J. Phys. Chem. 1983, 87, 3557-3562
the hydrophobic layer surrounding the clay particle than for the aqueous solution. Cetylpyridinium may replace the CTAB on the clay surface and reside in the bilayer surrounding the particle. In other clay systems studied and previously discussed, cetylpyridinium was found to be a very efficient quencher due to its adsorption on the clay and to act by a static and dynamic mechanism. Thallium and nitromethane are more soluble in the polar, aqueous environment and do not readily quench pyrene. Thallium may be repelled from the positive charge around the double layer which also would decrease its ability to quench PN+. Correspondingly, iodide ion is attracted to the positive charge on the quaternary ammonium ions which surround the particles and comprise the outer border of the double layer. The very large value of K,," for iodide in these systems in contrast to homogeneous aqueous solution comments on the location of pyrene on these colloidal particles. Evidently, pyrene resides in the region of positive charge close to the double layer surface. In this environment, it is readily quenched by iodide ions associated with the CTAB. Conclusion Adsorption of the cationic fluorescent probe PN+ onto either colloidal montmorillonite or kaolin causes a clustering of probe molecules on the surface. This leads to pyrene excimer formation as evidenced by its characteristic emission spectrum. Fluorescence polarization measurements indicate that the molecule is rigidly bound to the surface. The PN+ can be dispersed over the surface of the mineral by the addition of quaternary ammonium surfactants which cause the emission from the pyrene monomer to increase and the excimer to decrease. The monomer is quenched by the clay mineral surface but its
3557
interaction with the surface can be significantly decreased by coadsorbing a quaternary ammonium surfactant with a hydrocarbon chain length of 10 carbons or more. Excited PN+ exhibits a double-exponential decay of its transient fluorescence, suggesting that it is located in two different regions of the particle. Quenching studies demonstrate that the surfactant cetyltrimethylammonium bromide (CTAB) reduces the accessibility of quenchers not adsorbed by the clay to the PN+ on montmorillonite but it has a negligible effect when PN+ is located on kaolin. For molecules that are adsorbed by the montmorillonite particles, the quenching rate constants are reduced by at least 1order of magnitude compared to homogeneous solution due to the limited accessibility of PN+ and the reduced diffusion of molecules on the particle surface. All of the quencher molecules used exhibited a mixture of static and dynamic quenching. The addition of CTAB to a montmorillonite colloid in an amount that is twice the cation-exchange capacity of the colloid reverse the charge on the particles due to the formation of a CTAB bilayer around their surface. The emission spectrum and steady-state quenching studies comment on the location of pyrene in this environment as well as on the nature of these colloidal particles.
Acknowledgment. We thank the Army Research Office via grant No. OAAG29-80-KO007for support of this research. Registry No. CH3NO2, 75-52-5; PN', 81341-11-9; CTAB, 57-09-0;montmorillonite, 1318-93-0;NJV-dimethylaniline,12169-7;nitrobenzene, 98-95-3;cetylpyridinium chloride, 123-03-5; hexyltrimethylammonium bromide, 2650-53-5; octyltrimethylammonium bromide, 2083-68-3; decyltrimethylammonium bromide, 2082-84-0; dodecyltrimethylammonium bromide, 1119-94-4.
Nature of Isotherms for Adsorption on Pair Sites Consisting of Adjacent, Dissimilar Atoms M. Tinkle and J. A. Dumeslc' Lbpartment of Chemical Engineering, Unlversi@' I n Final Form: March 1, 1983)
of Wisconsin, Madison, Wisconsin 53706 (Received: November 29, 1982;
Through statistical arguments, approximate isotherms for adsorption on pair sites of a two-componentsurface (e.g., acid-base pairs of a metal oxide) were derived and compared to Monte Carlo simulations at different concentrations of the two surface constituents (e.g., anions and unsaturated metal cations). These isotherms for both associative and dissociativeadsorption are non-Langmuirian. For associative adsorption, the deviation from Langmuirian behavior is caused by a decrease in the number of pair sites blocked per adsorbed molecule as the coverage increases, with the maximum deviation occurring for a surface containing equal numbers of the two constituents. For dissociative adsorption, the deviation of the adsorption isotherm from the Langmuir equation is due to the limitations imposed by the surface constituent present in smaller concentration, these becoming more important as the concentrations of the two surface constituents become more unequal. The deviations from Langmuirian behavior for both associative and dissociative adsorption on pair sites may be observed experimentally when surface coverages are measured with an uncertainty less than about 5-10% of the saturation coverage.
