich rate-coverage dependence; otherwise, there are grounds to say that the surface heterogeneity reveals itself neither in the kinetics of catalytic reactions nor in chemisorption. Thus, the paradox does not exist because no reliable grounds to say about surface heterogeneity can be revealed. In connection with the paradox, Boudart and Dje´gaMariadassou5 considered also the following feature peculiar to the kinetics of heterogeneous reactions: “Mathematically, two different kinetic equations based on the same set of elementary steps, can approximate the same experimental data, depending on how far equilibrium are elementary steps.” This feature is not considered by us in this paper. I.2. Notions on Surface Heterogeneity: Origins and Earlier Discussions. Langmuir was the first to publish the notion on a possible heterogeneity in the adsorption properties of different surface centers. However, he believed that the surface centers of amorphous glasses only could be characterized by a continuous

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distribution by their adsorption ability (see ref 10, p 1371) and never supported the notion on heterogeneity of the surface centers of crystal solids. Meanwhile, Langmuir’s idea on a possible distribution of surface centers by their adsorption ability was extended to crystal bodies by Taylor.11 He assumed that the surface atoms located at crystal faces, edges, and corners, near crystal defects, etc. are different in their properties (the so-called “geometric” or “biographic” surface heterogeneity). This means that individual characteristics of the surface atoms, namely, the degree of their “valence unsaturation”, prevail over the collective characteristics of the surface as a whole. As noted above, Langmuir never considered the surfaces of crystal bodies as if they were heterogeneous. His measurements of high-temperature adsorption of Cs and Th at W show that the tungsten surface is quite homogeneous with respect to its adsorption ability.12 Langmuir showed that such adatoms depolarize each other to an increasing extent as the surface coverage increases. He presented a recognized proof for the electrostatic nature of the forces determining the metalvapor deposition at metal surfaces in similar systems.12,13 In other words, the chemical affinity is of nonprime importance for such systems. This is a principal difference of the systems similar to those studied in refs 12 and 13 from the catalyst-gas reactant systems where adsorption of the gases by the solid catalysts is controlled by the forces of the chemical affinity. The decisive importance of the chemical affinity causes the use of the term “chemisorption” to the processes proceeding at catalysts lying in reacting mixtures. Apparently, the most suitable term for adsorption of vapors in the systems similar to those studied in refs 12 and 13 is “electrosorption”. The systems studied in refs 12 and 13 and the usual gascatalyst and gas-adsorbent systems have in addition some other distinctions. First, the adsorption and desorption temperatures in refs 12 and 13 ranged between 1900 and 2000 K, i.e., much higher than those in the usual catalytic and adsorption processes. Second, according to ref 12, adsorption of alkali elements occurs on conditions that the metal adsorbent affinity to electrons exceeds the ionization potential of the alkali metal. Thus, the valence electrons of the adsorbate atoms pass into the adsorbent at the moment of adsorption. However, different gases and vapors may be adsorbed independently of their ionization potential and, generally, the mechanisms of gas adsorption may differ principally from the mechanism proposed by Langmuir. Third, in the experiments by Langmuir, all collisions of the molecules of the metal vapors with the free surface centers led to adsorption. Thus, adsorption proceeded with no activation energy and, therefore, the activation energy of desorption was equal to the heat of adsorption. Chemisorption of gases at the usual adsorbents and catalysts proceeds usually, by contrast, with some activation energy. Therefore, ref 12 by no means proves that mutual depolarization of adparticles should be forthcoming under conditions of gas chemisorption and should influence chemisorption and catalytic processes. Nevertheless, Langmuir’s idea on the mutual depolarization of adatoms was then multiply modified by different authors as applied to the adsorptiondesorption phenomena proceeding in the course of catalytic and adsorption processes (see, e.g., refs 14-

16). These modifications were intended for an analytical description of notions on the so-called “induced surface heterogeneity”. Such notions were developed to explain the coverage-dependent MHCs measured in some experimental works and, in turn, led to the conclusion on a coverage dependence of the adparticle activity in the subsequent transformation into products of catalytic reactions. Since the late 1930s, notions on surface heterogeneity were applied to descriptions of kinetic experiments relating to the reactions of ammonia synthesis at Fe/K2O/ Al2O3 catalysts,17 SO2 oxidation at Pt,18 shift-reaction at ferric oxide,19 methanol synthesis at ZnO-Cr2O3,20,21 oxygen exchange between carbon oxide and carbon dioxide at ferric oxide,22 and some other reactions.23 The notions are based on assumptions, among which the following ones are most principal: (1) The surfaces are heterogeneous and the MHCs decrease linearly with the coverage over the so-called range of mean coverages (or, more precisely, it is taken that the free energy of chemisorption decreases linearly and that the standard entropy does not change with the coverage). (2) When anybody arranges in his mind the surface centers in the order of decreasing adsorption ability, the changes in the activation energy of chemisorption are equal to half the changes in the MHCs at the corresponding centers (the “transfer factor” for chemisorption R ) 0.5). It is important to note that the term “mean coverages” has a special meaning: it is a coverage range where a0p . 1 and a1p , 1 [p is the equilibrium pressure of chemisorbed gas, and a0 and a1 are the adsorption coefficients (equilibrium constants of chemisorption) for those adsorption centers of this range that chemisorb most weakly and most strongly, respectively]. In order that such a pressure region could be wide the decrease in the MHCs over the coverage range should not be rather significant (according to estimations made by Boudart,3 it should be about 65 kJ/mol). Strictly speaking, realization of this condition should be proved for each system before application of these notions. The actual realization of these assumptions, all together, is, according to refs 17 and 23, the necessary condition of the so-called logarithmic isotherm, the use of which in combination with some assumptions on the reaction mechanisms is capable of giving the kinetic equations proposed for the reactions listed above. According to refs 17 and 23, these kinetic equations can in no way be interpreted on the basis of notions on homogeneous surfaces. Below, we will show that none of these assumptions was proved earnestly for any reaction. With the passage of time, it was stated that such a notion is insufficient for the description of a number of reactions. Therefore, it was taken that the distribution of the surface centers by their chemisorption ability can be described by the function containing two constants, C and Θ, instead of one constant taken earlier.24,25 No experimental data on the MHCs and chemisorption rates were proposed in furtherance of this conception. In a special case, when Θ f ∞, this notion leads to the logarithmic isotherm and does not differ from the notion considered above. Meanwhile, Temkin considered the NH3 synthesis and other above-listed reactions, for the kinetics of which he took R ) 0.5, as his central works. Nevertheless, he declared in refs 26 and 27 that “...physical

