Ind. Eng. Chem. Res. 1992, 31, 2257-2264
2257
KINETICS AND CATALYSIS Adsorbate Interactions and Site Exclusion in Catalytic Rates and Reactor Design+ Asha S. Datar and Sudarsan D. Prasad* Physical Chemistry Division, National Chemical Laboratory, Pune 411 008, India
The role of adsorbate interactions is clearly reflected in a model reaction, the activated complex of which needs one occupied and one vacant site as nearest neighbors. The maximum surface rate is observed at half surface coverage both for LangmukHinshelwood (LH) and Bragg-Williams (BW) models. For the quasi-chemical approximation (QC) model this value is enhanced and retarded by repulsive and attractive forces, respectively. The P" a t which the maximum rate occurs is identical for BW and QC models. Also, the corresponding space-time values attain minimum values a t critical inlet pressure. The key assumption involved in the analysis is that adsorbate interactions shift the energy levels of reactants and activated complex to the same extent. When this is not true, the surface rates can be corrected by invoking a colony of six sites. Repulsive forces enhance and attractive forces retard the rates, respectively. The rate curves also display pronounced asymmetry.
Introduction The role of adsorbate interactions in causing surface phase transformations and the formation of ordered structures has been the subject of extensive theoretical and experimental investigations (Ertl, 1983, 1985; Woodruff et al., 1983). However, there have been only a few studies that analyze their primal role in determining catalytic rates and selectivity (Bhat et al., 1985; Zhdanov, 1981; Silverberg et al., 1985; Gland et al., 1982; Silverberg and Benshaul, 1987). Much of this has been due to the lack of finegrained experimental data or a suitable model reaction system, wherein these effects can be unequivocally demonstrated. The present study is an attempt in that direction. The role of adsorbate interactions can be best demonstrated by enumerating the probability that a pair of nearest-neighbor sites are occupied by molecules. Attractive interactions enhance this probability, and repulsive forces act in a detrimental manner by strongly excluding a neighboring site from taking part in adsorption. What would be the expected behavior of a catalytic reaction in which the activated complex needs a pair of sites, only one of which is occupied? We hope to demonstrate that interaction effects are clearly transparent in such a simple reaction model. We tried hard to find a real system to which our model is applicable. There has been a mention in the literature that a model very similar to ours has been used for describing CO oxidation in a very narrow range of partial pressure (McCarthyet al., 1975; Sheintuch and Schmitz, 1977; Gea and Ibanez, 1981; Engel and Ertl, 1978). However, it is to be borne in mind that CO oxidation is a LangmuirHinshelwood (LH) reaction between adsorbed CO molecules and oxygen atoms. Even in this complications can arise due to oxygen islanding which results in nonclassical rate laws (Silverberg et al., 1985; Gland et al., t NCL
Communication No. 4933.
1982; Silverberg and Benshaul, 1987). Thus, even though extensive data are available, overall applicability of the model is open to question. Ethanol dehydrogenation (Frankerts and Froment, 1964) to acetaldehyde at low conversion levels (initial rate data) can be described by a kinetic model analogous to our rate expression. However, it is best to view this model as a hypothetical reaction system 80 as to make the theoretical analysis tractable and to focus on interaction effects in dynamics. Finally, how adsorbate interactions affect the choice of optimal operating variables (such as pressure) and of course the main concern of the design engineer, i.e., the space time needed to effect a given conversion, is to be analyzed. The paper is arranged as follows: First we discuss the adsorption models (in increasing order of sophistication) to deal with localized adsorption with interactions. Next we consider the methods of enumerating pair probabilities, especially the pair probability (reaction rate) which goes through a maximum as the partial pressure is increased. Finally a simple application to a plug flow reactor model is to be explored especially to understand the qualitative differences between the operation of the repulsive and attractive forces in determining space-time values. The analysis is carried out for a square lattice, with coordination number z = 4. This can be a (100) plane of a fcc crystal. Pll,Plo,Poodenote probabilities of a pair of nearest-neighbor sites with 2,1, and 0 adsorbed molecules, respectively, as shown in Figure la. Two models are employed, viz., the Bragg-Williams lattice gas model (BW)and the quasi-chemical approximation model (QC).In the former, random distribution of adsorbed molecules is assumed, while in the latter molecular pairs are thought to be randomly distributed. Each of these is considered in turn. The Bragg-Williams Lattice Gas Model (Fowler and Guggenheim, 1939; Hill, 1956). Since the molecules are randomly distributed, all pair probabilities can be constructed from the probabilities of an occupied site (6)
0888-5885f 92f 2631-2257$03.00f 0 0 1992 American Chemical Society
2268 Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992 0
0
0
0
0
0.0.0 0
.
