Znd. Eng. Chem. Res. 1992,31, 2497-2502 2497

Weekman, V. W. Lumps, Models, and Kinetics in Practice. Chem.

Wei, J.; Prater, C. D. The Structure and Analysis of Complex Re-

Received for review October 28, 1991 Revised manuscript received March 25, 1992

Accepted July 13, 1992

Robinson, C. J.; Cook, C. L. Low Resolution Mass Spectrometric Determination of Aromatic Fractions from Petroleum. Anal. Chem. 1969,41,1648-1564.

Sullivan, R. F.; Boduszymki, M. M.; Fetzer, J. C. Molecular Trans- formations in Hydrotreating and Hydrocracking. Energy Fuels 1989,3,603-612.

Tiesot, B. P.; Welte, D. H. Petroleum Formation and Occurrence; Springer Verlag: Berlin, 1984; pp 379-403.

Eng. B o g . Monogr. Ser. 1979, 75,

action Systems. Adv. Catal. 1962, 13, 203.

Rate of Heterogeneous Catalytic Reactions Involving Ionic Intermediates

Slavik Kasztelan Institut Franqais d u Pgtrole, B.P. 311,92506 Rueil-Malmaison, Cedex, France

A solution to the problem of the computation of the rate law of a catalytic reaction involving ionic intermediates and taking into account the ionic nature of the surface species is proposed in this work. It is shown that such a rate law is computable within the Langmuir-Hinshelwd-Hougen-Watson (LHHW) theory when an electric charge conservation equation is added to the usual set of equations. Using the approximation that the number of sitea equal the number of negative charges of the surface species and neglecting the contribution of the bulk, a rate law can be computed exactly. This is illustrated by considering four examples of catalytic reactions involving ionic intermediates, namely the isomerization of unsaturated hydrocarbons by Bronsted acids, the hydrogenation of unsaturated hydrocarbons involving the heterolytic dissociation of hydrogen, the generation of methyl radicals in methane oxidative coupling, and the partial oxidation of hydrocarbons, With no approximations the rate law obtained is complex and includes, in an intricate manner, several initial concentrations of surface and bulk species characterizing the catalyst.

Introduction Rate laws for catalytic reactions performed by non-

metallic catalysts are often established using the Lang- mukHinshelwood-Hougen-Watson (LHHW) formalism (Yang and Hougen, 1950). For redox reactions, redox models such as the well-known Mm-van Krevelen model (1954) are also often used. Rate laws obtained using these different models have been found very useful for the analysis of experimental resulta of kinetic studies (Smith, 1982) although in each case the formalism suffers from a number of simplifications and, in particular, the assump- tion that all of the sites are equivalent. Nevertheless, the rather simple equations obtained using these models are superior to empirical power rate laws as they are based on a description of the reaction mechanism and of the catalyst even though both are not, in general, known in great detail.

The various types of adsorption considered for the computation of LHHW rate laws are nondissociative or dissociative homolytic adsorption. Thus adsorbates are considered in the form of either a molecule, a fragment of a molecule, radicals, or atoms. However, a number of catalytic reactions on nonmetallic catalysts involve or are thought to involve ionic intermediates.

Catalytic reactions involving ionic intermediatea can be eliminations, additions, or substitutions (Noller and Kladnig, 1976). A simple example is that of an olefin transformed into a carbocation by association with a proton of Bronsted acid catalysta such as zeolites. Thus a first way to generate an ionic intermediate on a catalytic surface is by an associative reaction of a reactant with a preexisting ionic surface species.

Heterolytic dissociation is another means of generating ionic intermediates, namely one negative and one positive species on the surface. The heterolytic dissociation of a number of molecules has been shown or is thought to occur on nonmetallic polar catalysta. Hydrogen has been shown

to dissociate heterolytically on ZnO forming a hydride ion H- bonded to Zn2+ and a proton associated with an oxygen anion forming a hydroxyl ion OH- also bonded to Zn2+ (Kokes and Dent, 1972). Olefins have been also found or proposed to dissociate heterolytically on a number of ox- ides giving r-allyl carbanion or r-allyl carbocation (Burwell et al., 1969; Stone, 1990). Other molecules may also dis- sociate heterolytidy upon adsorption on an oxide catalyBt such as H20 (Burwell et al., 1969).