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grounds making heterogeneity of the surfaces to be uniform (linear coverage dependence of the MHCs) could hardly be seen”. To explain the uniform surface heterogeneity, which is necessary for deduction of the logarithmic isotherm used for substantiation of the kinetic equations obtained for these reactions, Temkin assumed the occurrence of a surface electron gas in solids.25,26,28 (A similar idea was expressed by Eley.29) He took that the near-surface and surface electrons are as if they are located in an infinitely deep potential well and that this feature retains them in the near-surface layer. It was taken that adsorption influences the electron concentration and the kinetic energy of the near-surface electron layer and that this influence causes a decrease in the MHCs with the coverage. Several additional assumptions were taken. The electrons are energetically independent from each other, and the following parameters are coverageindependent: the effective electron mass (m*) of the near-surface electrons (taken to be equal to the effective mass of the electron of the volume electron gas), thickness (l) of the layer of localization of the nearsurface electron gas (taken to be equal to the lattice constant), and ionicity (η) for the adsorption bond. (It is useful to note that the mass (m) of a free electron has no physical meaning for crystal bodies, because of the electron interaction with the periodic potentials of lattices.) With these assumptions, the following equation was obtained for the difference ∆Q between the MHC (Q) corresponding to a minor coverage and the current MHC (Qθ) corresponding to the coverage θ (h is the Planck constant):

∆Q ) η2h2Lθ/4πm*

(1)

It seems that this Q-θ dependence is capable of explaining the logarithmic isotherm of chemisorption. However, the assumptions of the theory came repeatedly under criticism (e.g., refs 30-35). The same assumptions supplemented with an interelectron interaction considered in the framework of the perturbation theory give a cofactor 0.5 - 8b/9π, where b ) (2πn0a02l)0.5 (n0 ) N0 + Ne, where N0 and Ne are the volume electron density of the “surface electron gas” before chemisorption and of the additional electron gas, introduced into the electron subsystem with the adatoms, respectively).30 Therefore, according to ref 30, ∆Q depends on θ because Ne ) f(θ), and thus eq 1 does not lead to the logarithmic isotherm. Besides, this result is the least of all evils. The criticism of refs 31-35 consists of the following principal points. (i) In the actual crystals, the surface electron gas together with the infinite reservoir of a three-dimensional electron gas form a continuous equilibrium distribution. (ii) Equation 1 reflects no specificity of catalytic action. (iii) If Temkin’s theory would be correct, the heat of hydrogen dissolution in metals would be coveragedependent as a result of a similar effect of the threedimensional gas. The authors of refs 33 and 34 pointed out several examples when the heat of hydrogen sorption was found to be constant. Later measurements showed that the heats of H2 sorption by Ce, Dy, Lu, Er, Tm, and Yb are also coverage-independent.36 One more remark can be made. Under the condition that all assumptions are correct, eq 1 allows no estimation of the ∆Q value, because there are neither experi-

mental nor theoretical prerequisites for sound estimations of the m* value for the “surface electron gas” in any real system. It is well-known that the m*/m value can vary in crystals from more than 10 down to less than 0.01 and can also be negative. For the numerous estimations of the ∆Q values, the m/m* values were taken from data on the cyclotron resonance for threedimensional metals. However, the m* value is determined by the interaction between electrons and lattice potential and, even if the surface electron gas exists in three-dimensional crystals, the m* value for its electrons differs from the m* value for the crystal as a whole by a factor that is indeterminable. The considerations presented above exhaust in no way all attempts of the application of different theoretical approaches to revealing the causes of the MHC decrease with the surface coverage. Isotherms that could be caused by surface interactions of different natures were obtained, for example, in refs 31 and 37-39 (all are reviewed in ref 15). The prime statements of these works are clever but have insufficient experimental support. These isotherms were not used for deduction of kinetic equations, and therefore we do not consider them in this work. I will end this section by quoting the available opinions presented by several researchers analyzing the notion on surface heterogeneity. Boudart’s conclusion was expressed by the following: “Cautious optimism in the use of ‘classical’ surface kinetics has been at the heart of my comments over the years.... In the meantime, the search for cases where the uniform surface approach is valid and the reasons for this validity will continue to be explored...”.3 Stoltze and Nørskov concluded that the surface heterogeneity proposed in earlier works for an explanation of the kinetics of NH3 synthesis is not observed in experiments with monocrystals, and their own computations show that the notion on surface heterogeneity is not necessary to explain the kinetics of ammonia synthesis under high pressures.40 The kinetic modeling performed by Dumesic and Trevino for the pressure range of 0.1-2 MPa led to an analogous conclusion.41 Reference 42 contains the following general conclusion: “The differential isotopic method based on the use of hydrogen, nitrogen, and their isotopes showed that all active centers formed by the surface atoms of crystal faces of porous platinum and graphite have the same adsorption activity in the relation to each of the studied gases, i.e. the surfaces of these adsorbents-catalysts are uniform.” Weller and his colleagues in their fundamental study collected 19 literature examples of the kinetics of catalytic reactions and examined them “for goodness to fit to either the Langmuir, Temkin, or Freundlich rate equations”.43 In this work, it was stated that “in 10 of 19 cases, the fitting is comparably good for all models; in 6 cases, the Langmuir model gives a better fit; and in 3 cases, the fit by the Langmuir model is much poorer than that by either the Temkin or Freundlich models”. This result speaks for itself, and I forbear from making detailed remarks. A lot of fundamental philosophical and experimental conclusions force a number of experts in surface sciences to doubt that the geometrical (biographic) heterogeneity and the induced heterogeneity of surfaces influence principally the adsorption and catalytic properties of real adsorbents and catalysts. It is difficult, if at all

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possible, to bring the notions on a significant effect of surface heterogeneity into coincidence with the following two fundamental Boreskov conclusions:44 (1) The specific catalytic activity [i.e., the turnover frequency (ref 5, p 7)] determined for the same experimental conditions is approximately the same for surfaces of powders, films, wires, and filaments chemically identical but produced on the basis of radically different methods. (According to several works, e.g., refs 45 and 46, the properties of metals change grossly when the crystal size is smaller than 3-5 nm; these results are considered below.) (2) Under the influence of reacting mixtures, the surface layers of catalysts change and a surface-layer state, which is the equilibrium or quasi-equilibrium one relative to the reacting mixture, forms; in other words, the structure of the catalyst is influenced by the reacting gaseous mixture and is transformed by it. Besides, if the MHCs for the surface intermediates were dependent widely on the coverage, the understanding of the phenomenon of specificity of the catalytic action would be hampered and the accepted thesis that the catalytic activity and selectivity of a solid in any catalytic process is determined by the possibility of the formation of definite intermediates should be abandoned. A truly paradoxical situation arose. On the one hand, a great number of theoretical deductions of logarithmic and exponential isotherms and equations for the ratecoverage dependences responding to heterogeneous surfaces with different distributions of the surface centers by their adsorption and catalytic ability are developed, and the kinetic equations thus obtained allow the description of a great number of data sets on the stationary reaction rate for different catalytic processes. On the other hand, the equilibriums and rates of chemisorption and the catalytic kinetics for the same processes can be described on the basis of a simple notion on uniform surfaces and simple equations deduced from this notion. Which are the catalysts functioning in reality in stationary catalytic processes? Do they manifest themselves as uniform or heterogeneous relative to the reaction ability of their surface centers? Are kinetic experiments and measurements of isotherms and rates of chemisorption capable of answering this question or measurements of the MHCs necessary? Which conclusion on these subjects can be made on the basis of the available data on MHCs? We will make an attempt to clarify these questions. II. Whether the Form of the Kinetic Equation Can Be Proof That the Surface Is Heterogeneous The main symbols used by us are as follows: r ) (1/A) (dn1/dt) ) 1/LA (dn1/dt) L ) vtL, where r is the areal rate; A is the area of the catalyst; n1 is the number of turnovers; t is the time; L is the number density of sites, i.e., the number of active catalytic sites per unit area; and vt is the turnover frequency.5,47 Following available works, in which the kinetics considered below were first discussed, and our earlier works (e.g., refs 48-52), we also take θi ) Li/L [i is the free surface catalytic site Z or an intermediate particle U, V, ..., adsorbed at this site (ZU, ZV, ...)] and θZ + θZU + θZV + ... ) 1. (Note that, for the deduction of kinetic equations, the authors of refs 5 and 47 use the relation L ) * + U* + V* + ... (* is the surface catalytic site, and U, V,