0
.
.
.
0
.
0
0
a )
o-o
b)
M
p, I
0
0 0 0 . 0 C )
0
0 Filled Site 0 Vacant Site Figure 1. (a, top) Adsorbed molecules on a square lattice and three types of pairs. (b,bottom) Colony of three sitea adjacent to a PI,,pair
EO
considered for activation energy computations.
and a vacant site (1 - e) through simple multiplicative formulas. pll = e 2
(1)
P,, = 2 q 1 - e)
(2)
poO = (1- e12
(3)
They also sum to unity, as they should: P11 PI0 + Po0 = 1
+
(4)
Finally the equilibrium partial pressure P is related to the surface coverage, by means of an adsorption isotherm (BW isotherm)
where bo, Q, and w represent the ratio of adsorbate to gas-phase partition functions, the heat of adsorption, and the interaction energy between a pair of nearest neighbors. The sign convention followed in the present work is that w > 0 represent repulsive forces; obviously w < 0 means attractive forces. The Quasi-ChemicalApproximationModel (Fowler and Guggenheii, 1939; Hill,1986). A further refinement of interaction effects is achieved in the quasi-chemical approximation model (QC) in which random distribution of all molecular pairs of the type 1 1 , 10, and 00 is assumed. ks the name implies, the pair probabilities are also related by a quasi-chemical equilibrium relationship (Fowler and Guggenheim, 1939; Hill, 1956): p 11p00 --
- -1 exp(
g)
so2
In addition, the following mass conservation relations hold good between the surface coverage 0 and the pair probabilities: 2pll+ pl0= 2e (7) 2p00 + pl0= 2(1-
e)
(8)
\\\
b.
.
+
Win
\
\
\, \
= w4* = w
BE
REACTION COORDINATE
4
Figure 2. (a) Energy diagram illustrating no change in the level of activated complex. (b)Energy diagram illustrating equal change in the level of activated complex as the reactants.
What remains is an isotherm expression which relates the surface coverage and equilibrium partial pressure:
0 = [I - 48(1 - B)(1
- e~p(-w/RT))]~.~
(9c) Using eqs 6-9 all the quantities of interest can be computed in a straightforward manner.
Shift of Activation Energy Due to Adsorbate Interactions The effect of adsorbate interactions on surface dynamica can be best understood with reference to Figure lb. The immediate neighborhood of a 10 pair is mapped therein. The 10 pair can become an activated complex for the surface reaction. In principle this can also become the activated complex for desorption, but since desorption is already modeled in the derivation of adsorption isotherms (which can be done through statisticalmechanical methods independent of any kinetic arguments), it is not considered further. It is further assumed that since surface reaction is rate controlling the activated complex for desorption will be placed at a different energy level in comparison to reaction. Once again, reference to Figure l b shows that the occupation states of the sites marked by 1,2, and 3 (three in number for a square lattice) figure in the overall ener-
Ind. Eng. Chem. Res., Vol. 31, No. 10,1992 2289 getics. When an adsorbed A molecule in the 10 pair becomes an activated complex, if n denotes the number of occupied sites, nuM interactions become nWA*A interactions in the excited state. Figure 2 lists the two possibilities. In one case the reactant level shifts due to adsorbate interactions, repulsive and attractive forces pushing it upward and downward respectively on an energy level diagram (Figure 2a). On the other hand, the activated complex is unaffected by adsorbate interactions as WA*A = 0 (Zhdanov, 1981; Silverberg et d., 1987) and hence activation energy is decreased by repulsive forces and enhanced by attractive forces. In other words the activated complex is an extremely loose molecule in which every bond is appreciably stretched. The second case corresponds to equal shift of the activated complex and reactant levels due to adsorbate interactions ( F i i e 2b). The consequence of this is analyzed first in the paper. Equal Energy Level Shift of Activated Complex and Reactants As already mentioned, WA*A = o ~so ,in effect interaction effects have virtually no effect on the activation energy Eo. This in turn means that the adsorbate interactions have predominantlydispersion terms, which are pairwise additive and which in turn can be broken in to a series of bond sum8 (Baker and Everett, 1962; Sinanoglu and Pitzer, 1960; McLachfan, 1964). Also, if the assumption of minimal bond breaking is made and that too only involving the reaction coordinate, it is conceivable that wA*A = w u . The real situation may be intermediate between wA*A = 0 and WA*A = w u , which can also be taken care of by defining a switching parameter u, 0 < u < 1. In our work u = 1 throughout. The basic equation for surface rate is rl0 = ~ l o ~ ~ ~ ~ ~ - ~ ~ / R T exp(-W/RT) ) l P l o C P l o(10) ,~ I