A third means of generating surface ionic intermediates is evidently by electron transfer from the adsorbed mole- cule to the catalyst or vice versa. This process is partic- ularly important in oxidation reactions.

The ionic nature of the adsorbed species on the catalytic surface is usually not taken into account in the computa- tion of rate laws according to LHHW or redox models. Then, from both a fundamental and an applied point of view, it appears of importance to be able to express, in a simple way, rate expressions for catalytic reactions in- volving ionic surface intermediates where the ionic nature of the species is explicitly taken into account.

The aim of this paper is to demonstrate that this can be done for a variety of catalytic reactions by adding an electric charge conservation equation to the classical me- thod of computation of LHHW rate law. The method wi l l be illustrated by selected examples of different reactions on nonmetallic catalyst involving heterolytic dissociation or electron transfer such as the isomerization of unsatu- rated hydrocarbons by Bronsted acids, the hydrogenation of unsaturated hydrocarbons via heterolytic dissociation, the generation of methyl radicals in methane oxidative coupling, and the partial oxidation of hydrocarbons.

Results and Discussion Description of the Catalyst. A nonmetallic perfect

oxide Mn+=02; with the cation symbolized by M and the

0888-5885 f 92 f 2631-2497$03.00 f 0 0 1992 American Chemical Society

2498 Ind. Eng. Chem. Res., Vol. 31, No. 11, 1992

anion represented by 02- is considered as the catalyst in this work. The different surface and bulk species of this catalyst are defined in this section.

The electroneutrality of the catalyst is expressed by the following relationship:

where [MI", and [O2-Iot represent the initial cation and anion concentrations of the solid, respectively.

The cations will be separated into bulk Mb and surface cation M, hence

[MI", = [MI"b + [MI0 (2) The symbol for a cation surface site will be M and the total surface site concentration will be [MI". All the sites M are taken as being equivalent and each surface species will be associated to one site.

Similarly, the oxygens of the catalyst 0:- will be either bulk anions ob2- or Surface anions M-02-, hence

(3) In this work a free surface site is defined as an empty surface anion position (symbol "V") linked to one surface cation M. The symbol for the free surface site is therefore M-V.

Many different species will coexist on the catalyst sur- face. There will be species with two formal integer electric charge such as the surface oxygen M a 2 - and species with one negative charge symbolized by M-Y; such as M-OH-, M-O*-, etc. Species with no formal electric charge such as an adsorbed reactant M-R or the unoccupied site M-V will be also present. In the following, the unoccupied site will be considered, by extension, as a surface species and called a vacancy although it does not contain electrons as the usual meaning of an anionic vacancy suggests.

With the notation employed above, it can be seen that, in terms of global negative charge, the surface species M-Y; is equivalent to M--Yi or M-"-Y;@ (with a + /3 = 1) as the total negative charge of the group site-adsorbed species is unity. Thus in this work no distinction is made between different cases of partial charge repartition for the same species. However, this could be done by using the notation M-Yi+ and defining different partial charges -j3 of the species YcB.

For redox reactions we shall consider that only one electron transfer occurs and that only oneelectron-reduced sites are formed. On the surface, the reduced site will either be a vacancy with one electron M-V- (Vo' in the Kroger-Vink notation) or a reduced cation M--V. In the bulk, reduced cations Mb'- may be found as well as a va- cancy with one electron Vb'-. In addition to these species, bulk vacancy v b and bulk oxygen species Ob*- will be ab0 considered as example of defects.

Computation of Rate Laws. The method of com- puting rate laws used in this work is that of the LHHW formalism (Hougen and Watson, 1943; Yang and Hougen, 1950). A set of elementary reactions is first written and one rate-determining step is chosen. Then the quasi- equilibrium conetanta for all the reactant adsorptions (KJ, surface reactions (KsJ or exchange reactions between bulk and surface species W e i ) are written as functions of con- centrations.