..., are the intermediate adsorbed particles) instead of θZ + θZU + θZV + ... ) 1.) The rest of the symbols will be given below; they correspond to those applied in refs 51 and 52. Meanwhile, the nomenclatures being employed by different authors are rather dissimilar. Therefore, I accepted with sincere thanks a recommendation of one of my reviewers for supplying a Nomenclature section (at the end of the paper). In this section, we will show that the kinetic equations for the catalytic reactions of NH3 synthesis at Fe/K2O/ Al2O3 catalysts,17 SO2 oxidation at Pt,18 shift-reaction at ferric oxide,19 methanol synthesis at ZnO-Cr2O3,20,21 and oxygen exchange between carbon monoxide and carbon dioxide at ferric oxide22 deduced on the basis of the notion on surface heterogeneity can be obtained on the basis of the notion on uniformity of surface centers relative to their reaction ability. The kinetic equations for these five reactions are usually presented as the proofs of the importance of surface heterogeneity for catalysis and of the utility of the notion on surface heterogeneity in describing the kinetics of catalytic reactions. Beginning from ref 17, Temkin and Kiperman wrote repeatedly that the equations considered in this section could not be obtained on the basis of Langmuir’s approach. For the near-equilibrium NH3 synthesis at Fe/K2O/ Al2O3 catalysts according to the reaction

N2 + 3H2 ) 2NH3

(2)

the following kinetic equation was proposed:17

r ) r + - r- ) dPNH3/dt ) k+PN2PH23/2/pNH3 - k-PNH3/PH23/2 (3) [the rates r+ and r- of reaction (2) in the directions from left to right (direct reaction) and from right to left (reverse reaction), respectively, are of the same order; PN2, PH2, and PNH3 are the partial pressures (under high pressures, they should be replaced by the activities) of N2, H2, and NH3, respectively; and k+ and k- are the rate constants for the direct and reverse reactions, respectively]. We apply no notions on surface heterogeneity and believe that the surface is covered almost entirely by nitrogen atoms [i.e., the adsorbed nitrogen atoms represent the most abundant reaction intermediates (mari)] and, therefore, the rate-determining step (rds) is chemisorption of nitrogen molecules (the specific rate of N2 chemisorption is high; however, there are scarcely any free surface centers) and the coverages by other chemisorbed particles are negligible. The mechanism considered just below is given in ref 48 and justified in refs 49 and 50. As follows from refs 51 and 52, different forms of its presentation that are identical in their essence can be written. They lead to the same kinetic equation

N2 + Z T ZN2

(4)

2ZN + 3H2 ≡ 2NH3 + 2Z

(5)

ZN2 + Z ≡ f 2ZN

(6)

(the identity sign (≡) means the equilibrium; f means that the equilibrium is shifted to the right, i.e., to a neartotal surface coverage by nitrogen atoms, and the sign T means the rate-determining reversible step).

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According to step (4) of the mechanism (4)-(6)

r ) r+ - r- ) k+1θZPN2 - k-1θZN2

(7)

Taking the law of mass action and θZN = 1, we have 2

K5 ) θZ2PNH32/PH23

(8)

K6 ) 1/θZθZN2

(9)

where K5 and K6 are the equilibrium constants of eqs 5 and 6, respectively. Now, the set of equations (8) and (9) gives θZ and θZN2. Insertion of θZ and θZN2 into eq 7 gives the required kinetic equation

r ) k+1K50.5(PN2PH23/2/PNH3) (k-1/K50.5K6)(PNH3/PH23/2) (10) Consider now SO2 oxidation to SO3 at Pt according to the reaction

SO2 + 0.5O2 ) SO3

(11)

For the kinetics of this reaction, the following equation was obtained earlier on the basis of the notion on heterogeneous surfaces:18

r ) k+PSO2PO20.25/PSO30.5 - k-PSO30.5/PO20.25 (12) We will show that this equation can be deduced on the basis of the notion on surface uniformity. According to ref 7, the rds is SO2 chemisorption and all other steps are equilibrium. We will keep all of these assumptions. It was supposed that SO2 is chemisorbed at the oxidized surface. We do not rule out the possibility that Z ) (PtO)surf. The following mechanism allows deduction of eq 12.

SO2 + Z T ZSO2

(13)

ZSO2 + Z ≡ f Z2SO2

(14)

Z2SO2 + 0.5O2 ≡ SO3 + 2Z

(15)

Taking θZ2SO2 = 1, we obtain equilibrium constants for eqs 14 and 15:

are heterogeneous and adsorption of CO is the rds, authors of ref 19 obtained the following equation:

r ) k+PCOPH2O0.5/PH20.5 - k-PH20.5PCO2/PH2O0.5

(20)

We will keep the last assumption and will take that the catalyst surfaces are uniform. Consider the mechanism as follows:

CO + ZO T ZOCO

(21)

ZOCO + ZO ≡ f Z2O + CO2

(22)

Z2O + H2O ≡ H2 + 2ZO

(23)

Let θZ2O be near unity, i.e., the surface be covered with Z2O almost entirely (Z2O is the mari). The deduction procedure similar to the above-demonstrated ones gives the following kinetic equation:

r ) (k+K310.5)(PCOPH2O0.5/PH20.5) (k-/K30K310.5)(PH20.5PCO2/PH2O0.5) (24) It is seen that eqs 20 and 24 are similar to each other. The following reaction is the methanol synthesis at the zinc-chromium oxide:

CO + 2H2 ) CH3OH

(25)

The kinetic equation for eq 25 is deduced in ref 21 under the assumptions that the rds is H2 chemisorption and surfaces are heterogeneous. The equation is confirmed by experiments in ref 20:

r ) k+PH2PCO0.25/PCH3OH0.25 - k-PCH3OH0.25/PCO0.25 (26) The following mechanism gives the same equation if θZH4 = 1.

