Langmuir 1997, 13, 1089-1094
1089
A Simultaneous Description of Equilibria and Kinetics of Adsorption on Flat Heterogeneous Solid Surfaces: Single Gas Adsorption at Low Surface Coverages† W. Rudzin´ski* Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie-Sklodowska University, 20-031 Lublin, Poland
C. Aharoni‡ Department of Chemical Engineering, TECHNIONsThe Israel Institute of Technology, Technion City, Haifa 32000, Israel Received December 15, 1995. In Final Form: April 1, 1996X Based on the statistical theory of the rate of interfacial transport, a standard procedure is proposed for developing equations for adsorption kinetics and equilibria, corresponding to some model of a heterogeneous solid surface. By replacement of a certain function by another, in a kind of master equation, the isotherm equation is transformed into corresponding equation for adsorption kinetics, and vice versa. The detailed form of that master equation is related to the form of the adsorption energy distribution. The theoretical origin of the popular empirical equations for adsorption kinetics is reexamined and the range of their applicability is shown. The present theoretical treatment is limited to lower surface coverages where collective character of adsorption can be ignored, along with the readsorption kinetics.
Introduction In the theories of adsorption-desorption kinetics, the mass balance of the adsorbate over the entire heterogeneous solid surface is usually written in the following form
∑i
Rd ) -M
Xi
∂θi
where M is the total number of sites on a studied solid surface, θi means the fractional coverage of sites of ith type, Xi is the fraction of these sites on the solid surface, and t is the time. Most frequently, ∂θ/∂t was taken to be the expression offered by the theory of activated adsorption-desorption, (TAAD)
{ }
-a -d ∂θ - Kdθs exp ) Kap(1 - θ)s exp ∂t kT kT
(2)
where s is the number of adsorption sites involved in an elementary adsorption-desorption process, p is the pressure in the gas phase, a and d are the activation energies for adsorption and desorption, respectively, and Ka and Kd are slightly temperature dependent constants. Further, T and k are the absolute temperature and the Boltzmann constant, respectively. Now, let us consider for simplicity physisorption and the case s ) 1. So, at the equilibrium when ∂θ/∂t ) 0, eq 2 yields the Langmuir isotherm equation
{ kT} (p,T) ) 1 + Kp exp{ } kT Kp exp
(e)
θ
(3)
where K ) Ka/Kd, ) (d - a), and where the superscript * Author to whom the correspondence should be addressed: e-mail,
[email protected]; fax, (48)-8133669. † Presented at the Second International Symposium on the Effects of Surface Heterogeneity in Adsorption and Catalisis on Solids, held in Poland-Slovakia, in autumn 1995. ‡ E-mail:
[email protected]. X Abstract published in Advance ACS Abstracts, September 15, 1996.
S0743-7463(95)01547-2 CCC: $14.00
θ(e)(p,T) )
∑i
{} {}
Kip exp
(1)
∂t
{ }
e refers to equilibrium. At equilibrium, the assumption of a discrete distribution of the fraction X of adsorption sites among corresponding values of , expressed in eq 1, leads to the following expression
Xi
i
kT
1 + Kip exp
i
(4)
kT
where now θt(e) means now the “total” (average) fractional occupancy of all adsorption sites. In the case of the actual (real) solid surfaces, one usually deals with a dense spectrum of adsorption energies which should be represented rather by a continuous function χ(), so that
{ kT} (p,T) ) ∫ χ() d 1 + Kp exp{ } kT Kp exp
(e)
θt
Ω
(5)
where χ() fulfills the normalization condition
∫Ωχ() d ) 1
(6)
and Ω is the physical domain of . For mathematical convenience, Ω was frequently assumed to be the interval (-∞,+∞) or (0,+∞). It was shown that replacing the true physical domain Ω ∈ (l,m) (l and m meaning the lowest and the maximum values of for a heterogeneous solid surface) by (-∞,+∞) or by (0,+∞) does not greatly affect the behavior of the calculated isotherm θt(e)(p,T), except when extremely low or high surface coverages are not considered.1 Hundreds of papers have been published showing how much the behavior of the actual gas/solid adsorption (1) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: New York, 1992.
