1208
Znd. Eng. Chem. Res. 1995,34, 1208-1218
Modeling of Adsorption and Kinetics in Catalysis over Induced Nonuniform Surfaces: Surface Electronic Gas Model Dmitry Yu. Murzin' Laboratory of Industrial Chemistry, Abo Akademi, FIN-20500, Abo, Finland
Several aspects of adsorption and reaction kinetics in catalysis over induced nonuniform surfaces are considered. They are discussed in terms of the surface electronic gas model, which accounts for the case of inhomogeneous surface. The adsorption equilibrium of gas mixtures was described, and adsorption isotherms were obtained. The reaction rate for the two-step sequence was deduced, and for that scheme the problems of optimum catalyst and metal-support interactions were considered. Structure sensitive and insensitive reactions with nonuniformly reactive adsorbates are also discussed in terms of the surface electronic gas model using the Polanyi parameter. The procedure for the rate law computation is given. Examples include a reaction graph with a pendent vertex and irreversible deactivation and reaction mechanisms with nonlinear elementary steps.
Introduction The theory of the reaction kinetics of heterogeneous catalysis is an important branch of the science of catalysis, widely used for process optimization and reaction design. In recent years new concepts based on the surface science of single crystal surfaces, such as surface reconstruction and ordered overlayers, have been introduced (Ertl, 1990; Madix, 1983; Somorjai, 1991,1993;van Howe and Somorjai, 1989; Woodruffet al., 1983). For instance chemisorption can lead to substrate restructuring of the metal catalyst surface and there is experimental evidence that the catalyst surface restructuring occurs on the time scale of catalytic reactions. It was also shown that adsorbate-adsorbate interactions play important roles in chemisorption and catalysis. The coadsorption of two-electron donors or two-electron acceptors can lead to surface-phase separation and island formation (Somorjai, 1991,1993). Such results have given us the possibility to develop microkinetic models based on reaction rate constants and binding energies deduced from surface science experiments (Somorjai, 1991). However, the kinetics of heterogeneous catalytic reactions is often described using either LangmuirHinshelwood or Rideal-Eley concepts (Smith, 1982) without taking into account the above mentioned phenomehon. According to the first concept, the most likely mechanism involves a surface reaction between adsorbed species. In contrast to this, the Rideal-Eley mechanism assumes that the surface reaction occurs as a result of the collision between a fluid-phase molecule and an adsorbed one. One of the approaches, which is widely used, is the Hougen-Watson formalism (Holland and Anthony, 1979). In this case one of the elementary reactions is considered to control the rate. Frequently, with such simplifications, rate equations cannot reflect essential features of the reaction mechanism. This is also due to the fact that ideal adsorbed layers are considered and the nonuniform character of the real + Current address: Laboratory of In4ustrial Chemistry,Ab0 Akademi, Biskopsgatan 8, FIN-20500, Abo, Finland. Telephone +358 21 654 431. FAX: +358 21 654 479. E-mail:
[email protected] address: Department of Chemical Kinetics and Catalysis, Karpov Institute of Physical Chemistry, Obukha 10, Moscow, 103064, Russia.