Z + H2 T ZH2

(27)

4ZH + CO ≡ CH3OH + 4Z

(28)

ZH2 + Z ≡ f 2ZH

(29)

K21 ) 1/θZSO2θZ

(16)

K22 ) θZ2PSO3/PO20.5

(17)

(k-/K360.25K37)(PCH3OH0.25/PCO0.25) (30)

Now, we obtain θZ and θZSO2 from the set of equations (16) and (17) and insert them into the equation constructed analogously to eq 10. The kinetic equation for SO2 oxidation at Pt is as follows:

The last example to be considered in this section is the CO2-CO oxygen exchange.

0.5

r ) (k+K22 )(PSO2PO2

0.25

r ) (k+K360.25)(PH2PCO0.25/PCH3OH0.25) -

CO2 + CO18 ) COO18 + CO

(31)

0.5

/PSO3 ) -

(k-/K21K220.5)(PSO30.5/PO20.25) (18) Equations 8 and 12 are identical in their forms. Consider now the shift reaction at ferric oxide:

In ref 22, the following kinetic equation was deduced under the assumption of surface heterogeneity and was successfully applied for the description of the experimental reaction rates.

(19)

r ) k+PCO18 PCO20.5/PCO0.5 - k-PCOO18 PCO0.5/PCO20.5 (32)

On the basis of the assumption that catalyst surfaces

We consider this reaction on the bases of the notion on

CO + H2O ) CO2 + H2

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surface uniformity and of the following mechanism (θZ2O = 1):

CO18 + ZO T ZOCO18

(33)

ZOCO18 + ZO ≡ f CO18O + Z2O

(34)

Z2O + CO2 ≡ CO + 2ZO

(35)

This mechanism is associated with the following kinetic equation.

r ) (k+K430.5)(PCO18 PCO20.5/PCO0.5) (k-/K42K430.5)(PCOO18 PCO0.5/PCO20.5) (36) Equation 36 is adequate to eq 32. We showed by five examples that the equations deduced earlier on the basis of the notion on surface heterogeneity are also deducible on the basis of the notion on surface uniformity. Boudart and Dje´ga-Mariadassou showed for different cases that “rate equations are similar [though they are not the same (my remark)] for a given mechanism on a uniform or nonuniform surface”, and “this result justifies the common practice of neglecting nonuniformity of catalytic surfaces in kinetic studies” [ref 5 (p 141)]. Our consideration shows, in agreement with ref 5, that the form of kinetic equations cannot be considered as proof of surface heterogeneity. The earlier procedure of deduction of the above kinetic equations is based on the use of the so-called logarithmic isotherm to describe chemisorption at heterogeneous surfaces. It is agreed that the applicability of the logarithmic isotherm for the description of chemisorption equilibriums shows that the surfaces are heterogeneous. Therefore, the problem that is considered in the following section is topical. In relation to the topic of this section, I present one more simple and clear example showing that kinetic data are quite insufficient for conclusions on the reaction mechanism. In 1979, it was shown53 that the kinetic equation deduced earlier19 for the reaction

CO + H2O ) CO2 + H2

(37)

on the basis of the two-step oxidation-reduction mechanism

Z + H2O / ZO + H2

(38)

ZO + CO / Z + CO2

(39)

is equivalent to the kinetic equation for this reaction deduced on the basis of the following two-step “displacing” mechanism

ZCO2 + H2O / ZH2O + CO2

(40)

ZH2O + CO / ZCO2 + H2

(41)

(/ means that the step is partially reversible). The mechanisms (38)-(39) and (40)-(41) differ principally from each other. Thus, any mechanism consisting of two reversible steps gives the same kinetic equation. Later,

this result was formulated in ref 5 (p 120) for the general case. III. Whether the Applicability of the Logarithmic Isotherm Can Be Proof That the Surface Is Heterogeneous This problem can be analyzed on the basis of comparison of the form of the logarithmic isotherm with the forms of the isotherms deduced for chemisorption of molecules occupying several surface centers of uniform surfaces. The following isotherms will be intercompared: (1) the logarithmic isotherm for chemisorption of two-atom molecules with their dissociation and formation of two surface atoms, each occupying one surface center of a linearly heterogeneous surface:54

θ ) (1/f) ln{[1 + (a0P)0.5]/[1 + (a1P)0.5]2}

(42)

(P is the equilibrium pressure; f, a0, and a1 are the constants); (2) the isotherm for multicentered chemisorption at uniform surfaces with formation of mobile films consisting of nondissociated molecules or of dissociated ones and migrating as a single whole15,55

P ) [θ/ac(1 - θ)n]{[z - (n - 1)θ]/θ}n-1 ) 1/acf1(θ) (43) [a is the constant adsorption coefficient, z is the number of surface centers that can be covered by rotation of a chemisorbed molecule in the plane of the surface through 360° around a center occupied by it (the center of rotation should not be counted), n is the number of surface centers covered by the chemisorbed molecule, and c is the constant dependent on z and n]; (3) the isotherm for chemisorption of two-atom molecules with their dissociation and formation of uniform chemisorbed films, where each chemisorbed atom occupies two surface centers and the atoms migrate on the surface independently from each other:15,55-57

P ) Kθ2[1 + (z - 2)θ]2/(1 - θ)4 ) Kf2(θ)

(44)

(4) the isotherm for two-atom molecules chemisorption at two surface centers of uniform surfaces without dissociation of the admolecules or with their dissociation and migration of the two atoms together, or with dissociation of the admolecules and without migration of the adatoms:15,57

P ) Kθ(z - θ)/z(1 - θ)2 ) Kf3(θ)

(45)

(5) the isotherm for dissociative chemisorption of twoatom molecules with formation of mobile films, where each atom occupies one center at uniform surfaces:10

P ) θ2/a(1 - θ)2 ) 1/af4(θ)

(46)

We perform this comparison on the basis of Figures 1-3. The points in these figures represent experimental data on the chemisorption isotherms for the N2-Fe/K2O/ Al2O3 system.58,59 These results were described with the logarithmic isotherm (42), and the authors considered this fact as the proof of heterogeneity of such catalysts and dissociative chemisorption of nitrogen. Therefore, the points in Figures 1-3 reflect a run of the curve representing the logarithmic isotherm (42) related to the coordinates of these figures. Authors of the notion on

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Figure 1. Isotherms of chemisorption at uniform surfaces (curves) and experimental data of ref 58 on N2 chemisorption at an Fe/ K2O/Al2O3 catalyst (points). Chemisorption with the formation of mobile films consisting of nondissociated molecules or of dissociated ones and migrating as a single whole: (1) n ) 4, z ) 8; (2) n ) 4, z ) 6.