© 1997 American Chemical Society
1090 Langmuir, Vol. 13, No. 5, 1997
Rudzin´ ski and Aharoni
systems is influenced by the adsorption energy distribution in these systems. Finally, two monographs treating these surface heterogeneity effects have been published.1,2 At small surface coverages, the second term on the righthand side of eq 2 can be neglected. Many papers have been published on experimental studies of adsorption kinetics,3 but the measured data did not always obey the Langmuirian kinetics represented by the first term on the right-hand side of eq 2. Thus, various empirical laws were formulated to correlate the experimental data for adsorption kinetics. The first attempts by Roginski and Zeldovich4,5 to provide a theoretical explanation for these exmpirical laws employed the concept of adsorption on an energetically heterogeneous solid surface, characterized by a certain distribution of the activation energy for adsorption. Later, that concept was elaborated more thoroughly by Aharoni and co-workers6-8 and more recently by Cerofolini.9,10 Studies of desorption kinetics have been carried out even more extensively. These were stimulated by the wide application of temperature programmed desorption (TPD) experiments to study the energetic properties of catalysts and catalyst supports.11 The lack of applicability of the Langmuirian desorption kinetics represented by the second term on the right-hand side of eq 2 was found very soon. It was observed that the activation energy for desorption d changes with surface coverage, so that the theoretical analyses of TPD desorption spectra started from the following equation
-
{
}
-d(θ) ∂θ ) Kdθs exp ∂t kT
(7)
In most cases, the dependence of d on θ was explained as originating from the energetic heterogeneity of the actual solid surfaces, characterized by the distribution of the activation energies for desorption. Although every adsorption process is accompanied by a simultaneous desorption process, and vice versa, studies of adsorption-desorption kinetics proceeded historically along two separate routes. One group of scientists studied kinetics of adsorption at low surface coverages and neglected desorption phenomena in their studies. This group of scientists described the surface energetic heterogeneity as a distribution of the activation energies for adsorption, across a solid surface. The second group of scientists investigated desorption at high surface coverages, mostly in TPD experiments, and described the surface heterogeneity as the dispersion of the activation energies for desorption. Provided that we accept TAAD, we should consider the energetic surface heterogeneity as a simultaneous variation of a and d values, from one adsorption site, to another. (2) Jaroniec, M.; Madey, E. Physical Adsorption on Heterogeneous Solids; Elsevier: Amsterdam, 1989. (3) See the review by Low, M. J. D. Chem. Rev. 1960, 60, 267. (4) Roginski, S.; Zeldovich, Ya. Acta Physicochim., URSS 1934, 1, 554. (5) Roginski, S.; Zeldovich, Ya. Acta Physicochim., URSS 1934, 1, 595. (6) Aharoni, C.; Ungarish, M. J. Chem. Soc., Faraday Trans. 1 1977, 73, 1943. (7) Aharoni, C.; Ungarish, M. J. Chem. Soc., Faraday Trans. 1 1978, 74, 1507. (8) Aharoni, C.; Suzin, Y. J. Chem. Soc., Faraday Trans. 1 1982, 78, 2329. (9) Cerofolini, G. F. In Adsorption and Chemisorption on Inorganic Sorbents; Dabrowski, A., Tertych, V. A., Eds.; Elsevier: Amsterdam, 1996. (10) Cerofolini, G. F. Langmuir, in press. (11) For most recent and exhaustive review see, Bhatia, S.; Beltramini, J.; Do, D. D. Catal. Today 1990, 7, 309.