Then a number of conservation equations have to be writtan. This is done in the following for the species which have been considered in this work. For the bulk, there will be the bulk cation site conservation equation:

(4)

n[M]", = 2[02-]", (1)

[O2-]"t = [O2-]"b + [M-02-]"

[Ml'b = [Mlb + [M-lb and the anion site conservation equation:

[O2-]"b [O"]b + [o'-]b + [VI, + [v'-]b (5)

For the surface, the conservation equation of the total number of surface sites [MIo will be

[MI" [M-VI + [M-02-] + [M-Y;] + [M-R] + [M-V-] + [M-O*-] + [M--VI (6)

In order to solve this set of equations, an additional equation is needed. This equation is the negative charge conservation equation (Kasztelan, 1990, 1991): 2[O2-]"t 2[02-]b + [o'-]b + [v'-]b + [M'-]b +

2[M-02-] + [M-Y;] + [M-V-] + [M-O'-] + [M--VI (7)

This set of equations can be solved as is usually done for classical LHHW rate law computation because there are as many unknown variables as there are equations. The rate law will be a function of the quasi-equilibrium con- stants Ki, Ksi, and Kei, the partial pressures Pi of all the reactants i, the kinetic constant k, and the initial con- centrations of species and defects ([ ]") characterizing the catalyst. However, a complex formula of the form of eq 8 will be obtained for the rate law and a numerical solution may be required in most cases.

r = kf(Ki , Ksi, Kei, Pi, [ 1") (8) To simplify the rate expression, a first approximation

can be used, namely, that the bulk does not intervene in the reaction. Then the negative charge conservation equation becomes the surface negative charge conservation equation: 2[M-02-]" = 2[M-02-] + [M-Y;] + [M-V-]

+ [M-OO-] + [M--VI (9) In this case the rate law will be a function of two initial concentrations, [MI" and [M-O"]".

Another approximation can also be made, namely, that the surface is either initially composed of half unoccupied sites M-V and half M-02- species or is composed of M- OH- species only. In other words, the sum of the negative charges of surface species equals the total number of surface sites, i.e.

(10) This approximation will make it possible to solve exactly the set of equations and the only parameter characteristic of the catalyst remaining will be [MI", the concentration of sites, as it is found in classical LHHW rate laws.

Reactions Involving Heterolytic Dissociation. The heterolytic dissociation of one reactant on the surface during a catalytic reaction will create a perturbation of the surface array of M-02- anions and free sites M-V. For example, on a catalyst such as an acidic oxide, H20 may play an important role by adsorbing on Lewis acid sites M-V and then transforming these sites into Bronsted acid sites (M-O"H+) upon heterolytic diaaociation. When such a dissociation occurs for water, a species carrying formally a double charge, M-02-, is transformed into a species carrying a single charge M-OH- (equivalent to M#-H+). Simultaneously, a free site with no electric charge is con- verted to a species, M-OH-, Carrying a single charge. Such a change of the number of electrical charge of a surface species is usually not taken into account in classical kinetic models. Two examples will be taken to illustrate the method of

computing rate laws of reaction involving heterolytic dissociation without electron transfer and with all the elementary steps of the reaction being surface processes.

2 [ M-02-] " = [MI "

Ind. Eng. Chem. Res., Vol. 31, No. 11,1992 2499

Table I. Set of Elementary Reactionr for the Iromerization of Olefin R into P on a Bronsted Acidic Catalyst

1.1 R + M-OH- M-ORH- KR 1.2 M-ORH- - M-OPH- rI80M 1.3 P + M-OH- = M-OPH- KP 1.4 HzO + M-V + M-0'- * 2M-OH- KHP

The first example (example 1) will be the isomerization of an olefin R by a Bronsted acid catalyst where heterolytic dissociation of water is considered. The second example (example 2) will be the hydrogenation of an unsaturated hydrocarbon where both hydrogen and water dissociate heterolyticall y .