Figure 2. Isotherms of chemisorption at uniform surfaces (curves) and experimental data of ref 58 on N2 chemisorption at an Fe/ K2O/Al2O3 catalyst (points). Chemisorption of two-atom molecules with their dissociation and the formation of chemisorbed films, where each atom occupies two surface centers and the atoms migrate independently from each other: (1) z ) 8, n ) 4; (2) z ) 6, n ) 4. Chemisorption of two-atom molecules at two surface centers without dissociation or with dissociation and migration of two atoms together or with dissociation and without migration: (3) z ) 6, n ) 2; (4) z ) 8, n ) 2. (5) Dissociative chemisorption of two-atom molecules with the formation of mobile films, where each atom occupies one center.

surface heterogeneity attached much importance to these results; for example, ref 58 is cited in ref 23 for 14 cases. In Figure 1, the lines correspond to isotherm (43) with n ) 4 and z ) 8 (curve 1) and n ) 4 and z ) 6 (curve 2). Curves 1 and 2 describe the equilibrium of N2 chemisorption with the formation of the surface Fe2N at {100} and {111} faces of cubic crystals, respectively. Keeping in mind the experimental errors and the approximate character of the isotherms, we conclude that no notion on surface heterogeneity is necessary to explain these experimental data. In Figure 2, the same experimental results are given on the background of isotherms (44)-(46). Curves 1 and 2 [isotherm (44)] correspond to z ) 8, n ) 4 and z ) 6, n ) 4, respectively. Both of these curves approximately describe the formation of mobile chemisorbed Fe2N films. It is seen that such an approach to this chemisorption phenomenon also allows a rather good description of the experimental data. These curves show that the n value (chemisorption of each molecule at four centers) rather than the z value (crystal-face index) is of principal importance. Curves 3 and 4 correspond to isotherm (45) with z ) 6, n ) 2 and z ) 8, n ) 2, respectively, and curve 5 corresponds to isotherm (46). Thus, all isotherms for chemisorption of gas molecules at more than one surface center represent the functions similar in their behavior to the function characteristic for the logarithmic isotherm. Namely, each of them has a middle-coverage section, which can be approximated

Figure 3. Isotherms of N2 chemisorption at an Fe/K2O/Al2O3 catalyst: points are experimental data from refs 58 and 59, and the numbers of the lines correspond to the numbers of experimental series in ref 59; the axes correspond to the isotherm of chemisorption of two-atom molecules with dissociation at uniform surfaces, when each atom occupies two surface centers and the atoms migrate independently from each other (z ) 4, n ) 2).

by a straight line, and each of them turns up and down in ranges of low and high coverages, respectively. Surface dissociation of admolecules and also increases in the numbers of surface centers (n) occupied by each chemisorbed molecule and of neighboring surface centers (z) decrease the slope of these isotherms. Figure 3 demonstrates, by the example of isotherm (44), the possibility for linearization of the isotherms of multicentered chemisorption. The lines in the figure are numerated in accordance with the experimental series numeration given in ref 59; in ref 58, series 3 is not presented and series 2 is numerated as 12. Series 3 and all other series are performed at 673 and 623 K, respectively. The isotherms of Figure 3 are computed with z ) 4 and n ) 4 (dissociative adsorption and each atom occupies two surface centers). Isotherm (44) describes well the data of refs 58 and 59. The results of this section bring out rather clearly that the applicability of the logarithmic isotherm to the description of any data set is by no means the proof of surface heterogeneity. Our consideration is approximate; however, the deduction of the logarithmic isotherm is also approximate and its basis is rather questionable. Meanwhile, in order for chemisorption of any one molecule to be possible, the necessity of several free surface centers is, on frequent occasions, doubtless. IV. Whether the Applicability of the Exponential Equation to the Description of Chemisorption Rates Can Be Proof That the Surfaces Are Heterogeneous To turn from descriptions of chemisorption equilibriums by the logarithmic isotherm to deductions of kinetic equations for the rates of catalytic reactions, followers of the notions on surface heterogeneity postulate the existence of a linear relationship between the activation energy and MHC. In particular, in first presentations (e.g., ref 54) of the notion on surface heterogeneity, the

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following relation was taken:

∆Ea ) -R∆q

(47)

where ∆Ea and ∆q are the increases in the activation energy and MHC, respectively, between any two centers of the heterogeneous surface and 0 e R e 1. The linear relationships are also postulated for all steps of catalytic processes. Kiperman (ref 23, section 7.3, p 112) describes such a turn as follows: “In the general case, the relations of linearity represent the interdependence between activation energies and heat effects of the elementary steps of processes proceeding at different centers of heterogeneous surfaces of solids. This relation means the occurrence of proportionality between rate constants and equilibrium constants for the elementary steps of any process proceeding at different centers of heterogeneous surfaces.” A combination of the linear relationship with some additional assumptions allows the deduction of the following equation for the chemisorption rate r+ at the hypothetical linearly heterogeneous (relative to the differential-heat-coverage dependence) surfaces over a so-called range of middle coverages determined by the inequalities a0p . 1 and a1p , 1:

r+ ) ka′a0RP exp(-Rfθ) ) k+P exp(-gθ)

(48)

[θ and P are the parameters and all other symbols are constants (R is the constant from eq 47; in the initial and most simple version of this notion, it was taken that R ) 0.5)]. The right-hand side of eq 48 represents the Elovich equation. The problem formulated by the title of this section will be analyzed on the basis of comparison of the form of eq 48 with the forms of the equations deduced in refs 57-59 for the r+-θ dependences that should be expected for chemisorption of molecules, each of which occupies several centers at a uniform surface. We consider the equations

r+ ) k+P(1 - θ)4z2/(z - θ)2 ) k+Pf ′1(θ)

(49)

r+ ) k+Pz(1 - θ)2/(z - θ) ) k+Pf ′2(θ)

(50)

r+ ) k+Pz(1 - θ)2 ) k+Pf ′3(θ)

(51)

Equations 49-51 respond to the same chemisorption mechanisms as eqs 44-46, respectively. Figure 4 presents log[f ′(θ)] vs θ for f ′1, f ′2, and f ′3 corresponding to eqs 49-51, respectively. It is seen that these functions can be approximated by linear dependences in the coverage range from 0.1 to 0.5-0.6. This result means that, over this range of coverages, the r+-θ dependences translated, on the one hand, by the Elovich equation and, on the other hand, by eqs 49-51 are of the same form. Thus, the applicability of the exponential equation (48) to the description of chemisorption rates over some θ range cannot be proof that the surfaces are heterogeneous. V. Available Data on MHCs: Whether They Confirm the Notion on Surface Heterogeneity Thus, a formal description of neither the kinetics of heterogeneous catalytic reactions, the chemisorption