Let us consider the generalization of eq 2 for a heterogeneous solid surface, characterized by a continuous spectrum of adsorption and desorption energies. Provided that every adsorption site is characterized by a certain pair of values, a and d, for the whole heterogeneous surface, we have
∂θt ) ∂t
[
] { }]}
∫Ω ∫Ω { Kap(1 - θ) exp{ kTa) a
d
-
[
Kdθ exp
-d kT
-
χ(a,d) da dd (8)
where χ(a,d) is a two-dimensional differential distribution of the fraction of the surface sites among corresponding pairs of the values {a,d}. Following the theoretical results obtained for the case of adsorption equilibria, we will neglect the changes in Ka and Kd from one adsorption site to another. Apart from the mathematical problems posed by the solution of the integral equation (8) with respect to χ(a,d), one faces problems of a fundamental nature. Namely, first of all, one must know the analytical form of θ as the function of a and d. In order to make use of the statistical approaches to adsorption, one must establish the relationship between a and d. Seeking the relationship beween d and a on different adsorption sites seems to represent a difficult fundamental problem. Finally, the lack of applicability of Langmuirian adsorption kinetics was reported for typical physisorption systems in which the sense of the activation energy for adsorption seems to be difficult for interpretation. All the above mentioned difficulties disappear when, as the starting point, one applies statistical theory of the rate of interfacial transport.12 Theory The starting point of our consideration is the equation developed by Ward et al.12
[ {
}
{
}]
-(µg - µs) µg - µs ∂θ - exp ) Kgs′ exp ∂t kT kT
(9)
where µg and µs are the chemical potentials of the gaseous and adsorbed (surface) molecules, respectively, and Kgs′ is a constant. They assumed next that the transient configurations of adsorbed molecules are close to the equilibrium ones. Now, let us consider the Langmuir model of adsorption, i.e., one-site occupancy (localized) adsorption when no interactions exist between adsorbed molecules. Then
µs θ ) ln s kT q (1 - θ)
(10)
where qs is the molecular partition function of the adsorbed molecules. Using the ideal-gas approximation for µg, we have g µg µ0 ) + ln p kT kT
(11)
Now, let us consider the region of lower surface coverages, i.e., neglect the second term within the brackets in eq 9. As the adsorption kinetics are essentially nonequilibrium processes, we introduce the superscript (n) at the surface coverages appearing in the equations for adsorption (12) Ward, C. A.; Findlay, R. D. J. Phys. Chem. 1982, 76, 5615.
Adsorption Kinetics and Equilibria on Surfaces
Langmuir, Vol. 13, No. 5, 1997 1091
kinetics. Then
{ }
µ0g 1 - θ(n) µg - µs ∂θ(n) s ) Kgs′ exp ) Kqs′pq exp ∂t kT kT θ(n) (12) The main effect of the energetic heterogeneity of the actual (really existing) solid surfaces is related to the dispersion of the minima in the gas-solid potential function across a solid surface. In the case of localized adsorption, these local minima are called “adsorption sites”, and the value of the gas-solid potential at these local minima, taken with a reverse sign, is called “the adsorption energy”, and usually denoted by . Thus, while considering the kinetics of adsorption on a heterogeneous solid surface, we will write the molecular partition function qs as the following product
{kT }
qs ) q0s exp
(13)
where q0s is the same for all the adsorption sites and varies from one to another site. We will further denote by K the following product
Figure 1. Temperature dependence of the function θ(,c). The dimensionless temperatures are τ ) 1 (;), τ ) 5 (- - -), and τ ) 10 (- - -).
Both eq 16 and eq 18 can be written in the same form
{ }
K ) q0s exp
µ0 kT
(14)
θt(i) )
[
-1
(15)
∫-∞
[
+∞
1+
{ }]
(∂θ(n)/∂t) - exp Kgs′pK kT
θ
χ() d (16)
{ }]
1 - ) 1+ exp Kp kT
[
-1
(17)
and the experimentally measured surface coverage θt(e) is given by
θt(e) )
1 - exp{ }] ∫-∞+∞[1 + Kp kT
(19)
(∂θ(n)/∂t) K′gspK
-1
χ() d
(18)
where the subscript (e) in θ(e) and θt(e) refers to equilibrium conditions.