Introduction In modeling adsorption on uniform surfaces it is generally assumed that the number of surface sites blocked by each adsorbed molecule is a constant, independent of surface coverage. For example, associative adsorption is 0022-3654/83/2087-3557$01.50/0
assumed to involve one surface site per adsorbed molecule and dissociative adsorption is described in terms of two or more identical surface sites per molecule. The corresponding adsorption behavior is represented by well-known Langmuir isotherms. In the present paper, adsorption of 0 1983 American Chemical Society
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Tinkle and Dumesic
The Journal of Physical Chemistry, Vol. 87,No. 18, 1983
+#*lebri?
FIgum 1. Model suface Consisting of 50 anions (0)and 50 unsatwated cations (*): (a) adsorbed molecule with no neighboring adsorbed species; (b) two neighboring adsorbed molecules; (c) isolated anion; (d) anion which becomes isolated during adsorption.
molecules on pair sites consisting of dissimilar atoms on a two-component surface will be demonstrated to be non-Langmuirian. For illustration, adsorption on an oxide surface is considered. The relative numbers of surface anions and coordinatively unsaturated metal cations depend on the stoichiometryof the oxide, the structure of the surface, and the oxidation state of the surface. The latter may be dependent on the composition of the gaseous atmosphere over the sample.’P2 Adsorption of a base (e.g., NH,) may take place on a single unsaturated metal cation and adsorption of an acid may be associated with a single oxygen anion (e.g., adsorption of COz to give a monodentate surface species)?~~ Langmuir isotherms would be expected to describe these types of adsorption in the absence of surface energetic heterogeneities. In contrast, adsorption of molecules with both basic and acidic functions may require acid-base pair sites on the surface. This may include, for example, adsorption of C02 to give bidentate surface species, adsorption of CO to give surface carboxylate species, and adsorption of Bronsted acids to give surface conjugate bases and proton^.^^^ For a random distribution of anions and metal cations on the surface, the number of pair sites blocked by each adsorbed molecule will be shown below to be a function of the surface coverage. Simple statistical arguments and Monte Carlo simulations are presented to estimate the shape of the adsorption isotherm for this non-Langmuirian behavior. The majority of this paper considers the case of associative adsorption, where the two functions of the adsorbed molecule are confined to adjacent acid and base sites on the surface. It will be shown at the end of this paper, however, that adsorption is also non-Langmuirian when the two parts of the molecule dissociate and migrate to separate acid and base sites on the surface. Blocking of Pair Sites by Adsorbed Molecules Consider a square surface lattice with the lattice points occupied by a random distribution of anions and coordinatively unsaturated metal cations. This can represent in an approximate manner, for example, the surface of an irreducible oxide (e.g., A1,0,, MgO) for a given extent of (1)Kubsh, J. E.;Chen, Y.; Dumesic, J. A. J. Catal. 1981, 71, 192. (2) Kubsh, J. E.; Dumesic, J. A. AlChE J. 1982, 28, 793. (3) KnBzinger, H. Adu. Catal. 1976, 25, 184. (4) Tanabe, K. “Solid Acids and Bases”,Academic Press: New York, 1970.