rl0, klo, E,, and hEi denote the surface rate, rate constant (which includes the nonconfigurational part of the ratio of adsorbed- to gas-phase partition functions), activation energy for zero coverage (no interaction effede), and level shift contribution to activation energy. More detailed considerations are given in D a h and Prasad (1991). The sum denotes the sum over all the configurations, which is equal to 23 = 8. When AEi = 0, eq 10 reduces to a simple rate formula which is proportional to the pair probability p10:
= kl0 exP(-Eo/RT)Plo (11) Evaluation of Plois trivial for the BW model. A little more algebra is involved for the QC model in computing Plo using eqs 6-9. At first it may be thought that the removal of 10 pair is analogous to desorption. However, the activated complex for desorption is different from that of reaction. Besides occupation states of four sites in the immediate neighborhood of an adsorbed molecule matter, in all there are 24 = 16 coni5gurationsin contrast to 23for the reaction. If we invoke the assumption WA*A = wAA for desorption, then the desorption rate will be proportional to surface couerage (instead of Pl0)and hence show a monotonic increase, instead of nonmonotonic behavior observed in the latter case. It is to be noted that in both cases the summation term in the right-hand side of eq 10 is unity. r10
Results and Discussion On the basis of what has been said above, the behavior of the surface rate can also be understood by looking at
1,
1
t Figure 3. Plot of Plova 6 for different interaction energiea for BW, LH, and QC models. Q = 17.0 kcal mol-'; bo = 1 X 108 Torr.
Ploexcept for a scale factor. One feature that is immediately obvious is that PI, is symmetric with respect to B irrespective of whether the Langmuir, BW, or QC model is employed, and hence all display a maximum at B = 0.5. This is evident from Figure 3, wherein all the pair probabilities are plotted as a function of B for both attractive and repulsive interactions. For the LH and BW models, since both m u m e random distribution of adsorbed molecules, the maximum value of Plois 0.5 (in scaled unita) as is obvious from eq 2. This is true for both attractive and repulsive interactions. In sharp contrast to this,the maximum value of the Plo depends on the value of the interaction energy parameter for the QC model. Thus o>o w < o
> 0.5 Pi0 < 0.5 Pi0
for the QC model. Also w > 0,0 >> 1: PI0 = 1 (12) and w < 0, 1 0 1 >> 1: (13) P,, z 0 Thus when refined treatment of interactions is incorporated, as in the QC model, the maximum value of the surface rate can be enhanced for repulsive interactions and is almost twice that predicted (see eq 12) by the mean field theory (BW model). The behavior of the Plowhen attractive interactions is present is diametrically opposite to what has been discussed above. Once again the BW model predicts a maximum value of 0.5, which is same as that of the Langmuir model. If it is desired to get a maximum rate for the rl0 (see eq ll),it is obvious that the strong site exclusion (in the sense that an occupied site excludes ita neighboring sites from taking part in adsorption), when repulsive interactions predominate, favors the formation of P,, Thus the probability of an occupied site surrounded by z vacant neighbors reaches its peak value for repulsive interactions. On the other hand, when attractive interactions are operative, molecules tend to cluster together to form islands (Gland et al., 1982; Silverberg and Benshaul, 1987). If the islands have no holes (vacant sites) in them, it implies that the probability of any vacant site occurring adjacent to a filled site is almost zero, and Plo> predicts 0. Refinements are thus important.