(20)
whereas when equilibrium is attained, c * c(e)
c(e) ) -kT ln(Kp)
(21)
When T f 0, θ(i) tends to the step function θc(i)
{
0, for < lim θi ) θc(i) ) 1, for g c and θt(i) ) c Tf0
∫∞ χ() d (i)
c
(22)
for both equilibrium and nonequilibrium conditions. This is shown in Figure 1, where the function θ(c(i),T), written in the following form
-1
When equilibrium is attained, θ ) θ(e) is given by the condition µs ) µg, i.e. (e)
χ() d
For the nonequilibrium conditions c(i) * c(n)
c(n) ) kT ln
The above equation describes the rate of adsorption on adsorption sites having adsorption energy equal to . The experimentally measured surface coverage θt(n), and the mean rate of adsorption (∂θt(n)/∂t), are the values θ(n) and (∂θ(n)/∂t) defined in eq 15, averaged over all kinds of adsorption sites taken with an appropriate statistical weight. With a dense spectrum of adsorption energies on a solid surface, that statistical weight becomes practically a continuous differential distribution of the number of adsorption sites among corresponding values of adsorption energy, χ(). Thus, in the case of a heterogeneous solid surface, the experimentally determined values θt(n) and (∂θt(n)/∂t) are to be related to the following average
θt(n) )
-1
i ) e, n
{ }]
(∂θ(n)/∂t) - exp Kgs′pK kT
(i) c - kT
∫-∞+∞ 1 + exp
and rewrite eq 3 in the following form
θ(n) ) 1 +
{ }]
[
g
exp θ((i) c ,T) )
{} {} r τ
r 1 + exp τ
(23)
is shown, as the function of the dimensionless variables r ) ( - c(i))/kT0, and τ ) T/T0. For the reduced temperature τ ) 1, function 23 is very close to the step function 22. In the theories of adsorption equilibria on heterogeneous solid surfaces, the step function defined in eq 22 is usually called the “condensation isotherm”. The application of condensation approximation (CA) has been thoroughly studied in theories of equilibria of adsorption on heterogeneous solid surfaces.1 It was shown there that, in the case of adsorption on a heterogeneous solid surface, the essential condition for the applicability of the CA approach is that the variance of χ() must be at
1092 Langmuir, Vol. 13, No. 5, 1997
Rudzin´ ski and Aharoni
least 10% larger than the variance of (∂θe/∂). Of course, the same will be true also for adsorption-desorption kinetics. Replacing the true kernel θ(i) (i ) e,n) in eq 19 by its corresponding step function 22 means that it is assumed that the adsorption proceeds gradually on various adsorption sites in the sequence of decreasing adsorption energy . At a given temperature T, pressure p, and (∂θ(n)/ ∂t), adsorption “front” is on the sites whose energy is equal to c given in eq 20. Thus, the overall adsorption rate (∂θt(n)/∂t) is, in fact, governed by the local rate of adsorption, on sites whose adsorption energy is equal to c(n), through the obvious relation
( )
∂θt(n) ∂θ(n) ) constant χ(c(n)) ∂t ∂t
(24)
)c(n)
Equation 28 is convenient to demonstrate that it is not essentially the condition T f 0 for the condensation approximation to be applicable. The essential condition is that the variance of the adsorption energy distribution must be larger than kT. In the limit T f 0, or more generally when kT/(m - l) is small, eq 28 reduces to the following
θte )
{ } {
∂θtn m m - l exp ) KgspK exp θt ∂t kT kT
( ) (n)
∂θt ∂t
(25)
χ(c(n))K ˜ gspK
where K ˜ gs ) constant Kgs′. One may argue that because eq 9 does not apply at very small, θ f 0, and very high, θ f 1, surface coverages, one cannot accept the step isotherm in our consideration. However, it is obvious that the picture of a sharp “adsorption front” will be a very good representation for the true isotherm (23) at not too high temperatures, and the true isotherm (23) does not violate the condition that θ cannot be very small or very high at ) c(i), i.e., on the adsorption sites where the kinetics of the “local” adsorption governs the kinetics of the “total” adsorption through relation 24. For some distributions χ(), the integration of eq 22 can be performed in an exact analytical way. This, for instance, is the case of the rectangular adsorption energy distribution
{
1 for l e e m χ() ) m - l 0 elsewhere
kT ln m - l
[ [
{ {
}] }]
m - c(e) 1 + exp kT
(27)
The corresponding equation for adsorption kinetics is obtained from eq 27 by replacing c(e) by c(n). Then
∂θt(n) ) ∂t
{ } {
l exp KgspK exp kT
}
{
}
m - l (n) m - l - exp θ kT t kT m - l (n) 1 - exp θt kT
{
}
θt(n) )
(28)
˜ gs(m - l)-1. Equations 27 and 28 are not where Kgs ) K subject to any temperature limits and are also valid at high temperatures (provided that the adsorption can still be considered in terms of localized adsorption).