surface hydr~xylation,~ or the surface of a reducible oxide (e.g., Fe304)for a given extent of surface oxidation.’Y2 The model surface shown in Figure 1is composed of an equal number of anions, “0”,and coordinatively unsaturated cations, ”*”. Lines connecting adjacent anions and unsaturated cations show the possible pair sites on this surface (Le., 0-* pairs). For a square surface lattice containing N points, there are 2N adjacent points. If one denotes 80 and 8, as the fractions of these N lattice points which are anions and unsaturated cations, respectively, then the probability that any adjacent lattice points comprise a pair site is 28& (The factor of two is included since both 0-*and *-O pairs must be counted.) For the surface of Figure 1which contains 100 lattice points with Bo and 6, equal to 0.5, the most probable number of possible pair sites, Npair,is 100, and in general Npdr= 4NB08, (1) The adsorptive properties of a collection of pair sites can be understood qualitatively by counting the number of pair sites blocked by each adsorbed molecule. Consider first the case of very low surface coverages by adsorbed species. This is shown in Figure l a for a single adsorbed molecule on the surface. This molecule not only blocks directly the pair site on which it is adsorbed, but it also blocks indirectly three other 0-*pairs from serving as sites for further adsorption. In general, for surfaces of low coverage, it can be shown that the most probable number of pair sites indirectly blocked is three, regardless of the values of 80 and 8,. Specifically, by blocking the “0” of an 0-* pair on which a molecule is directly adsorbed, three nearestneighbor lattice points can no longer form a pair site with that “0” and thus are indirectly blocked. The probability that each of these neighbors actually had been a pair site in conjunction with the “0” is 8,. Similarly, by blocking the “*” of the 0-* pair on which the molecule is directly adsorbed, three adjacent neighbors are indirectly blocked; and, the probability that each of these comprised an 0-* pair site in conjunction with the “*n is 80. Thus, the average number of 0-*pair sites which are indirectly blocked by an adsorbed molecule is given by (36, + 38J, and this is equal to 3 since 6, and 80 must sum to unity (for a pure metal oxide). The most probable total number of pair sites blocked (directly and indirectly) by an adsorbed molecule at low coverage is, therefore, equal to 4. At higher coverages, however, the number of pair sites which are indirectly blocked by an adsorbing molecule must decrease, since some of these sites are blocked by previously adsorbed, neighboring molecules. This is shown in Figure lb. Approximate Isotherms for Associative Adsorption on Pair Sites As discussed above, by blocking the “0” of the 0-* pair on which a molecule is directly adsorbed, three nearestneighbor lattice points are indirectly blocked. The approximate probability that a neighboring lattice point is an unsaturated cation not associated with an adsorbed molecule is equal to (6, - e), where 8 is defined as the number of adsorbed molecules on the surface, n, divided by the number of lattice points, N . This result follows from the assumption that each adsorbed molecule is associated with one cation and one anion. Analogously, the approximate probability that each lattice point adjacent to the “*” of an 0-* pair is an anion which is not covered by an adsorbed molecule is (8, - 8). Thus, the predicted average number of 0-* pair sites which are indirectly (5) Peri, J.
J. Catal. 1976, 41, 227.
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983 3559
Isotherms for Adsorption on Pair Sites
+
blocked by an adsorbing molecule is given by [3(&- 0) 3(00- e)], and this is equal to (3 - 60) since 0. and Bo must sum to unity. The total predicted number of pair sites blocked by an adsorbing molecule at a specific coverage is therefore equal to (4 - 60). It is important to note that the above expression is approximate since the lattice points adjacent to the “0” and n** of the 0-* pair were treated independently, and this is not strictly true for associative adsorption. So that the shape of the adsorption isotherm for a collection of pair sites could be generated, the blocking function, (4 - 60), times 6n is summed from zero coverage to a coverage 0, where 6n is the increase in the number of adsorbed molecules. This product is equal to the number of pair sites blocked at a specific value of 0. The fraction F of the pair sites blocked is obtained after normalization by 4NOO0,,the total number of pair sites: n
C(4 - 60)6n
F=
0
TABLE I: Monte Carlo Simulations of Adsorption on a Model Surface Consisting of 515 Anions and 509 Unsaturated Cations
a
Replacing the sum by an integral and noting that 0 = n/N gives
where K is the adsorption equilibrium constant and P is the pressure. Substitution for F gives the adsorption isotherm: 0 KP = (5) 4608, - 40 + 302 It will be shown that the functional dependence of the blocking function on 0 is too weak and that the isotherm of eq 5 predicts a value of 0 which is too low at a given value of KP. Another approach for generating an isotherm for adsorption on a collection of pair sites is to estimate directly the number of unoccupied pair sites for a given number of adsorbed molecules. It was shown earlier that for N lattice points, the number of pair sites at zero coverage by adsorbed molecules is equal to 4N000. (see eq 1). By analogy,the number of vacant pair sites at a given coverage 0 is approximately equal to 4N(Oo- e)(& - 0). In this case, it is noted that the numbers of anions and unsaturated cations which are not associated with adsorbed molecules are equal to N(Bo- 0) and N(&- O ) , respectively. With reference to eq 4, the corresponding adsorption isotherm is 0 0 KP = (6) 4@0- @(e* - 0) 4e0e. - 48 + 482 It can be argued that the above isotherm overestimates 0 at high values of KP. In short, the above analysis assumes that the probability that a randomly chosen lattice point is a given type of surface site (i.e., vacant anion, vacant unsaturated cation, or a site associated with an adsorbed molecule) is equal to the fraction of the surface composed of that type of site. Therefore, any lattice point has a fiiite
0 40 86 121 148 169 187 203 222 237 246
6a
no. of unoccupied pair sites
KPb
0 0.0391 0.0840 0.118 0.144 0.165 0.183 0.198 0.217 0.231 0.240
1080 914 740 624 526 473 421 379 3 27 295 273
0 0.0438 0.116 0.194 0.281 0.357 0.444 0.536 0.679 0.803 0.901
Surface coverage.