catalytic surface (either the existence of energetically different sites or lateral interactions) is not taken into account. One of the simplest physical models that is used in catalytic kinetics is the model of an ideal adsorbed layer. In this case all surface sites are considered as identical and the interactions between adsorbed particles are neglected. Such a surface was defined as uniform. However, this model of an ideal adsorbed layer disagrees with a number of experimental data. For example, the differential heat of adsorption is not constant as a rule but decreases with surface coverage (Wedler, 1976) and the rate of adsorption and adsorption equilibrium are not described by the Langmuir equation or the Langmuir isotherm correspondingly (Temkin, 1979). In several cases the kinetics of catalytic reactions disagrees with the equations obtained on the basis of the model of ideal adsorbed layers. For example, the kinetics of ammonia synthesis obeys the well-known TemkinPyzhev equation, and the kinetics of some hydrogenation, oxidation, etc. reactions were explained in terms of a nonuniform surface (Temkin, 1979; Kiperman, 1979; Kiperman et al., 1989). In the present paper the treatment of the adsorption and kinetics of some heterogeneous catalytic reactions from the viewpoint of real adsorbed layers will be given and several general problems of catalysis, such as structure insensitivity and volcano-shaped curves will be addressed. Examples of the rate computation in different cases will be also presented. Models of Real Adsorbed Layers. Two different assumptions are generally used for the description of the physical chemistry of real adsorbed layers: either surface sites are different o r there is a mutual influence of adsorbed species. The first case is defined as a biographical nonuniformity and the second one as an induced nonuniformity (Burwell, 1977; Temkin, 1979). On biographically nonuniform surfaces a certain distribution of properties is considered. Such nonuniformity can be either chaotic, when the adsorption energy on a given site is independent of the neighbor site, or discrete. However, if in an elementary surface reaction only one adsorbed particle is involved, the difference in these distributions cannot be observed (Temkin, 1972). Different functions determining the fraction of the total number of surface sites characterized by the definite adsorption energy of a given substance were proposed,
0888-5885/95/2634-1208$09.QOIQ 0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995 1209 and one, which accounts for the Temkin adsorption isotherm, corresponds to an evenly nonuniform surface. However, there are no physical reasons for such a distribution, and not only on polycrystalline surfaces but also on single-crystal faces the isosteric heat of adsorption depends on the coverage (Boudart 1986; Somorjai and Carazza, 19861, thus indicating nonuniform behavior of adsorbed species as a function of coverage. In order to describe such data, another type of model can be used, namely, induced nonuniformity. Frequently, what is defined (Burwell, 1977; Temkin, 1979) as induced nonuniformity is discussed in the literature as a uniform surface with variation of the binding energies of adsorbed particles with coverage due t o adsorbate-adsorbate interactions. Effects of interactions between adsorbed particles were addressed in several studies. Chemisorption and reaction kinetics (Benziger and Madix, 1979; Lombardo and Bell, 1991; Shachovskaya et al., 1970, Snagovskii, 1978; Zhdanov, 1991) as well as some engineering aspects of lateral interactions , such as selectivity patterns (Bhat et al., 19841, the effect of interaction on fixed-bed reactor performance (Doraiswamyand Prasad, 19871, and catalyst effectiveness (Shendye et al., 19931, were discussed. It is also interesting to note that surface heterogeneity can be also responsible for such an interesting phenomenon as concentration oscillations (Myshlyavtsevet al., 1992; Pikios and Luss, 1977; Schwartz and Schmidt, 1987). In these studies different models of interaction were applied. The simplest representation of adsorbateadsorbate-adsorbent interactions is the Fowler-Guggenheim (1939) isotherm. This isotherm differs from the Langmuir one in the exponential term, accounting for the interactions. A model of that type was implied for the cases of one-component adsorption (Bhat et al., 1984; Doraiswamy and Prasad, 1987; Shendye et al., 1993). Although the model was used