(30)
kT ln[Rpt + 1] m - l
(31)
{ }
(32)
where
R)
l - c(e) 1 + exp kT
}
Equation 30 is just the Elovich equation which is probably the most popular one to analyze experimental data for adsorption kinetics. Originally, it was proposed as an empirical equation, but later, it was associated intuitively with a constant (rectangular) distribution of adsorption (desorption) energies. Attempts to derive it theoretically always started with the Temkin isotherm for adsorption equilibria. Additionally, several additional assumptions had to be made, to arrive at the Elovich kinetic equation. This made all these derivations obscure and the nature of the Elovich equation still remained semiempirical. The general procedure proposed here leads to the kinetics equations, in which the parameters have a precise thermodynamic interpretation. That procedure allows adsorption kinetics to be predicted from the adsorption equilibria and vice versa. The integration of eq 30 leads to the following expression
(26)
leading to the generalized Temkin adsorption isotherm, in the case of adsorption equilibria.
θt(e) )
(29)
which is obtained for the adsorption energy distribution (26), by adopting the condensation approximation formulated in eq 27. The corresponding equation for adsorption kinetics reads
In other words, c(n) in eq 23 can be considered as
c(n) ) kT ln
m - ce m - l
m m - l KgsK exp kT kT
The comparison of eqs 30 and 31 leads to the conclusion that investigating linearity of the plot ln(∂θt(n)/∂t) vs θt(n)
ln
∂θt(n) m - l (n) ) constant θt ∂t kT
(33)
is to be recommended to check whether the adsorption kinetics are Elovichian. The linearity of the plot θt(n) vs ln(Rpt + 1), predicted in eq 31 involves the necessity of making an always suspicious assumption concerning the value of the R-parameter. As eqs 27 and 28 represent the exact results of the integrations of eqs 18 and 16 when χ() is the function defined in eq 26, they correctly reduce the appropriate equations for a homogeneous solid surface when (m l)/kT f 0. (To check it one must apply de l’Hospital rule.) Although the shape of χ() for an actual solid surface is expected to be a complicated curve, in general, its
Adsorption Kinetics and Equilibria on Surfaces
Langmuir, Vol. 13, No. 5, 1997 1093
“smoothed” version can, to a first crude approximation, be represented by the Gaussian-like curve
{ } { }]
- 0 1 exp c c χ() ) - 0 1 + exp c
[
(34)
2
centered at ) 0, the dispersion of which is related to the heterogeneity parameter c. This function has a physical background related to some universal features of energetically heterogeneous solid surfaces.1 When equilibrium is attained, we obtain from eqs 21, 22, and 34
[
{
}]
,
0 < kT/c < 1
c(e) - 0 c
θt(e) ) 1 + exp
-1
(35)
or, in another form
θt(e) )
(K0p)kT/c 1 + (K0p)kT/c
(36)
where
{ }
K0 ) K exp
0 kT
(37)
Equation 36 is the well-known “generalized Freundlich” (Bradley’s) isotherm which is commonly used to correlate experimental adsorption isotherms. It reduces to the Freundlich isotherm at low surface pressures (coverages)
limθt(e) ) (K0p)kT/c
(38)
pf0
Now, let us consider the form of θt(n) and (∂θt(n)/∂t) corresponding to the Gaussian-like adsorption energy distribution (34). For obvious physical reasons, there must be a certain minimum and a maximum value of the adsorption energy on a heterogeneous solid surface, l and m. Thus, the Gaussian-like function (34) should, for various values of the parameter c, be viewed correctly in the way shown in Figure 2. All the functions shown in Figure 2 are normalized to unity and given by the equation
χ() )
1 FN
{ } { }]
- 0 1 exp c c - 0 1 + exp c
[
(39)
2
where the normalization factor FN reads
[
{
FN ) 1 + exp
}] [
l - 0 kT
-1
{
- 1 + exp
}]
m - 0 kT
-1
(40) Figure 2 shows that when c f ∞, the function in eq 39 becomes the rectangular (constant) energy distribution. It means, that the Elovich equation is likely to be a limiting form of all kinetic equations when the surface is strongly heterogeneous. With finite integration limits, the applicability of the condensation approximation has been discussed in the recent monograph by Rudzin´ski and Everett.1
Figure 2. Effect of the heterogeneity parameter c on the shape of the adsorption energy distribution χ(), given in eq 39. Here EU is a certain energy unit comparable to kT. As can be deduced from the figure, l ) 2EU, 0 ) 5EU, and m ) 8EU. The values of the parameter c are 0.5EU (- - -), 1.0EU (- - -), and 5EU (;). One can see that as the heterogeneity (parameter c) increases, χ() tends to rectangular adsorption energy distribution.