b
See eq 4.
TABLE 11: Monte Carlo Simulations of Adsorption on a Model Surface Consisting of 218 Anions and 806 Unsaturated Cations
4NOoO.
At adsorption equilibrium, the following holds: {surfaceconcentration of adsorbed species) KP = {surfaceconcentration of remaining pair sites) n (4) 4NOoO*(1- F)
no. of adsorbed molecules
a
no. of adsorbed molecules 0 30 52 73 88 97 116 122 130 142 151
6”
no. of unoccupied pair sites
KPb
0 0.0293 0.0508 0.0713 0.0859 0.0947 0.113 0.119 0.127 0.139 0.148
676 562 482 406 374 319 256 239 20 8 177 153
0 0.0534 0.108 0.180 0.235 0.304 0.453 0.510 0.625 0.802 0.987
Surface coverage.
See eq 4.
probability of being a member of a pair site. On an actual surface, however, some anions (or cations) may not be adjacent to unsaturated cations (or anions) and therefore can never serve as adsorption sites. This will be described in further detail later in this paper. Monte Carlo Simulation of Associative Adsorption on Pair Sites To test the quantitative utility of the two approximate isotherms of eq 5 and 6, we conducted Monte Carlo simulations to determine the number of pair sites remaining after various extents of adsorption. Model surfaces were drawn from a list of random numbers. One of the surfaces was for Bo and 0, each equal to 0.5 and the other was for Bo equal to 0.20 and 0, equal to 0.80. The model surfaces were square, with 32 lattice points on each side (Le., 1024 lattice points). The left and right edges and the top and bottom edges were treated as adjacent columns or rows to approximate a continuous surface. The grids were numbered so that random adsorption could be simulated. A random number generator was used to specify the coordinates and orientation of each molecule brought to the surface. If that position on the surface corresponded to a vacant pair site, then the molecule was allowed to adsorb. Tables I and I1 show the results of these simulations for the surfaces with 0, equal to 0.5 and 0.2 at various extents of adsorption. Figure 2 shows one of these surfaces at selected extents of adsorption. Taking the ratio of the number of adsorbed molecules to the number of unoccupied pair sites for each surface coverage allows points on the adsorption isotherm to be generated (see eq 4). These points are shown in Figure 3 for the model surfaces with Bo equal to 0.5 and 0.2. Also plotted in the figure are the approximate adsorption isotherms given in eq 5 and 6.
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The Journal of Physical Chemistry, Vol. 87, No. 18, 1983
0.08-
0
0.05 0.10 0.15
0.20 0.25 0.30 0.35 0.40 0.45 0.50
e
Flgure 4. Fraction of the surface consisting of isolated anions, eob. as a function of surface coverage by adsorbed molecules, 8, for surfaces with 8, equal to 0.5, 0.45, 0.4, and 0.3.