eo"
eo"
e
A pseudohomogeneous rate model is assumed. Adsorption equilibrium is thought to prevail between gas and surface phases and surface reactions are much slower. T ~ xo, xe, and Plodenote the space time, inlet and exit conversion levels, and the pair probability for a half-occupied pair of sites, respectively. As is customary for a PFR, the conversion in a volume element x , the preasure (concentration)at any point P, and the inlet pressure Po are related by P = Po(1 - x ) The surface coverages and pair probabilities can be readily computed using eqs 1-9. For the computation of surface coverage as a function of pressure in eqs 5 and 9 at every point, a library package involving Muller's iteration (DRTMI, PDP-11) was used (Doraiswamy and Prasad, 1987). The space time computation in eq 14 is most conveniently carried out using an eight-point Gauss-Legendre quadrature by the use of following substitutions [for details see Doraiswamy and Prasad (1987)l: zi = + x0)/2 + [ ( x e - ~0)/214i (15) i=8
710
=
- x0)/2 CAi/PlO (zi,4i) i=l
(16)
As before, 4i denotes the root of the Legendre polynomial and Ai the coefficients of the Gauss-Legendre procedure. Thus the computation of T~~ involves the evaluation of eight surface coverages and the associated pair probabilities. Use of a higher order Gauss-Legendre
~
,
Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992 2261 Table I. Space-Time Values and Models BW model W
:i-.- /
IO? 86
7 0
W.I.0
P
IF?
for BW and QC
QC model
P
e
-0.25 -0.5 -0.75 -1.0
Attractive Interaction Forces 5.3934 2.0919 5.4042 3.2163 2.3724 3.2510 1.9033 3.0440 1.9407 1.1985 4.5732 1.1557
2.2045 2.5708 3.2008 4.3830
0.25 1.0 2.0 3.0
Repulsive Interaction Forces 14.6926 1.9065 14.7032 65.2365 1.8404 65.6936 479.0057 1.8176 479.9794 3529.7220 1.8098 3534.859
1.7987 1.4787 1.2414 1.1050
I
----
-
4-
1' / ,
2-
./'
----- _ ---/._______ ---_ _ _ _
OC MODEL(REWLSIVE ) 01
0
1 ~ ~ * - - ~5 11 % , 1 1 1 * 1
0.1
I
10
' ' 1 1 8 1 1 1 1
IO'
8
~~~~~~~1
0
I0 '
*
~
~
1' 0
'~
~~ "
~ '
L ~
Id
y
1
SW MODEL (ATTRACTIVE 1
0
0.1
I
I
I
10
I
d
5 Fmre 9. Variation of T~~ with Po for attractive interaction energiea for BW model.
Once again eighbpoint Gam-Lengendre quadrature @
to be invoked for evaluation of eqs 17-19 to find the
e".
The resulting nonlinear eqs 17-19 have been solved by the library package DRTMI for Po. The space-time values and are presented in Table I. For repulsive interactions the T~~ decreases monotonically with increasing w . The decrease is more noticeable in the case of the QC model in comparison to the BW model. At lower values of interaction energy parameter o,the divergences between these models are not significant. The maximum difference between BW and QC model predictions occur at the highest value of the interaction parameter chosen in the present study, namely, w = 3, where it almost amounts to a deviation of 60%. Thus if we were to choose the more crude BW model, we would be overestimating the space time (weight of catalyst needed) by the same extent. It is to be noted that the divergences between the model predictions increase monotonically with w , but for a typical temperature of 500 K (RT= 1kcal/mol) w = 3 kcal/mol represents a maximum upper bound on interaction energy parameter (Silverberg et al., 1985; Silverberg and Benshaul, 1987; Datar et al., 1989; Einstein and Schrieffer, 1973; Binder, 1976) and this denotes the largest improvement on space-time value predictions by a higher degree of refinement. As in the case of Ploplota mentioned earlier, the region of pressure over which the appreciable variation of T~~ occurs gets shrunk by several orders of magnitude when we consider attractive interactions. Figures 9 and 10 illustrate the behaviour for the BW and QC models. $I sharp contrast to the operation of repulsive forces, the increases as the magnitude of IwI increases for attractive forces. This directly follows from the smaller valuea of Plo when attractive forces are present. However, the differences between the QC and BW model predictions are not as appreciable as in the case of repulsive interactions, the
e eh
L
[ $IQC -lF(i =
- 2B)F3[(1- ,8)2P]-1 dx = 0
e
2262 Ind. Eng. Chem. Res., Vol. 31, No. 10,1992 - W
1.0
10.0
0
0.1
5.0
c
OC MODEL (ATTRACTIVE 1
E
I
I
1
10
L
no 5 d
Po
Figure 10. Variation of iI0with Pofor attractive interaction energies for QC model.