As it was demonstrated above, in the case of the rectangular energy distribution, the condition for the CA approach to be applied is not solely T f 0, but the condition (m - l)/kT f ∞. Practically, that condition means that the ratio of variance of (∂θ(e)/∂), or (∂θ(n)/∂) to the variance of χ(), should be smaller than 0.9. In the case of the function χ() defined in eq 34, that ratio is represented by kT/c. As function 39 resembles, to some extent, the rectangular distribution (26), we will now use relation 25 as an approximate one in eq 22. Namely, we will treat χ(n(n)) as a constant, while calculating integral 22. For the function defined in eq 34, the expression for θt(n) reads, (n)
θt
{
[
}]
c(n) - 0 ) 1 + exp c
-1
(41)
Considering eq 25 and solving eq 41 with respect to (∂θt(n)/ ∂t), we obtain
( )
∂θt(n) 1 - θt(n) ) KgspK0 ∂t θt(n)
c/kT
(42)
though the constant Kgs in eq 42 is not identical to that in eq 28. Comparison of eqs 42 and 12 shows that the Gaussianlike dispersion of adsorption energies causes the term (1 - θt(n))/θt(n) to be rised to the power c/kT. Now, let us investigate the simplified form of eq 42 valid at low surface coverages
∂θt(n) ) KgspK0(θt(n))-c/kT ∂t
(43)
In other words, let us investigate the adsorption kinetics corresponding to a Freundlich adsorption isotherm. After integration we obtain
[(
θt(n) ) 1 +
c K K pt kT gs 0
)
]
1/(1+c/kT)
(44)
In the case of a homogeneous solid surface, instead of Freundlich region we have Henry’s region in the limit of
1094 Langmuir, Vol. 13, No. 5, 1997
Rudzin´ ski and Aharoni
Figure 3. Black circles (b) are the data for the kinetics of hydrogen adsorption on chromia drawn by Aharoni and Ungarish13 in log(q/N0) vs log[(A/N0)t] coordinates, where q is the adsorbed amount, and N0 and A are some constants. The dotted line (- - -) drawn here by us shows that at smaller surface coverages (∂ log q/∂ log t) ) 1/2.5.