The agreement of the Monte Carlo points with the two approximate isotherms is good to the highest coverage to which the Monte Carlo simulations were carried out. Equation 6 seems to describe the behavior of the system better than eq 5. An important characteristic of an isotherm, besides its shape, is the saturation coverage. According to eq 5, the predicted saturation coverage Pt for a surface with 0, equal to 0.5 is 0.33, while the saturation coverage predicted by eq 6 is 0.5. The Monte Carlo simulation suggests that Pt for this surface is approximately 0.4. The reason that Pt does not reach 0.5 on this surface is clear upon examination of Figure 1. In order for an anion to be available for adsorption, it must have as a neighbor at least one unsaturated cation which is not associated with an adsorbed molecule. Even on a surface with no adsorbed molecules, some anions are isolated (i.e., have no neighbors which are vacant unsaturated cations, as shown in Figure IC).As coverage increases and more unsaturated cations are covered by adsorbed molecules, more anions become isolated (see Figure Id). For a surface with fewer anions than unsaturated cations, the number of pair sites remaining at any coverage will be limited by the number of vacant anions remaining which are isolated. (Analogous reasoning applies to surfaces containing fewer unsaturated cations than anions.) The fraction of the surface consisting of isolated anions eoisois given approximately by the probability that a given lattice site is occupied by an anion without an associated adsorbed molecule (e, - 0) times the probability that all of the four adjacent lattice sites are occupied by anions or by cations with associated adsorbed molecules (e,
+
:I
i
I
0.20
8
o.lot//-/
0.151
ord.1
A A
i
Oi2
d.3
OI.4
d.5
KP
d.6
d.7
08
d.9
J
1.0
Figure 3. Isotherms for adsorption on pair sites for surfaces with Bo equal to 0.5 and 0.2. Circles are points from Monte Carlo simulations. Lines are isotherms predicted by eq 5 (-.-), 6 (-), and 9 (--).
This relationship is shown in Figure 4 for surfaces having different values of Bo less than 0.5. At low coverages, the value of eoisois given by eos, which corresponds to the fraction of the anions on the surface which are surrounded by four other anions. As the surface coverage increases, Bois0 increases because anions adjacent to unsaturated cations at low coverages become isolated as more cations become associated with adsorbed molecules. A t higher coverages, Bois0 reaches a maximum value, denoted as BOisoma,, and as the surface coverage approaches eo, the calculated value of Bob decreases to zero. This is an artifact of the approximate calculation. First, the value of 8 cannot reach 8, because some of the anions on the surface are isolated; and second, the number of isolated anions should increase constantly throughout the adsorption isotherm. However, this approximate calculation yields a useful result. Most of the sites which are isolated at saturation
The Journal of Physical Chemistry, Vol. 87,
Isotherms for Adsorption on Pair Sites
TABLE 111: Properties of a Model Surface Consisting of 1024 Lattice Points no, of pair sites
e sat
at zero coverage
eo:e*a
eq 1
Monte Carlo
eq 8
Monte Carlo
pair sites blockedb
50:50 1024 1080 0.42 0.39 2.7 40:60 983 0.37 30:70 860 0.29 655 676 0.20 0.20 3.3 20:80 Average number of pair a Composition of surface. sites blocked per adsorbed molecule at saturation.
become isolated at low coverages in the adsorption isotherm. Therefore, the maximum value of Ooh in Figure 4 for each surface, Ooeoma,, can be taken as an approximation for the number of isolated anions at saturation coverage. The saturation coverage can then be estimated from pat
= 0, - ~ o ~ o m a x
(8)
In general Oohom%isequal to 2.62OO5.For the surface with 0, equal to 0.5,00 mar. is equal to c a 0.08and the saturation coverage by adsorbed molecules is thereby estimated to be 0.42. This value is in good agreement with the results of the Monte Carlo simulations. The above discussions about the saturation surface coverage allow an empirical adsorption isotherm to be proposed based on eq 5 and 6. The values of Ptpredicted by eq 5 and 6 for 0, equal to 0.5 are 0.33 and 0.5, respectively, while the Monte Carlo simulation gives a value of 0.4. Since the only difference between eq 5 and 6 is the coefficient of O2 in the denominator, this coefficient is taken to be an adjustable parameter which is chosen such that Pt equals 0.4. The following empirical isotherm results: 0 KP = (9) 46& - 40 3.75~9~
+