Table 11. Comparison of Actual Approximate P-: QC Model
e'"
w
-0.25 -0.5 -0.75 -1.0 0.25 1.0 2.0 3.0
5.4042 3.2510 1.9407 1.1557 14.7032 65.3966 479.9794 3534.859
Values with
e'"/€-
P" 2.5109 1.5229 0.9237 0.5602 6.8225 30.5899 226.030 1670.154
2.1522 2.1346 2.1010 2.0627 2.1546 2.1376 2.1231 2.1164
W
Figure 11. Plot of In models.
maximum divergence being about 10%. The critical choice of the inlet pressure on the other hand depends sharply on w. In fact, as w varies from 0.25 to 3, the varies over several orders of magnitude, suggesting an exponential dependence on F. An equation of the following type can be fitted to
ein
w
w:
In (I$'")= c1 + czw
(20)
In fact, Figure 11illustrates that this can be an accurate representation of facta for both attractive and repulsive interactions. Critical Inlet Pressure: An Approximate Formula. It was mentioned in a previous section that Ploreaches a maximum at B = 0.5 and a corresponding P" can be computed by putting B = 0.5 in eqs 5 and 9. Denoting this pressure by P"
It is interesting to note whether there is a correlation which is the critical inlet between this Paand pressure needed to achieve the A reference to Table I1 shows that the ratios are nearly constant and can be put equal to 2 to a fair degree of accuracy. This is a real shortcut to a computation of pfp'" and is independent whether the more refined QC model or the BW model is employed. Thus for practical comeq 21 is highly advantageousin providing putations of a shortcut, even though an approximate answer to the problem of hding the optimal choice for the inlet pressure Po such that the r$ is attained.
ein, e$".
Energy Level Shift of Reactants As mentioned in the beginning of the section on shift of activation energy, we analyze the case explicitly mentioned in Figure 2a. The basic assumption is that adsorbate interactions shift only the energy level of the reactants. The other assumptions involved are (1)the con-
e'" vs interaction energies for QC and BW
centration of activated complex is so small that it can be neglected in evaluating surface coverages and pair probabilities and (2) the activated complex is sort of a loose molecule such that the interaction energy of it with surrounding adsorbed molecules wA*A = 0. A number of authors have used such an assumption in their work (Zhdanov, 1981; Silverberg et al., 1985; Silverberg and Benshaul, 1987). A reference to Figure l b shows that only the three sito the left of the occupied site containing an A molecule interact with it and thus contribute to terms in the activation energy change. Using the standard treatment available (Zhdanov, 1981; Silverberg et al., 1985; Silverberg and Benshaul, 1987), it is possible to write down the conditional probability that three sites (z - 1sites) are occupied by exactly j molecules: 72-1-J
Equation 22 has a simple interpretation: given a central site as a site containing an A molecule, the conditional probability that immediate neighborhood (bond) contains another is (Pll/B).The probability that it is vacant is given by 1- (Pll/B).Since the pairs are randomly distributed, we use a simple binomial formula, with the combinational factor (j'- '1 to arrive at eq 22. The surface rate will be given by the sums of all these probabilities weighted by the interaction energy of the configuration j w , i.e.
C J-2-1
Fl0 = ilo
(: j
p l , ( ~ l l i e ~-
(1 - P1,/v-1-Jv
(23a)
1-0
where h = exp(w/RT)
(23b)
h o = klo exp(-&/RT) (234 Bragg-Williams Lattice Gas Model. A considerable simplication results if we assume that the molecules are
Ind. Eng. Chem. Res., Vol. 31, No. 10,1992 2263 randomly distributed. In the present case Pll/e = 0 and eq 23a neatly reduces to a simpler equation: = h102e(1- e)[(i - e)
(i&W
+ 0x13
(24)
As mentioned in the previous section, we should clearly identify the removal of 10 pairs with a surface reaction and not desorption. First of all, the activated complex for desorption is different from that of reaction and situated on a much lower energy level. Second, occupation states of four nearest neighbors affect the activation energy instead of three for the reaction. By proceeding along the same lines as for the reaction, we get an analogous functional relationship: rd
=
- e) + X8I4
(25)
For repulsive interactions the desorption rate shows a monotonic dependence on the surface coverage and hence once again shows a quilitatively different behavior than the reaction rate. However, for attractive interactions the desorption rate can display a maximum the location of which is given by
er= = i/[5(i - A)]
0
a -
Figure 12. Plot of log Flom 0 for attractive interactions for BW and QC models. -O 5
/
(26)
which is again different from the location of the maximum of the rate curve. The Quasi-ChemicalApproximation Model. For the quasi-chemical approximation similarly we have
-10I 0
(fio)gc = kiJ'ioP
- (%/e) +- X(p11/e)l3
(27)
Once again for comparison purposes a desorption rate expression rd
kde[l - (%/e)
+ w1l/e)i4
IO
I
--
w .-IO _..._Q C MODEL
- B W MODEL
-
01
-1 5 -
(28)
can be written which clearly shows that desorption is different from reaction events. Since the stress is not on desorption, we will not discuss it further. In the event when adsorbate interaction energy is zero X = 1 and we recover
-2 0
1 0
e -
I O
Figure 13. Plot of log fl0vs 0 for repulsive interactions for BW and QC models.