low pressures (coverages). Then instead of eq 44, we have
[
{kT }]
θ(n) ) 2KgsKpt exp
1/2
(45)
Thus, in the case of a homogeneous solid surface, the double-logarithmic plot ln θ(n) vs ln t should (at small pressures), be a straight line with a slope (∂ ln θ(n)/∂ ln t) ) 1/2. In the case of a heterogeneous solid surface an analogous double-logarithmic plot should also be linear, but the slope (∂ ln θ(n)/∂ ln t) ) (1 + (c/kT))-1. Because the values of c/kT are larger than unity, the slopes (∂ ln θ(n)/∂ ln t) smaller than 1/2 indicated adsorption kinetics on a heterogeneous solid surface. Values c/kT ) 3/2 are very typical for many of the actual solid surfaces, so the slope (∂ ln θ(n)/∂ ln t) ) 1/2.5 should also be typical. This can be seen in Figure 3 in the paper by Aharoni and Ungarish,13 which is redrawn in Figure 3. The (“condensation”) approach becomes less and less accurate with increasing temperature. Rudzin´ski and Jagiełło have shown that, for typical heterogeneous surfaces and typical physical regimes at which the gas-solid adsorption experiments are carried out, the CA approach can be safely applied. If a question arises whether the CA approach is sufficiently accurate in a particular case, one may apply the more accurate RJ approach1 to represent the result of the integration in eq 19. As we focus our attention here on most fundamental problems, we will use the CA approach for the sake of simplicity. Let us consider why the model of localized adsorption along with the CA approach (assumption of a strong surface heterogeneity) leads to these generally (13) Aharoni, C.; Ungarish, M. J. Chem. Soc., Faraday Trans. 1 1976, 73, 456.
observed laws for adsorption kinetics (Bangham’s power and the Elovich equations). It is now widely realized that there is the geometric nonuniformity of real solid surfaces, which is the main source of their energetic heterogeneity for adsorption. The surface geometrical distortions lead to a dispersion of the value of the local minima in the gas-solid potential function described by χ(). If these local minima are the consequence of geometrical distortions, the creation of various energy barriers separating these local minima will be an unavoidable accompanying effect. These energy barriers will hinder the translational movement of adsorbed molecules across the solid surface. In other words, the stronger the surface heterogeneity, the stronger the tendencies to localized adsorption. This is why the assumption of localized adsorption along with the CA approach work so well together. The observations of the actual adsorption systems provide strong support for this intuitive hypothesis. By taking into account the dispersion of the local minima in the gas-solid potential functions (e.g., the dispersion of ), we have arrived at a more accurate result that the rate of adsorption is proportional to [(1 - θ(n))/θ(n)]c/kT. For low surface coverages, the above statement leads to Bangham’s empirical power law. The proportionality [(1 - θ(n))/θ(n)]c,kT is likely to be valid in the case of moderately heterogeneous surfaces. In the case of strongly heterogeneous surfaces, the adsorption rate will be proportional to exp{-[(m - l)/kT]θt(n)}, because, in this limit, every (one-modal) adsorption distribution degenerates into a rectangular adsorption energy distribution. However, as demonstrated in Figure 2, this degeneration proceeds gradually, so that the rate of adsorption will be a hybrid between that described by the term [(1 - θ(n))/θ(n)]c/kT and a rate of adsorption described by the term exp{-[(m - l)/kT]θt(n)}. Thus the experimental data can be correlated by both the power laws and the Elovich equations, but always with a limited success. One can only say, that this hybrid behavior more resembles that predicted by the term [(1 - θ(n))/θ(n)]c/kT than that predicted by the term exp{-[(m - l)/kT]θt(n)}, or vice versa. The purpose of the present publication has been to provide a standard routine for a simultaneous description of equilibria and kinetics of adsorption on a nonporous but energetically heterogeneous solid surface, characterized by a certain distribution of adsorption energy. To illustrate the new procedure we have chosen some expressions for the adsorption energy distribution, which lead to compact, well-known equations for adsorption kinetics. In general, the functions θt(e)(p,T) and ∂θt(n)/∂t ) f(θt(n),p,T,t) will have to be evaluated numerically. In the case of adsorption kinetics, numerical calculations will involve solving, for an assumed function χ(), the systems of the two equations (22) and (25) to eliminate c(n), and to obtain ∂θt(n)/∂t as the function of θt(n),p,T. Ward and co-workers12 have proposed that µs in eq 9 be expressed by appropriate functions of coverage and temperature developed for adsorption equilibria. Their proposal has been fully supported by the computer simulations of adsorption kinetics, published recently by Talbot et al.14 While assuming that adsorption is accompanied by desorption, they found that the transient configurations of adsorbed particles can be well approximated by the corresponding equilibrium configurations at the same coverage. LA9515478 (14) Talbot, J.; Jin, X.; Wang, N.-H. Langmuir 1994, 10, 1663.