A comparison of this isotherm with the points from the Monte Carlo simulations for 0, equal to 0.5 is shown in Figure 3. The agreement between this isotherm and the Monte Carlo result is within 5%. It should be noted that the Monte Carlo simulations are strictly valid only for the case of irreversible adsorption. I t is possible to rearrange adsorbed molecules on the surfaces saturated by these simulations (as if by desorption and readsorption) and thereby increase the coverage. For a surface with 0, equal to 0.5, this rearrangement leads to an increase in the saturation coverage from the value of 0.39 to 0.43. Indeed, this latter value is approximately equal to the value of Pt calculated from eq 8. As the pressure approaches infinity, the maximum saturation coverage should be reached for reversible adsorption. This suggests that the coefficient of O2 in eq 9 be increased from 3.75 to the value of 3.9 to reflect this increase in saturation coverage. In short, the precise value of the coefficient of O2 in eq 9 is not known at present; however, the above arguments suggest that this value lies in the interval from 3.75 to 3.9. Comparison of Associative Adsorption on Pair Sites with the Langmuir Isotherm Table I11 shows the saturation surface coverages and the number of pair sites present at zero coverage for several values of 0, less than 0.5. These values of Ptand Nw were determined from Monte Carlo simulations and/or from eq 8 and 1. I t is apparent from this table that, for those surfaces where 0, is small (e.g., less than ca. 0.3), Onat is
No. 18, 1983 3561
nearly equal to 0,. This is because the probability of having adjacent anions decreases as 0, becomes smaller; therefore, the number of isolated anions decreases with decreasing 0,. In addition, as the value of 0, decreases, the average number of pair sites blocked per adsorbed molecule, B,at saturation increases. For example, B increases from 2.7 to 3.3 as 0, decreases from 0.5 to 0.2. Since the number of pair sites blocked per adsorbed molecule is equal to 4 at zero coverage on all of these surfaces, the number of pair sites blocked is more nearly constant over the entire adsorption isotherm for surfaces with smaller values of 0,. Accordingly, the isotherm becomes more Langmuirian as 0, decreases. It should be noted that the above conclusions regarding changes in the saturation coverage and shape of the adsorption isotherm with decreasing 0, below 0.5 are exactly analogous to the effects of decreasing 0, below 0.5. Specifically,one considers isolated cations in the latter case instead of isolated anions. Thus, in general, the maximum deviation from the Langmuir isotherm for associative adsorption on a collection of pair sites occurs for 0, and 0, equal to 0.5. An important question for consideration is what would the consequences be of describing the adsorption behavior on a collection of acid-base pair sites in terms of the Langmuir equation instead of the equations developed in this paper? For example, consider the comparison of the Langmuir isotherm with eq 9. I t was shown above that the saturation coverage for 0, less than 0.5 is approximately equal to 0,. Thus, the Langmuir isotherm for 0, less than or equal to 0.5 is of the form: 0 K$=0, - 0 (An analogous expression can be written for 0, less than or equal to 0.5.) Clearly, the observed value of KL must be a function of coverage to describe the non-Langmuirian behavior of eq 9. One can equate the two isotherms at a given pressure and solve for KL as a function of 0 and K, the equilibrium constant of eq 9: KL = K
+
4606*- 40 3.7502 0, - 0
(11)
I t can be seen from eq 11 that the deviation of the adsorption isotherm from the Langmuir equation decreases as Bo decreases. That is, the saturation coverage Pt decreases as 0, decreases, and thus the second-order terms in 0 can be neglected in comparison to first-order terms. In addition, as Bo decreases the value of OoO, approaches the value of 6,. With these approximations at small 0, (or O.), KL appears to be independent of coverage, demonstrating once again the apparent validity of the Langmuir isotherm under such conditions. One may now estimate the accuracy with which adsorption data must be measured in order to observe the deviations from the Langmuir isotherm predicted in this paper. For adsorption on a surface having an equal number of anions and unsaturated cations (such a surface showing the maximum deviation from Langmuirian behavior), one could attempt to fit the data using either eq 9 or 10. The correct saturation coverage (e.g., 0.40) would be obtained from eq 9 for adsorption on pair sites by setting 0, and 0. equal to 0.5, while the Langmuir isotherm of eq 10 could be made to give the correct saturation coverage by setting Bo equal to 0.40. Equations 9 and 10 could also be made to predict the same pressure at which the surface coverage equals half the saturation coverage (i.e., 0 = 0.2). This is achieved by making KL = 1.75K. It can then be shown that the values of 0 predicted by these