Example 1. The isomerization of an olefin R is de- scribed by the set of elementary reactions reported in Table I. An alkyl carbocation RH+ is first formed by an associative adsorption of the olefin on a proton in reaction L1 where the notation used for the carbocation is M-ORH- rather than M-Oa-RH+ in order to keep track of the global single negative charge of the surface species. Rearrange- ment of the adsorbed carbocation then occurs according to reaction 1.2, which is considered to be the rate-deter- mining step. The isomerized olefin P can then desorb according to reaction 1.3. The adsorption of water is taken into account by reaction 1.4.

For this set of elementary reactions, the conservation equation of the total number of surface sites [MIo is [MIo [M-VI + [Ma2- ] + [M-ORH-] +

[M-OPH-] + [M-OH-] (11) and the surface negative charge conservation equation is written 2[M-O2-I0 = 2[M-02-] + [M-OH-] +

A solution is obtained by making the approximation that the number of negative surface charge is equal to the number of sites (eq lo), hence

[M-ORH-1 + [M-OPH-] (12)

with ai = Kipi. This rate law has the classical form of a LHHW rate law and takes into account all of the reactants and in particular water. It can be easily found that if the water partial pressure is infinite, the Lewis sites are all converted into Bronsted si- and the concentration of site [MIo is equal to the proton concentration, [H+]O. Then the classical rate equation for proton-catalyzed isomeri- zation is found:

Example 2. The example of the hydrogenation of an olefrn with heterolytic dissociation of hydrogen is yet an- other demonstration of the interest in accounting for the ionic nature of the surface species. For such a reaction, the feed contains the unsaturated hydrocarbon R, the hydrogen to be dissociated heterolytically, and H20 also dieeociated heterolytically. This example will illustrate a more complex case of a catalytic surface in interaction with two reactants which can dissociate heterolytically and generate a common species, here M-OH-. A two-step reaction scheme, reported in Table 11, is

considered in this work starting with an associative ad- sorption of the olefin with a proton to form a carbocation RH+ (reaction E l ) , followed by the addition of an hydride ion (reaction 11.2). The latter reaction wil l be considered to be the rate-determining step. Evidently other reaction

Table 11. Set of Elementary Reactionr for the Hydrogenation of Olefin R with Heterolytic Disrociation of Hydrogen . -

11.1 R + M-OH- M-ORH- KR

11.3 RH2 + M-V * M-RHP K R H ~

11.5 HzO + M-V + M a 2 - 2M-OH- KHP

11.2 M-ORH- + M-H- - M-RHZ + M-0" rHm

11.4 Hz + M-V + M-0'- * M-OH- + M-H- KH2

schemes can be considered, such as a two-step hydrogen- ation through the addition of the hydride to form an alkyl carbanion RH- followed by the addition of a proton coming from an OH- group.

A simplified rate law for this hydrogenation reaction is established as before using a charge conservation equation and making the approximation of the equality of the total number of surface sites and the total number of charges of the surface species. The obtained rate (eq 15) has again a form similar to a LHHW rate law.

rm ~HM[M]O X ffRaH2cyH20

(aH2 + aH,O( l + a R ) + 2aH20"2(1 + aRH2)'/2)2 (15)

Hydrogenation with heterolytic dissociation of hydrogen is usually not considered for kinetic analysis of experi- mental results in the literature. It is then interesting to note that making the usual approximations for high-tem- perature reaction, Le., Kipi << 1, leads to the rate law rm = ~ H M [ M ] ~ K R P R K H ~ H ~ with an order 1 relative to the hydrocarbon as well hydrogen, as experimentally found for aromatic hydrogenation over sulfide catalysts for example (Ahuja et al., 1970). Then on the basis of an order 1 for the hydrogen and hydrocarbon partial pressure depen- dency of the rate of hydrogenation, an heterolytic mech- anism cannot be excluded.