in several studies, it is claimed (Shendyeet al., 1993)that the physical meaning of the interaction parameter still remains unclear. Effects of adsorbate-adsorbate interactions were also described using the more refined lattice gas model (Lombardo and Bell, 1991; Zhdanov, 1981,1991). The relationships between the rate of an elementary reaction and the coverage are complex and cannot be written in a closed form when this model is used. In the model each adsorbate is assumed to be localized on a twodimensional array of surface sites and the site is assumed to be either vacant or occupied by a single adsorbate. A given adsorbate can interact with another adsorbate on nearest-neighbor sites, next nearestneighbor sites, etc. (Lombardo and Bell, 1991). One of the most frequently used postulates in the model is the quasi-chemical approximation which assumes that the adsorbates maintain an equilibrium distribution on the surface. The lattice gas model with this approximation was used for the description of TPD spectra as well as in some instances for reaction of gases on metal surfaces (Lombardo and Bell, 1991; Zhdanov, 1991). Among the studied reactions was the steady state oxidation of CO over Ir, hydrogen over Pt, and the kinetics of the CONO and CO-02 reactions over Rh (Lombardo and Bell, 1991; Zhdanov, 1991). The lattice gas model was also applied for the investigation of adsorbate islanding (Silverberg et al., 1985), selectivity, catalyst effectiveness, and space times in the case of a simple reaction
scheme (Bhat et al., 1984; Doraiswamy and Prasad, 1987; Shendye et al., 1993). Unfortunately the cumbersome character of the lattice gas model reduces its application t o the kinetics of concrete catalytic reactions, especially when the reaction is a complex one. It is, however, interesting to note that regressing the parameters in both models of adsorption (Fowler-Guggenhem and lattice gas) can lead t o an equal description (Shendye et al., 1993). Among the other types of models accounting for lateral interaction, a surface electronic gas model was proposed (Shakhovskaya et al., 1970; Temkin, 1972). The model was advanced in order to describe the inhomogeneity of the catalyst surface, and it explains mutual interactions between adsorbed particles and their interaction with the catalyst determined by changes in the position of the Fermi level. The model is based on the assumption that a complete or partial ionization of the adsorbed particles takes place during chemisorption, with electrons being transferred to the surface layer. It means that chemisorbed particles feed their electrons into the surface layer of the solid or take electrons from it, forming at the surface a kind of twodimensional electronic gas. The changes in the electron concentration in the solid proceed only in the subsurface layer. The model can be used only when surface coverage is not small. A peculiar characteristic of the model is that the energy of the adsorbed layer is determined only by the total number of adsorbed particles and does not depend on their arrangement (Temkin, 1972). Another essential feature of the model is the statement that electrons in the subsurface layer can be treated as isolated from other metal electrons, as if they are put in a narrow and deep bath. The form of the model of the surface electronic gas provides possibilities for its application to chemisorption of gas mixtures and thus to modeling of kinetics of complex reactions. Therefore in the present paper some aspects of adsorption and catalysis over induced nonuniform surfaces will be treated using this physically reasonable model.
Adsorption on Induced Nonuniform Surfaces As it was shown in work by Temkin (1972) a simple calculation in the spirit of Sommerfeld's theory of metals, but for the two-dimensional case, leads t o the following equation: €=€
o
-7-2 h2L 4nm*
where 7 is the effective charge acquired by an adsorbed particle, 6 is the degree of covering, h is Planck's constant, L is the number of adsorption sites on the unit surface, E is the heat of adsorption, and m* is the effective electron mass. For the adsorption isotherm of gas A it was obtained (Temkin, 1972)
ln(aP)=
h2L
2 4nm*kT0
+ ln-1 -e 0
where a is the adsorption coefficient, P is the equilibrium pressure of gas A, k is the Boltzmann constant, and T is the absolute temperature. We point out that 7 can be either positive or negative. In this simplest case eq 2 corresponds to the Fowler-Guggenheim isotherm.