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The Journal of Physical Chemistry, Vol. 87, No. 18, 1983
two equations differ by less than about 0.02 at all pressures. Thus, the non-Langmuirian behavior of associative adsorption of pair sites can be observed if the surface coverages are measured with an uncertainty less than about 5% of the saturation coverage. Isotherm for Dissociative Adsorption on Pair Sites Having demonstrated the non-Langmuirian behavior of associative adsorption on pair sites, we show with the following brief derivation that this is also the case for dissociative adsorption on pair sites, where both parts of the molecule are free to migrate to different acid and base sites. At equilibrium, the rates of adsorption ra and desorption rd are equal. The rate of adsorption is proportional to the number of vacant pair sites. For a random distribution of the acidic and basic parts of the molecule on the surface anions and cations, respectively, the number of vacant pair sites is equal to 4N(OO- e)(& - 6). While this expression was shown earlier to be approximately valid for associative adsorption, it is exact for dissociative adsorption. Thus, r, is equal to r, = 4k,N(00 - e)(& - 6)P (12) where k, is an adsorption rate constant. The rate of desorption is proportional to the number of acidic parts of the molecule on the surface, BN, times the probability that an adjacent lattice point is occupied by a basic part of the molecule, 46. The factor of 4 is included since each lattice point has four neighbors. Thus, rd is equal to rd
= 4kdNo2
(13)
where kd is a desorption rate constant. The adsorption isotherm generated by equating eq 12 and 13 is KP =
62
(6, - 0)(6, - 6 )
where K = k,/kd. For comparison, the form of the Langmuir equation for dissociative adsorption with 6, less than 0.5 is
Thus, if the Langmuir isotherm is used to describe dissociative adsorption on a collection of pair sites, then KL will be a function of surface coverage, as given by 6, - 6
KL = K-
60
-6
(16)
It is interesting to note that the Langmuir equation is, in fact, correct when Bo and 6, equal 0.5. As Bo decreases
Tinkle and Dumesic
below 0.5 (and 6, increases accordingly), the adsorption isotherm becomes more non-Langmuirian, due to the fact that the constituent present in small amount becomes the limiting factor for adsorption. This is in contrast to the behavior of associative adsorption, where the deviation from the Langmuir isotherm is greatest when O0 and 6, are equal to 0.5. One can also estimate the accuracy with which experimental data must be measured in order to distinguish differences between eq 14 and 15 for dissociative adsorption. For a surface with Bo equal to 0.2, the minimum deviation between these two equations is achieved by setting KL equal to approximately 1OK. The values of 6 predicted by these two isotherms then differ by less than about 0.015 at all pressures. Thus, the nonLangmuirian behavior of dissociative adsorption on pair sites on such a surface can only be observed if the surface coverages are measured with an uncertainty less than about 8% of the saturation coverage. For a surface with more unequal numbers of the two surface constituents, the non-Langmuirian behavior may be evident at a higher uncertainty level, since the deviation will be stronger in such cases. Conclusions The results of statistical arguments and Monte Carlo simulations indicate the adsorption on pair sites of a two-component surface (e.g., acid-base pair sites on an oxide surface) does not follow the Langmuir isotherm. For associative adsorption, the deviation of the adsorption isotherm from the Langmuir equation is greatest when the surface is composed of an equal number of anions and unsaturated cations. For dissociative adsorption, however, the adsorption isotherm is less Langmuirian with more unequal numbers of anions and cations. So that these deviations from the Langmuir isotherm can be observed experimentally, surface coverages must be measured with an uncertainty less than about 5-10% of the saturation coverage. For the case of disssociative adsorption, the saturation surface coverage by adsorbed molecules is equal to the quantity of the surface constituent (Le., anions or unsaturated metal cations) which is present in lesser amount. This is only approximately true for associative adsorption due to the presence of isolated anions (or cations) which do not have any unsaturated cations (or anions) as nearest neighbors and cannot, therefore, serve as adsorption sites. The abundance of these isolated anions and cations increases as the numbers of surface anions and cations become equal.
Acknowledgment. We acknowledge the National Science Foundation for providing a Graduate Fellowship to M.T. and also for partial financial support of this work.