Figure 4. Rate-coverage dependences for chemisorption at uniform surfaces: (1) chemisorption of two-atom molecules with their dissociation and the formation of chemisorbed films, where each atom occupies two surface centers and the atoms migrate independently from each other (z ) 4, n ) 4); (2) chemisorption of twoatom molecules at two surface centers without dissociation or with dissociation and migration of two atoms together or with dissociation and without migration (z ) 4, n ) 4); (3) dissociative chemisorption of two-atom molecules with the formation of mobile films, where each atom occupies one center.

isotherms, nor the rates of chemisorption is capable of revealing the surface heterogeneity. It remains to be seen whether the relations of linearity, such as eq 47, apply to catalytic and chemisorption processes and to be seen as the main problem, namely, whether the measurements of the MHCs give grounds to contend that surfaces of catalysts are heterogeneous. To substantiate eq 47, the followers of the notion on surface heterogeneity refer to the Brønstad relation. However, this relation does not have unambiguous interpretation, and the conditions where it was revealed are far from the conditions occurring at the surface catalyzing any reaction between gas components. For catalytic reactions proceeding in solutions under the action of different one-type acids, Brønstad found that the logarithms of the rate constants are in a linear relation with the dissociation constants of these acids.60 As for heterogeneous catalytic reactions, neither the relation (47) nor other linear relations between ∆Ea and ∆q were proved for any surface. Consider now data on the MHCs. Available microcalorimeters are rather sensitive for reliable measurements of the molar heat effects of gas chemisorption at powders (see, for example, refs 8 and 61), films, and filaments (see, for example, ref 62). However, correct measurement of the heats of chemisorption or desorption of small portions of gases is a rather complicated problem. It is evident that the heat effects should be related to the amounts of individual substances reacting in a strictly determined process. Meanwhile, during chemisorption, the following side processes are most frequent: interaction of the adsorbate with substances previously chemisorbed and formation of surface or gaseous products; interaction of the chemisorbed grease vapor with the adsorbent; adsorbate diffusion into the solid body; interaction of the adsorbate with impurities or with solid components diffusing from the body to the surface; adsorbate adsorption on the walls of the adsorption apparatus; chemical interaction between the adsorbate and walls of the adsorption apparatus, metal taps, and balances; and so on. In the course of chemisorption experiments, each of these processes may influence the values of the heat effects measured, the gas amounts actually chemisorbed or desorbed, and the surface-body distribution of the adsorbate in the solid. Therefore, during experiments,

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it is necessary to preclude or to take into account all side phenomena. The correctness of the measurements of chemisorbed or desorbed amounts depends not only on the skill of experts in calorimetry but also on the current state of the techniques and procedures for adsorption studies. Computations of the MHCs on the basis of indirect methods add, on frequent occasions, the errors caused by inadequacy of these methods. Therefore, the presentation of all available data on the MHC with no critical analysis of them makes no sense. On the other hand, a detailed analysis of such data is impossible in this paper, and interpretation of the available data on the MHC at oxides is associated with revealing and discussion of some features specific for each concrete system (see, for example, ref 8). On frequent occasions, available reviews on MHC are not capable of helping because they are not critical. Therefore, we restrict our consideration of this subject to a list of metal-gas systems where constant or nearconstant MHCs were revealed over rather wide ranges of surface coverages and demonstrate the temporal tendencies in refinement of the MHC data as a result of improvement of chemisorption apparatuses, techniques, and procedures. A list of such systems is given below with no pretence of its completeness: H2-Fe,63,64 H2-Ni,65 H2-Pt (black),66 H2-Pt deposited at SiO2,67 H2-catalyst containing Pt and W deposited at SiO2,67 O2-Nb,68 O2-W,68 O2-Mn,68 O2-Ni,68 O2-Co,68,69 O2Fe,68 O2-Mo,68 O2-Ta,68 O2-Ti,68 O2-catalyst containing Pt and W deposited at SiO2,67 O2-W deposited at SiO2,67 O2-Cu,70,71 O2-Ag,72,73 O2-Ir deposited at SiO2,74 O2-Ir deposited at γ-Al2O3,74 N2-Ni,75 N2-Fe promoted with Al2O3 and K2O,76 CO-Ni,65,77 CO-Dy,78 CO-Ni promoted with K or Cs,77 CO-Pt,77 CO-Pt promoted with Rb or Cs,77 acrylic acid-catalyst containing V and Mo, acrylic acid-catalyst containing V, Mo, and Cu, acrylic acid-catalyst containing V, Mo, and P, acrylic acid-catalyst containing V, Mo, and Cs,79 HCtCHplatinum, H3C-CtCH-platinum, H2CdCdCH2-platinum,80 H2-Pt,81 H2-Ce, H2-Dy, H2-Lu, H2-Er, H2Tm, H2-Yb,38 etc. In the context of the problem that is under consideration in this section, the behavior of MHC measured at catalyst surfaces pretreated to be similar to those functioning in the course of any stationary catalytic process is of prime interest. We will consider two examples of such a kind. Figure 5 gives the heats of nitrogen chemisorption (Figure 5a) and of hydrogen chemisorption (Figure 5b,c) measured at 470 K at the Fe/K2O/Al2O3 catalyst surface. Before these measurements, the catalyst was reduced in the calorimeter up to a state when its contact with small portions of NH3 vapor led to desorption of a portion of hydrogen and chemisorption of nitrogen and residual hydrogen and did not lead to water desorption.8,76,82,83 Joining empty bulbs to the volume with the catalyst, we observed H2 desorption and were able to measure the molar heats of H2 desorption (equal in magnitude to the heat of chemisorption) as functions of chemisorbed amounts of nitrogen and hydrogen. Taking into account the heat of H2 chemisorption and the tabulated heat of NH3 decomposition to gaseous H2 and N2, we were able to compute the heats of NH3 chemisorption from the measured values of the heats of catalyst-NH3 interaction, leading to chemisorption of nitrogen and a measured portion of hydrogen. It is seen that the surface reveals itself as the uniform one.