1210 Ind. Eng. Chem. Res., Vol. 34,No. 4,1995
We will use the surface electronic gas model in the description of adsorption of gas mixtures on nonuniform surfaces. Let us assume that on the surface exists Ni adsorbed particles A, with the effective charge qi and N, particles of Aj with the effective charge qj , where i * j . First it can be assumed that q is not a function of the degree of covering. The total quantity of electrons in a subsurface layer Ne can be expressed as follows
in the following way:
The site balance is given as J
eo=i-ei-
C ej
(11)
j=lj#i
From eqs 9-11 and Hca/R = (eo - c)/kT we obtain the adsorption isotherm for adsorption of component i of a gas mixture where Neois the value of Neo at 8 = 0. The value of the maximum surface kinetic energy E , is therefore UiP,=
1
"
i-eiwhere s is the surface area. When an additional number of particles aNi is adsorbed, additional electrons vi N i will be in the surface layer, they will change the surface kinetic energy in comparison with the adsorption on the clean surface, and the adsorption energy will be
n (vivjejc~~ J
ei
exp(vi2eiC/T)
j=lj#i
Cej
where C = C'lk. Frequently, adsorption occurs with dissociation according to the scheme
where 2 is the surface site and /3 is the stoichiometric coefficient. In that instance instead of eq 9 we have
+
where go is the value of E a t 8 = 0. The sign corresponds to the case of a repulsive interaction and the - sign for an attractive one. Then the differential heat of adsorption can be expressed by the equation
Since 8i = Ni lLs and considering
c'= vQh2/4nm*
(7)
where Y Q can be either +1 (repulsive interactions) or -1 (attractive) we obtain
j=lj#i
The thermodynamic approach (Adamson, 1976; Clark, 1970; Snagovskii and Ostrovskii, 1976) results in
+
In Pa = (H",/RT) - Sca/R (H6,,lRT)- Sea,lR
(9)
where Hca,is the configurational enthalpy and S a is the configurational entropy of the adsorbed layer. According to Temkin (1972b) s",coincides with that for the simple adsorption, thus
eOeist eieost
S', = R In -
+
In Pa = (H",/RT)- SCa/R L
From (14) an adsorption isotherm for the case of gas mixtures can be easily obtained. Another case that is worth considering is the adsorption of polyatomic compounds when an adsorbate occupies several sites on the surface. A general approach to deriving these types of isotherms for ideal surfaces was discussed by Boudart and Djega-Marriadassou (19841, and this subject was treated in detail by Snagovskii (1972) by means of statistical mechanics. To avoid complication below we will consider only the case of one-componentadsorption. The adsorption isotherm for an ideal adsorbed layer for substances whose molecules occupy more than one elementary site on the adsorbent surface is given by the following equation (Snagovskii , 1972):
where p is the multiplicity of the system, n is the screening parameter, and 3t is the number of elementary sites that one molecule occupies. The parameter n depends on coverage, and an explicit expression for the variation of the screening parameter with coverage for different cases was proposed by Snagovskii and Ostrovskii (1976). Equation 15 corresponds to the following equation for the chemical potential in the adsorbed phase:
(10)
(16)
where Oist is the standard coverage, which can be defined
where fa is the partition function for the adsorbed layer.
Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995 1211 The chemical potential of the molecules in the gas phase (ug)is given (Snagovskii and Ostrovskii, 1976) as
Equation 12 can be given in the form
ai = O,fI/O$,
(24)
where where c is the concentration and ea is expressed according to eq 8, but for the one-component case:
n J
fI
= exp(vi20iC/Z7
(vivjOjC/T)
(25)
j=lj#i €,=E
0
2 -voac'
(18)
The equilibrium conditions for adsorption are equality of pg and pa. Therefore the isotherm takes the form
In the above treatment the effective charge of the adsorbed species was supposed to be constant (e.g., independent of coverage), thus the differential heat of adsorption linearly decreases (for repulsive interactions) with coverage, which corresponds to the logarithmic adsorption isotherm. Generally speaking the effective charge depends on coverage (Temkin, 1972a)than other types of adsorption isotherms can be obtained (Murzin, 1992).
Equation 24 corresponds to the law of surface action for the equilibrium Z+I=ZI
(26)
where fI is the activity coefficient of substrate I in the adsorbed condition. According to Temkin (1972a) the configurational enthalpy of the transition state in the case of onecomponent adsorption is
H", = v*qoc'
(27)
where v* is the effective charge of the transition state. Since Hctcan be expressed as follows J
wt,= (v*ivjeic')+j = l j # i (v*ivjojc')
(28)
we obtain
fi'
Equation 12 can be used for the determination of surface coverage in the case of catalytic reactions when the interaction effects of chemisorbed particles can be described in terms of the surface electronic gas model. Such a procedure was done (Snagovskii et al., 1978) in assuming that the surface coverage of adsorbed particles (except one) can be neglected. We will start from the case of the equilibrium replacement adsorption ZB + A = ZA
+B
n J
Kinetics of Elementary Surface Reactions
(20)
where Z is a surface site. Since
= exp(vSiviOiC/Z7
(r]*ivjB,C/T) (29)
j=1 j # i
In the case of the elementary reaction
A + ZI + Z
-t
S
(30)
where ZI are sites with adsorbed particles of substrate I , the reaction rate is expressed
(31) where fi and f ' can be obtained from eqs 25 and 29.