Reactions Involving Electron Transfer. Catalytic partial or total oxidation reactions also involves various charged species such as 02-, 0.-, and 02*-. It therefore appears of interest to extend the use of a charge conser- vation equation for the calculation of the rate of oxidation reactions. The difficulty will be that in catalytic oxidation the bulk intervenes in the reaction mechanism through, for example, the mobility of oxygen species, vacancies, and electrons. However, as will be shown hereafter, the use of a charge conservation equation, resulting from the electroneutrality condition, allows solving of the set of equations leading to the computation of all surface or bulk species. Approximations will be needed however to obtain simplified LHHW rate laws.

In the following, two examples of oxidation reaction will be taken: one in which the rate-determining step is the reaction of the molecule with an oxygen species in an Eley-Rideal mechanism (example 3) and one where the electron transfer governs the rate of reaction (example 4).

Example 3. The catalytic generation of methyl radicals in methane oxidative coupling appears to be favored by the use of basic p-type semiconductor catalysts composed of irreducible cations (Dubois and Cameron, 1990). Two reaction mechanisms are proposed in the literature. The first one proposed by Lunsford et al. (1984) emphazises the role of the 0'- species to abstract an hydrogen atom from methane. The second, proposed by Wuka and Jinno (1986) emphazises the role of a non-fully reduced form of diatomic oxygen species (0;- or 022-) to activate adsorbed methane.

The set of elementary reactions proposed by Lunsford et al. (1984) is chosen in this work in an adapted form and is reported in Table 111. Oxygen is adsorbed and trans-

2500 Ind. Eng. Chem. Res., Vol. 31, No. 11,1992

Table 111. Set of Elementary Reactions for the Generation of Methyl Radicals in Methane Oxidative Coupling (Adapted from Lunsford et al. (1984))

111.1 0 2 + M-V'- + Map'- KO2 111.2 M-02'- + M-V'- 2M-O'- K502 111.3 M-0'- + M-V'- * M-V + M-0'- K s O 111.4 CHI + M-0'- - M-OH- + CHB' r 111.5 HzO + M-0'- + M-V + 2M-OH- KHR 111.6 M-0'- + ob2- + Ob'- + M a 2 - Ke0

formed into anions in reactions 111.1-111.3. The notation used is a chemical notation where the 0.-, whether surface or bulk, is the positive charge carrier h+ of the p-type semiconductor as the cation is supposed to be irreducible. The reduced site M-V- is therefore a vacancy associated with an electron (Kroger-Vink notation Vo'), and the unoccupied site M-V is a vacancy which is not associated with an electron (Kroger-Vink notation Vo").

The rate-determining step is considered to be the hy- drogen abstraction from a methane molecule coming from the gas phase by a 0'- anion (Eley-Rideal mechanism, reaction 111.4). The desorption of water is taken as a recombination of two OH- (reaction 111.5). Finally, to be complete, an exchange between bulk and surface oxygen species should be included (reaction 111.6). It also corre- sponds to the positive charge migration responsible for the p-type semiconductivity.

To compute the species concentrations, the following three conservation equations are required first, the con- servation of surface sites,

[MI' [M-VI f [M-OH-] + [M-V'-] + [M-O,*-] + [M-0'-] + [M-02-] (16)

second, the conservation of bulk anionic sites as the bulk is involved with the mechanism through reaction 111.6,

(17) and, third, the charge conservation equation over the whole catalyst,

[O2-]'b = [o'-]b + [02-]b

2[0"]'t = 2[02-]b + [o'-]b + 2[M-02-] + [M-OH-] + [M-V'-] + [M-O,'-] + [M-O*-] (18)

This again can be solved as the number of independent equation matches the number of unknown concentrations. The rate law will now depend on three initial concentra- tions characterizing the catalyst: [MI', [Op]'b, and [02-]', For the sake of clarity the exchange of oxygen species with the bulk is neglected and the number of negative charges of surface species is assumed to be equal to the number of surface sites (eq 10). The obtained rate of reaction has the form of a LHHW rate law:

r = k[M]O PCH,8021'2

1 + ao2 + pO:/2 + ( ~ K s O ' / ~ + aH,01/2)~02'/4

(19)