Figure 5. Heats of chemisorption at an Fe/K2O/Al2O3 catalyst at 470 K: of N2 vs coverage by nitrogen (a) and of H2 vs coverages by hydrogen (b) and nitrogen (c).8,76,82,83

Earlier, Emmett and Brunauer obtained from isotherms that the heat of N2 chemisorption is coverageindependent (146 kJ/mol).84 Another example is the copper-zinc-alumina catalyst intended for the methanol synthesis from hydrogen and carbon oxides and for the shift-reaction. We studied the catalyst reduced to a state when it was capable of low-pressure decomposition of methanol vapors to the equilibrium gas mixture containing residual methanol, hydrogen, and carbon oxides. A full set of these works, including the process of reduction of the catalyst, were described in refs 85-89, and the studies of reduced catalyst were detailed in refs 51, 52, and 90. It was concluded that, under the conditions of the laboratory and industrial methanol synthesis, the catalyst consists of zinc oxide, alumina, and Cu4(OH2), the last having a surface similar in its features to that of the metal copper. The alumina component is the structure-forming one, Cu4(OH2) catalyzes mainly the shift-reaction, and ZnO catalyzes the methanol formation.51,52 The following results prove the uniformity of the Cu4(OH2) and ZnO surfaces: (i) The molar heat of H2 chemisorption (43.8 ( 3.3 kJ/mol in the isothermal experiments performed in the temperature range 298-353 K) does not depend on the adsorbed quantity. (ii) The isotherm of the heats of H2 chemisorption at ZnO at 293 K over the pressure range from 0 to 300 hPa is described well by the Langmuir isotherm deduced for dissociative chemisorption at homogeneous surfaces. (iii) The molar heat of CO2 chemisorption at ZnO (59 kJ/mol in the isothermal experiments performed at 298 and 323 K) does not depend on the chemisorbed quantity. (iv) The isotherm of CO2 chemisorption at ZnO at 323 K over the pressure range from 5 to 110 hPa is described well by the isotherm deduced for chemisorption of each molecule at two adsorption centers of homogeneous surface. (v) The heat of CO chemisorption (66 ( 4 kJ/mol at 303 and 353 K) does not depend on the chemisorbed quantity. (vi) The rate of O2 chemisorption is described well by the equation deduced for chemisorption of each molecule at two centers of homogeneous surface. Thus, there is not a grain of evidence to suggest that the surface heterogeneity reveals itself in any manner

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Figure 6. Heats of H2 chemisorption at Ni films at 273 and 298 K: (1) 1950,91 (2) 1953,92 (3) 1960,93 (4) 1964,94 and (5) 1966.95 Figure 8. Heats of O2 chemisorption at silver powders: (3) 1966, 373 K;99 (2) 1977, from isotherms, 450-615 K;73 (1) 1979, 465 and 496 K.72

Figure 7. Heats of H2 chemisorption at Fe films at 273 and 298 K: (1) 1950,96 (2) 1953,97 (3) 1972,63 (4 and 5) 1978,98 and (6) 1990.64

in the MHCs, chemisorption isotherms, or rates of chemisorption. The following figures are intended to demonstrate the long-term tendencies in transformation of the notion on the coverage dependence of the MHC. Figure 6 presents the data on the H2 MHC measured at 273-298 K under ultravacuum at evaporated Ni films. These data are obtained by different authors in a period of about 15 years.91-95 It is seen that the heatcoverage dependence over a rather wide range of middle coverages tended to be less pronounced with the passage of time and came to naught at 1966. According to the data obtained by Bro¨cker and Wedler,95 the MHCs of hydrogen are coverage-independent. An analogous tendency can be found for the hydrogen MHC at iron at 273-298 K (Figure 7). Authors of refs 96 and 97 obtained in 1950 and 1953, respectively, MHCs decreasing with the coverage. However, beginning from 1972, the values almost independent of the surface coverage were repeatedly obtained under ultravacuum for middle coverages of the Fe surface.63,64,98 The level of the constant MHC varies by about only (5% from work to work. It is clear that the most probable cause of this tendency is improvement of the apparatuses, techniques, and procedures. The same tendency occurs in the measurements of the heats of chemisorption of gases at powders. Before the start of our works in the 1960s, there were no available calorimetric data on MHC measured at elevated temperatures at powders. The first system to be investigated was the silver-oxygen one (Figure 8, curve 3).99 We used traditional apparatuses, techniques, and procedures for pretreatment of the samples and for measurement of the chemisorbed gases. In a length of time, we revealed the side heat effects and learned to eliminate them.7 Curve 1 and the points present the data obtained by us on the basis of an improved technique.72

Figure 9. Heats of O2 chemisorption at cobalt: (1) powder, 298 K;72 (2) films, 273 K;68 (3) tabulated heat (per mole of O2) of cobalt oxidation to CoO, 298 K; (4) tabulated heat (per mole of O2) of cobalt oxidation to Co3O4, 298 K.

It is seen that two forms of chemisorbed oxygen can exist at the surface and that the heat of formation of each of these forms is coverage-independent. A similar curve (curve 2) was computed by Czanderna73 on the basis of isotherms. For chemisorption at powders under the usual vacuum and at evaporated films under ultravacuum, constant MHC levels close to each other are obtained for a number of systems. An example is given in Figure 9. Constant levels of the heats of oxygen chemisorption at cobalt powder69 and film68 differ from the mean level by less than 5%, and this difference is within the measurement error. The analysis performed in this section allows us to conclude that no surface heterogeneity reveals itself in MHCs of gases at metal powders and films. This conclusion lays on the table the question on the applicability of the relation of linearity (47). VI. Conclusion: “True and Sufficient Features and No Others Should Be Taken as the Essence of Natural Phenomena; the Nature Is Simple and Does Not Luxuriate in Excesses” (I. Newton, “Principia”) The results of our consideration can be formulated as follows: (1) The assumptions of the principal effect of geometric and induced surface heterogeneities on chemisorption and catalytic phenomena were originally proposed 75-80 years ago as the heuristic ideas intended for clarification of the nature of catalytic and chemisorption processes and for an analytical description of experimental data, a portion of which was not confirmed later on the basis of instrumentation, techniques, and procedures developed in time.

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(2) A consensus between followers of the notions on uniformity and nonuniformity of catalyst surfaces was never achieved. On the one hand, formalistic methods and algorithms for the description of experimental data on the kinetics of catalytic reactions and equilibriums and rates of chemisorption were developed on the basis of the notion on surface heterogeneity and applied to a number of data sets. On the other hand, justified doubts were repeatedly cast on the likelihood of hypotheses on surface heterogeneity. Langmuir’s notion on surface uniformity was developed and applied to the description of chemisorption and catalytic phenomena so widely that Boudart had every reason to write3 that “... surface nonuniformity has been ignored for almost 50 years in catalytic kinetics by physical chemists, chemical engineers, and, more recently, surface scientists”. (3) The paradox formulated in the 1950s factually led the further development of the understanding of heterogeneous catalysis into a dead end. It was stated that kinetic equations deduced from the radically different notions are capable of describing the kinetics of any one reaction. The paradox generated a lot of questions, for example, the following: Are the notions under consideration mutually exclusive? What is prevailing: the individual properties of surface atoms or the properties of the crystal as a whole? If surfaces are heterogeneous, is the character of the heterogeneity reproducible for catalysts of any given composition? If so, why is it reproducible? If not, why are the kinetics of any process reproducible when different catalysts of any chemical composition are used? Is Boreskov’s rule on the constancy of the specific catalytic activity of powders, films, wires, etc., of any metal compatible with the notion on surface heterogeneity. What is the manner in which the notion on the principal effect of surface heterogeneity influences the methods of preparation of catalysts? The nature of the paradox should be unraveled because a success in the solution of this problem could help to answer the above-listed questions. For this to happen, it is necessary to state whether the surface heterogeneity influences principally adsorption and catalytic processes. Sections II-V show that there are no grounds to argue that surfaces reveal themselves in catalysis and adsorption as heterogeneous relative to their adsorption and catalytic activity. We see two possible causes of this fact. The former is the all-surface and surface-body energy exchanges resulting in leveling of all individual peculiarities of the surface atoms. The latter is as follows.48,52 It seems likely that, under conditions of adsorption-desorption equilibrium or stationary catalytic processes, surfaces are covered with the multitude of two-dimensional crystal-like chemadphase islands permanently varying in their form and size, appearing and disappearing in such a way that their total surface area remains unchangeable in time; during catalysis, such intermediate island chemadphases of different chemical compositions can exist. Each of them is in equilibrium with the gaseous medium. The occurrence of such island chemadphases is supported by a gain in the free energy evolving as a result of the twodimensional crystallization of individual adparticles. The vast majority of the two-dimensional crystals of each of the chemadphases consists of a great number of admolecules. Therefore, chemisorption or desorption leading to some decrease or increase in the size of such