Kinetics of Overall Reactions
Z+A=ZA
(21)
Z+B=ZB
(22)
Using eq 12 we obtain for the equilibrium constant (20)It follows from eq 23 that when adsorbed particles A
and B have the same effective charge the equilibrium constant will be the same as that in the case of a uniform surface.
Two-step Sequence. 1. Rate Expression. In heterogeneous catalysis quite often we face the situation where among the reaction intermediates one is the most abundant and all others are present a t the surface at much inferior concentration levels, and the two-step sequence which accounts for these cases will be discussed below. It was demonstrated (Boudart, 1972;Kiperman, 1979; Kiperman et al., 1989; Temkin, 1979)that in many cases the reaction mechanism can be viewed with this twostep sequence model, proposed first by Temkin and discussed in detail by Boudart (1972): step 1:
A+Z=ZI+F
step 2:
B
result:
A+B=F+D
+ ZI = Z + D (32)
Where A, B, F, and D are reactants and I is an adsorbed
1212 Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995
intermediate. This model was applied for the description of reaction kinetics in various cases (Boudart and Qega-Mariadassou,1984; Kiperman et al., 1989 Temkin, 1979). For the simplification in the following treatment we will assume that the effective charges of the transition states are equal and are in proportion with the effective charge of particle I. For a stationary reaction with the two-step sequence it was demonstrated (Murzin, 1992) that
The constant a is the Polanyi parameter (Ishikawa, 1988). The reaction rate is expressed as
For the region of medium coverage it is usually claimed that 81/(l - 81) = 1. Then we obtain from eq 33
of thermodynamic characteristics follows a curve with a maximum. Relation 37 corresponds to a linear relationship between the Gibbs activation energy and the standard change of the Gibbs energy. If the standard entropy of each kind of adsorbed species and activated complex in the case of nonuniformly reactive adsorbates does not differ substantially, this relation is equivalent to the linear dependence between the variation of the internal energy change and the activation energy. It was demonstrated (Temkin, 1957) that for an optimum catalyst an optimum surface coverage exists. Temkin considered (1957) the model, described by eq 32, and the surface of the catalyst was assumed to be uniform. The derivation was also simplified by the assumption that values of a are equal for both steps of eq 32 and that a = 1/2. It was shown (Temkin, 1957) that under the conditions when the overall rate is a maximum the optimum surface coverage is equal to 1/2. More recently (Temkin, 1984) the two-step reaction on uniform surface was considered without any simplification. An explicit expression was obtained for the case when values of a are equal for both steps; the surface coverage 8 on the optimum catalyst is given by the following equation (Temkin, 1984):
8=a 8, = T In U/y2C
(35)
+
where u = (k1PA +k-&d/ (k-1PF k2PB). From eqs 33 and 35 and with an approximation for the region of medium coverage 81 G 1 - 81
(38)
The rate of the elementary reactions of step 1 in the mechanism in eq 32 can be written (Murzin, 1992) as
and
Equation 36 is a form of the well-known equation for the reaction rate in the case of a two-stage reaction on biographically nonuniform surfaces (Temkin, 1979), which was widely used for the description of kinetics of heterogeneous catalytic reactions. 2. Optimum Catalyst. The relationship between thermodynamics and kinetics in chemical reactions is usually expressed by the Brprnsted equation
k=gP
where PAis the partial pressure of A, a is the Polanyi parameter, and A is a constant. This constant accounts for the degree of nonuniformity and it holds that
(41) The constants of the elementary reactions, k-1 and kl, are connected with the equilibrium constant K1 by
(37)