(304
ticle symmetry), but when we involve activation energy corrections, it may be only marginally asymmetric for an infinite lattice. However, for finite clusters the truncated terms (eqs 24 and 27) are asymmetric. The 15- at which Plogoes through a maximum also gets shifted appreciably. Om" > 0.5 for w > 0; 8"" < 0.5 for attractive forces. The eo= are once again appreciably different. The difference between the QC and BW models is more appreciable as far as the rate predictions are considered. However, the 19- are almost identical for the QC and BW models showing that the refinements are of secondary importance in the prediction of Om=.
Results Figures 12 and 13 summarize the kinetic consequences of the modified treatment of the surface rates. The first thing to note is the asymmetry in the rlo-B plots. This asymmetry is due to the nonzero X term, which directly depicts the role of the interaction energy parameter (see eqs 24 and 27). If we use an infinite lattice and sum over all the site configurations (in which every site has either 0 or 1 occupation states), then the average will be conveniently computed in the grand canonical ensemble. This Ploprobability will be symmetrical (due to the hole-par-
Summary The role of adsorbate interactions in dynamics (surface rates, pair probabilities, and activation energies) is analyzed for a square lattice. Two levels of approximation are employed (1) interaction shifta the energy level of activated complex and reactants equally; (2) only reactant levels are shifted by interaction. In the former case only the pair probabilities are affected, and the reaction rate displays a symmetrical plot around e = 0.5 illustrating the hole-particle symmetry. In the latter case this symmetry is destroyed, and the activation energy is considerably altered. Repulsive forces and attractive forces cause enhancement and decrease in rates in both models but to different degrees. The qualitative differences between BW
(f1O)BW
=
(f1O)QC
= k l P 1 0 = k1020(1 - e)
(29)
Equations 24 and 28 have a simple physical interpretation. When w > 0, X > 1 and the energy difference between the activated complex and the reactant adsorbed molecules is lowered, as repulsive interactions raise the average internal energy of the adsorbed molecule with consequent enhancement in the rates. An opposite trend follows for the case of attractive interactions. Finally, eq 27 may be written using eq 7 entirely in terms of PI0 as
w e = 1 - (plO/2e)
2264 Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992
and QC approximations are reflected in the rate behavior, and are appreciable to warrant attention in a reactor design strategy.
Nomenclature
Ai= coefficients of Gauss-Lengendre procedure
bo = entropy change factor, Torr klo = kinetic constant for a bimolecular reaction with an occupied site adjacent to a vacant site, mol g-' s-' klo = preexponential factor independent of adsorbate interactions, mol g-' s-' Po = inlet pressure, Torr Pm" = equilibrium pressure at which Plois maximum Plo= probability of a pair containingone adsorbed molecule and a vacant site Pll = probability of a pair containing two adsorbed molecules Poo = probability of a pair containing two vacant sites Plo,i= correction factor which takes into account the probabilities of various configurations around 10 pair Q = heat of adsorption, kcal mol-' rlo = rate for a bimolecular reaction with an occupied site adjacent to a vacant site, mol g-' 5-l ilo = same as rl0but with correction for activation energy, mol g-' 5-' R = gas constant, kcal mol-' deg-' x = conversion in a volume element zo, z, = inlet and outlet conversion levels z = coordination number for an adsorption site Zi= transformed variable defined by eq 15 Greek Letters ct = parameter defined by eq 19b
j3 = parameter defied by eq 9c B = surface coverage of adsorbed molecules X = parameter defined by eq 23b 7 = space time for bimolecular reaction, s 4i = roots of the Legendre polynomial o = adsorbate pairwise interaction energy parameter, kcal
mol-'
Literature Cited Baker, J. A.; Everett, D. H. High Temperature Adsorption and the Detenuination of the Surface Area of Solids. Trans. Faraday SOC. 1962,58,1608-1623. Bhat, Y. S.;Prasad, S. D.; Doraiswamy, L. K. Influence of Surface Heterogeneity on Selectivity with Multicomponent Chemisorption. AIChE J. 1985,31,1682-1694. Binder, K. P h e Transitions and Critical Phenomena; Domb, C., Green, M. S., Ed.; Academic Press: New York, 1976; Vol. 5A.