two-dimensional crystals changes almost not at all their molar energy, similarly to sublimation or evaporation not influencing the molar energy of large-sized threedimensional crystals, and no significant mutual influence of admolecules on their capability to react with molecules of another chemical nature occurs. Apparently, the former cause acts under the condition that the metal crystals are so large-sized that the number of volume atoms is well over the number of surface ones; otherwise, chemisorption associated with a partial polarization of a chemisorbed particle may change noticeably the number of electrons in the body of the crystal and thus change its characteristics, including the catalytic and chemisorption ones. For very fine crystals, a dependence of the MHC on the coverage may be expected. For example, for the crystals of sizes 20 × 20 × 20 atoms (about 5 × 5 × 5 ) 125 nm3), the numbers of the volume and surface atoms are in the ratio of about 3.5. Thus, it can be expected that the crystals of a linear size of 5 nm and less may be characterized by dependences of the integral MHC on the crystal size and of the MHC on the coverage. This result correlates rather well with the experimental results obtained in refs 45 and 46. However, it is unlikely that so fine crystals could exist in reality in the course of catalytic and adsorption processes proceeding stationary at elevated temperatures. Actually, the specific metal surface of the stationary-working supported metal catalysts does not usually exceed 2030 m2/g. An approximate computation shows that metals consisting of crystals of the mean size 20 × 20 × 20 atoms and having atomic weights from 50 to 200 should be characterized by specific surfaces from 220 to 55 m2/ g, respectively. 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. Their adequate understanding is hampered because it does not always happen that the specific surfaces, crystal sizes, and structure of the films are specified rather clearly. It is improbable that the above effect of very small crystals influences their chemisorption ability. In this connection, we note that the relative activities measured in different works at different faces of any one metal are not always the same. In addition, it is very important that the surfaces of catalysts working under stationary conditions are in equilibrium with the corresponding gas medium and with the catalyst body, while the degree of equilibrium of evaporated films is under question. Summation of the results of this consideration with the last two special remarks allows the following general conclusion. “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.” This conclusion was made in ref 49 on the basis of a data file that was significantly smaller than that presented in this paper. Today, I would like to add that the available data on the kinetics of catalytic reactions can be described on the basis of notions on surface uniformity of catalysts; i.e., surface nonuniformity does not reveal itself in the kinetics of catalytic reactions. This generalization is supported by available fundamental kinetic studies performed by a number of authors for the last 2 decades.

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Thus, no surface heterogeneity reveals itself in catalysis. This general conclusion means that no paradox exists. The paradox arose from an algebraic peculiarity lying in the fact that principally different sets of assumptions can sometimes lead to the same equations. The use of different more or less voluntary premises and oversimplifications on the way from the model to the solution accentuates this peculiarity. Nomenclature / ) partially reversible step or overall catalytic reaction T ) partially reversible rate-determining step ≡ ) equilibrium step or overall catalytic reaction ≡ f ) equilibrium step with the equilibrium shifted to the right a ) adsorption coefficient, i.e., constant of the adsorption equilibrium a0 and a1 ) in Temkin’s theory of heterogeneous surfaces (TTHS), adsorption coefficients responding to adsorption in a selected range of middle coverages at the surface centers with maximum and minimum adsorption abilities, respectively c ) numerical constant computed from the permutation theory, c )f(z,n) (see below) C ) constant in TTHS Ea ) in TTHS, the activation energy for adsorption at a center of the heterogeneous surface f ) constant in TTHS, the so-called breadth of heterogeneity g ) constant in the Elovich equation k+ ) rate constant for reversible reactions or simple steps including chemisorptions proceeding according to written stoichiometric equations in the direction from the left to the right (for direct reactions or steps) per unit surface of active component of catalysts or adsorbents k- ) same as the previous in the direction from right to left K ) equilibrium constant and that multiplied by a numerical coefficient not dependent on parameters of equations under consideration L ) number density of active centers, i.e., the number of active centers per unit surface area m ) electron mass m* ) effective electron mass n ) number of surface centers covered by a chemisorbed molecule n1 ) number of turnovers; number of times that the overall reaction takes place n0 ) parameter of Temkin’s theory of surface electron gas (TTSEG), n0 ) N0 + Nc (see just below) N0 and Ne ) in TTSEG, the electron density of the “surface electron gas” before chemisorption and of the additional electron gas, introduced into the electron subsystem with adatoms, respectively P ) pressure q ) in TTHS, the heat effect of adsorption at a center of the heterogeneous surface Q ) molar heat of chemisorption Qθ ) current molar heat of chemisorption corresponding to the coverage θ r ) areal differential rate of an overall reaction or a simple stage including chemisorption r+ ) same as the previous in the direction from left to right (for partially reversible reactions or reaction steps) r- ) same as the previous in the direction from right to left vt ) turnover frequency, number of turnovers per catalytic center and per unit time z ) number of surface centers that can be covered by rotation of a chemisorbed molecule in the plane of the

surface through 360° around a center occupied by it (the rotation center should not be counted) Z ) surface catalytically active center R ) “transfer factor” for chemisorption; a constant in Temkin’s theory of heterogeneous surfaces R0 and R1 ) constants in the same theory as the previous one, atm-1 Γ ) adsorbed amount per unit surface area, mm3 (NTP) m-2 η ) ionicity of chemisorption bonds; a constant in Temkin’s surface electron gas theory θ ) surface coverage θZ, θZCO, ... ) surface coverage by free catalytically active centers, carbon monoxide molecules, and so on Θ ) constant in TTHS

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Received for review January 24, 2004 Revised manuscript received April 1, 2004 Accepted April 7, 2004 IE049923J