Example 4. Other oxidation reactions are limited by electron transfer and are usually interpreted by a Mars- van Krevelen model or an LHHW model with two distinct sites, one oxidized and one reduced. As an example of such a reaction, the partial oxidation of an hydrocarbon, RH2, by oxygen to give an oxygenated product is considered. A reaction scheme has been reported in Table IV, which has been adapted from Grasselli et al. (1981). This scheme satisfies most of the well-known features of the partial oxidation of propylene to acrolein (Haber and Witko, 1981; Krenze and Keulks, 1980; Monnier and Keulks, 1981; Grasselli et al., 1981; Kung, 1989) except the use of an 0'-

with Pi = KsiKiPi.

Table IV. Set of Elementary Reactions for the Partial Oxidation of Hydrocarbon RH2 (Adapted from Grasselli et al. (1981))

IV.1 IV.2 IV.3 IV.4 IV.5 IV.6 IV.7 IV.8 IV.9

species to extract the second hydrogen from the hydro- carbon. This feature has been chosen to avoid a two- electron transfer to a single metallic center M in one step. Thus only one-electron transfer occurs and the only re- duced species formed is M--V. Here the reduced site M'--V is a reduced cation M("-l)+ as partial oxidation catalysta are often n-type semiconductors containing re- ducible cations. Then the oxidized and reduced sites are two different oxidation states of the same metallic cation, for example Mo8+ and Mo6+ in molybdenum based oxide catalysts.

The oxygen adsorption goes through the sequence 02*-, 0.-, and 0" and is described by reactions IV.1-IV.3. The rate of the oxidation step is determined by the production of 02-, Le., reaction IV.3. The adsorption of the hydro- carbon occurs through an hydrogen abstraction associatad with an electron transfer to form the radical intermediate RH'. This reaction IV.4 is assumed to determine the rate of reduction of the catalyst. The radical RH' is then ox- idized according to reactions IV.5 and IV.6, and the oxi- dized product desorbs according to reaction IV.7. Finally, the desorption of water is taken into account by reaction IV.8 and the exchange between bulk and surface oxygen species by reaction IV.9.

For the set of reactions in Table IV, the rate of reaction is computed by writing the adsorption constants and the equality of the rates of reduction and oxidation:

k,,[M-O*-] [M*--V] = kredP~~2[M-V] [M-02-] (20)

and three conservation equations, one for the total number of surface sites, [MI' = [M-VI + [M-02-] + [M*--V] +

[M-O*-] + [M-O,'-] + [M-OH-] + [M-OR-] + [M-ORH-] + [M--RH'] (21)

one for the conservation of bulk anion sites, (22)

and one for the conservation of the total negative electric charge of the catalyst

[O2-]'b = [02-]b + [M--V]b

2[O2-]'t = 2[02-]b + [M'--V]b + 2[M-02-] [M*--V] + [M-O'-] + [M-O2'-] + [M-OH-] +

[M-OR-] + [M-ORH-] + [M--RH'] (23) The approximation that all phenomenon occurs on the surface is then made in the following. This is admittedly a rough approximation in the case of partial oxidation where bulk oxygen mobility is oftan an important feature of selective oxidation catalysta. Using the approximation that the number of surface sites equals the number of surface negative charges, all of the surface species con- centrations can be calculated and the following rate of reaction is obtained:

A k r e d P ~ ~ ~ k o x P ~ ~ ' / ~ r = (24) (B(koxPo2'/2)'/2 + o(krdpRH,)'/2)2

with A [M]"(Ks~&,)'/~

B = + + ~ o ~ ~ ~ , ~ ' ~ ~ ~ ~ ~ ' ~ ~ K s ~ ~ ~ - ' ( ~ + KsRH-'&;'/~)