Datar, A. S.; Prasad, S. D. Adsorbate Clustering and Surface Rates above T,: Modified Mean-Field Approach. Chem. Phys. Lett. 1991,181,446619. Datar, A. S.;Prasad, S. D.; Doraiswamy, L. K. On the Identification of the Nature of Adsorbate Interactions. Chem. Phys. Lett. 1989, 159,337-341. Dorabwamy, L. K.; Prasad, S. D. Effect of Interaction and Mobility on Fixed Bed Reactor Performance. Chem. Eng. Commun. 1987, 58,63-68. Einstein, T. L.; Schrieffer, J. R. Indirect Interaction between Adatoms on a Tight-Binding Solid. Phys. Rev. B 1973,7,3629-3648. Engel, T.; Ertl,G. A Molecular Beam Investigation of the Catalytic Oxidation of CO on PD (111). J. Chem. Phys. 1978, 69, 1267-1281. Ertl, G. Chemical Processes on Well Defined Surfacea. In Catalysis Science and Technology; Anderson, J. R., Boudart, M., Eds.; Springer-Verlag: Berlin, 1983; Vol. 4,Chapter 3. Ertl, G. Catalysis and Surface Phase Transformations. Surf. Sci. 1986,152/153,328-337. Fowler, R. H.; Guggenheim, E. A. Statistical Thermodynamics; Cambridge University Press: Cambridge, 1939. Frankerts, J.; Froment, G. F. Kinetic Stydy of Dehydrogenation of Ethanol. Chem. Eng. Sci. 1964,19,807-818. Gea, J. R.;Ibanez, J. L. Multiplicity of Steady States in Heterogeneous Catalysis: The case of Langmuir's ntb order Kinetics. J. Chem. Phys. 1981,75, 1538-1543. Gland, J. L.; Fisher, G. B.; Kollin, E. B. The Hydrogen-Oxygen Reaction over the Pt(ll1) Surface: Transient Titration of Adsorbed Oxygen with Hydrogen. J. Catal. 1982, 77,263-218. Hill,T. L. Statistical Mechanics; McGraw-Hill: New York, 1956. McCarthy, E.; Zahrandnik, J.; Kuczynski, G. C.; Carberry, J. J. Some Unique hpecta of CO Oxidation on Supported Pt. J. Catal. 1975, 39,29-35. McLachlan, A. D. Van der Waals Forces between an Atom and a Surface. Mol. Phys. 1964,7,381-388. Sheintuch, M.; Schmitz, R.A. Oscillations in Catalytic Reactions. Catal. Rea-Sci. Eng. 1977,15,107-172. Silverberg, M.; Benshaul, A. Adsorbate Islanding in Surface Reactions: A Combined Monte Carlo Lattice Gas Approach. J. Chem. Phys. 1987,87,3178-3194. Silverberg, M.; Benehaul, A.; Rebentrost, F. On the Effecta of Adsorbate Aggregation on the Kinetics of Surface Reactions. J. Chem. Phys. 1985,83,6501-6513. Sinanoglu,0.; Pitzer, K. S. Interactions between Molecules Adaorbed On a Surface. J. Chem. Phys. 1960,32,1279-1288. Woodruff, D. P.; Wang, G. C.; Lu, T. M. Surface Structure and Order-Disorder Phenomena. In The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis; King, D. A., Woodruff, D. P., Eda.; Elsevier: New York, 1983;Chapter 4,p 310. Zhdanov, V. P. Lattice Gas Model for Desorption of the Adsorbed Molecules of Two Kinds. Surf. Sci. 1981,111,63-77.
Received for review June 19, 1992 Accepted July 7, 1992