where k is the rate constant, K is the equilibrium constant of the elementary stage, g and a (Polanyi parameter) are values constant for the series of reactions compared. Introduced first for homogeneous acidbase catalytic reactions by Brprnsted, it was extended to homogeneous reactions by Evans and Polanyi, gasphase chain reactions, as well as heterogeneouscatalytic reactions (Golodets, 1982; Ishikawa, 1988). The constants g and a are determined by parameters characterizing the elementary mechanism (composition and structure of activated complexes and so on) thus allowing the existence of an optimum catalyst, on which the rate of the catalytic reaction per unit of surface has maximum value. A n equation similar to 37 was used (Golodets, 1982) for the explanation of Balandin "volcano curves", when the catalytic activity as a function
The constants of the elementary reaction, kl and K - 1 , which correspond to the case of uniform surface can be expressed as k, =glKlal
(43)
and (44) Let us consider the case when values of the Polanyi parameter are equal for both steps: a1 = a 2 = a . In
Ind. Eng. Chem. Res., Vol. 34, No. 4, 1995 1213 ficients corresponding to MSI and to non-MSI conditions obeys
the steady state the reaction rate is given by
r = r1 = r-l = klPA(l - 0 ) exp[-d8] - k-,PF8 x exp[(l - a)M] (45)
as/a = exp[-AdkT
(51)
When the decrease of the initial heat of hydrogen adsorption on I " i 0 z is 3.5 kcaVmol even if q * = q we times lower than a. have that as is 3 x In the case of an elementary reaction, the activity ~ ) the activity coefficient coefficient of substance A ( f ~ and of the transition state Cp") under MSI conditions can be expressed as
We have therefore
where
(47) Setting arlae = 0 the condition for the maximum overall rate can be determined:
f" = exp[-AdkT
and therefore
O=a
(38a)
It is worth noting that the value of the optimum coverage is exactly the same both on a uniform surface (Temkin, 1984) and on a surface where there is a mutual interaction between adsorbed species. 3. Metal-Support Interaction. The problem of the metal-support interaction (MSI) in heterogeneous catalysis was a subject of considerable interest in the recent years (Haller and Resasco, 1989). Strong metalsupport interactions could occur between group YIII metals and reducible oxides. In such a case the catalysts are hydrogen activated at sufficiently high temperatures, which leads t o catalytic enhancement although the chemisorption of hydrogen and CO could be suppressed (Vannice, 1992). Although different aspects of the metal-support interaction were investigated mainly from the viewpoint of catalyst characterization, kinetic models, accounting for metal-support interactions, are still lacking. Let us now consider the possible influence of the support within the framework of the surface electronic gas model. We can assume that in this instance, either due to an electron transfer from the support to the metal (mainly for the smallest particles) or due to the shielding of the metal by the particles of the support, the initial heat of adsorption (eo)sas well as the effective charge of the adsorbed particles qs can be changed. This assumption is in good correspondence with the experimental data, as the decrease of the initial heat of adsorption of hydrogen on Pt/TiOz was reported (Hermann, 1989) to be 3.5 kcal/mol. Then we have for adsorption of gas A = E - A€
+ ($)2esC'
(49)
where Ae is the difference between (EO)" and eo. The adsorption isotherm under MSI conditions is expressed then as
+ (qs)28sC/T+ In e / ( l - 0 )
(50)
where usis the adsorption coefficient under MSI conditions. Now we have that the ratio of adsorption coef-
+ CO(~s)2/rl
(52)
+ Ce(f)(q*s)/rl
(53)
and
ff" = exp[-aAdkT
ln(asP) = -A€
+ Ce(($)2 - #)/TI
In the instance of a metal-support interaction we will take that ( v * ) ~= a(vs);therefore, for a stationary reaction
The reaction rate rs is expressed then as
rs = r1 - r-l = klPA(l - e,) exp[aAdkT a($)'Ce,/rl - k-lPFeIexp[(l - a)(~s)'Ce,/rl (55) For the region of medium coverage we have from eqs 54 and 55, that
From eqs 36 and 56 the relative activity under MSI conditions is expressed as
(57) When -In U E aAe/kT the reaction rate under MSI conditions will be negligibly small. However, in some cases (for example if qs