D = 1 + (yo2 + CXOR + &:J2

This rate law has a form similar to the Mars-van Krevelen equation (Mars and van Krevelen, 1954), although the model is notably different. At the reaction temperature usually employed for partial oxidation of hydrocarbons, the terms Kipi can be neglected compared to 1. Then in eq 24 the terms B and D are no longer dependent on ox- ygen partial pressure with B = and D = 1. This leads to the Mars-van Krevelen rate law:

r = (25)

It can be checked that the two limiting cases are r = krd'[M]OPRH2 and r = ko~[M]oP0~/2 , which are experi- mentally obtained for the partial oxidation of propylene over B i - M A oxide catalysts, for example (Monnier and Keulks, 1981).

Rate Laws and Parameter Characteristics of the Catalyst. T h e reaction mechanisms written for the four examples considered in this work are only proposed to demonstrate the utility of an electric charge conservation equation to compute rate laws of catalytic reactions in- volving ionic surface intermediates. Other reaction mechanisms can be written leading to different rate laws depending on the set of elementary reactions and the rate-determining step choeen. In all cases the computation of these rate laws will need an electric charge conservation equation. This additional equation generalizes the LHHW method of rate law computation to a large variety of catalytic reactions. Evidently this conservation equation can also be used when the steady-state approximation is preferred for rate law calculations.

For the sake of clarity the examples of computation of rate laws reported above have been treated using two ap- proximations, namely, no contribution from the bulk and the equality of the number of surface negative charges and the number of sites. However, if these approximations are not made, general rate laws will be obtained in the form of complex mathematical functions which may require numerical methods to be solved. These general rate laws will contain all the variables of the system: catalyst- reactant-experimental conditions. In particular, they will be functions of a number of initial concentrations ([ 1") characterizing the catalyst in an intricate manner (eq 8). Thus variables are no longer separated as is usually found in conventional LHHW rate laws of the form

r = k [MI "f(KjSj) (26)

This indicates that kinetic modeling of catalytic reactions, involving ionic intermediates, requires not only an esti- mation of the concentration of surface sites [MIo, but also an estimation of other species concentrations.

In addition, it appears that as more and more details are included in the reaction mechanism (different sites, different surfaces, structural defects, etc.), more initial concentrations [ 1" will be included in the rate law. These concentrations are dependent on the nature and the CrystaUographic structure of the catalytic phase as well as the morphology of the catalyst particles, and all these parameters will be needed to obtain a more comprehensive

kredpRH2kox(KsO$O~02) 'I2

(2k,,1/2(K~0$o~02)1/4 + (k,$RH,)'/2)2

Ind. Eng. Chem. Res., Vol. 31, No. 11,1992 2501

kinetic analysis or modeling of a heterogeneous catalytic reaction.

Conclusion In conclusion, it is shown in this work that the use of

an electric charge conservation equation, within the framework of the Langmu+Hinshelwood-Hougen-Wat- son formalism, allows the computation of rate laws for reactions involving ionic intermediates. Four examples have been taken to illustrate the usefulness of thia method. Two examples deal with surface reactions involving hete- rolytic dissociation and no electron transfer, and two other examples deal with reactions involving electron transfer and participation of the bulk. Using reasonable approx- imations, rate laws are computed exactly. The simplified rate laws obtained for the examples of catalytic reactions involving ionic intermediates taken in this work are slightly different than the rate equations obtained using nonionic surface species. They explicitly take into account the ionic nature of the surface species. With no approximations, the systems of equations to be solved will lead to general rate laws with a complex formula which may need to be solved numerically. These rate laws include several initial concentrations characterizing the catalyst. Thus a com- prehensive kinetic model for reactions involving ionic in- termediates will require, in addition to the number of sites, complementary species concentrations characterizing the catalyst.

Acknowledgment

and a review of the manuscript.

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I am indebted to Dr. C. Cameron for fruitful discussions

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Receioed for reuiew August 16, 1991 Revised manuscript received June 5, 1992

Accepted June 26, 1992