Langmuir 2002, 18, 6789-6795
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Kinetics of Multilayer Langmuirian Adsorption Igor A. Vinokurov and Jouko Kankare* Department of Chemistry, University of Turku, FIN-20014 Turku, Finland Received March 26, 2002. In Final Form: June 24, 2002 The kinetics of multilayer adsorption processes in solution onto an adsorbent has been explored theoretically. The basic assumption was the “Langmuirian character” of the adsorption process. The general analytical solution for multilayer coverage was obtained, which may be used to evaluate the time dependence of coverage in those cases, when the interfacial kinetics is relatively slow compared to the rate of diffusive transport. As a limiting case of equilibrium of the adsorption process, the general solution provides the expression for the well-known isotherm derived by Brunauer, Emmett, and Teller (BET isotherm). Additionally, useful approximate analytical solutions were derived based on the general solution for the very beginning and the very end of the adsorption process. The theoretical results of this work allow estimating the influence of the finite adsorption rate on the functional time dependence of coverage given a set of suitable constants.
Introduction The growth kinetics of thin films adsorbed on a substrate is of interest for many aspects of surface chemistry, from understanding of adsorption-desorption mechanism to more practical problems such as corrosion, catalysis, and growth of materials with a layered structure. Various kinetic models have been suggested for the formation of monolayers, the simplest of which is the Langmuirian adsorption. In this model no mutual interactions of adsorbate molecules are taken into account, but still, despite of its simplicity, the model fits surprisingly numerous cases. The adsorption may have as its ratedetermining step either the transport of adsorbate molecules to the adsorbent surface by, e.g., diffusion,1 or the rate of the adsorption process itself which may be chemisorption or purely physical adsorption. However, there is no reason to believe that the adsorption process stops on a monolayer. The first layer may be strongly chemisorbed but the adsorbate molecules may continue adsorption by weak physical interaction on top of the first layer, the third layer on top of the second layer, etc. This was the basic assumption of Brunauer, Emmett, and Teller2 when they derived their celebrated isotherm, now called the BET isotherm. The BET isotherm is one of the most popular adsorption isotherms used widely in multilayer adsorption studies and especially for determining surface areas of adsorbents. As the word isotherm implies, the BET equation is derived for the equilibrium of the adsorption process. In the original form the BET equation is
v ) vm
(
1-
)(
p γ p0
)
p p 1 + (γ - 1) p0 p0
(1)
Here p refers to the partial pressure of the adsorbent gas, p0 to the condensation pressure, i.e., the pressure when the condensation of the gas starts at this temperature, v * To whom correspondence should be addressed. E-mail:
[email protected]. (1) Kankare, J.; Vinokurov, I. A. Langmuir 1999, 15, 5591-5599. (2) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309-319.
to the volume of gas adsorbed, and vm to the maximum volume of gas adsorbed. The constant γ is defined as
(
γ ) exp
)
Q1 - Qv RT
(2)
where Q1 is the adsorption heat for the first layer and Qv the heat of condensation of gas, representing the heat of adsorption of the second and further layers. The most interesting feature of the BET equation is its singularity as p f p0. In many experimental cases indications of this singularity are seen as an abrupt increase in the adsorbed amount as the pressure approaches the condensation pressure. As a matter of fact, this abrupt increase is an indication of the formation of a new phase, which in principle would grow to an infinitely thick layer. Although originally derived for the adsorption from the gas phase, attempts have been made to extend the BET equation to the liquid phase.3 In this case the partial pressure is replaced by concentration, the condensation pressure by the saturation concentration, and the gas volume v by the adsorbed amount. The pioneers in this field were Hansen, Fu, and Bartell4 who showed that various sparingly soluble organic compounds gave BETtype isotherms when adsorbed from water to carbon. The data were best treated by using a reduced concentration of adsorbate, meaning the ratio of concentration to the saturation concentration in complete analogy with p/p0. In many cases the signs of singularity were observed as the saturation concentration was approached. In the original publication2 the BET equation was derived for equilibrium conditions and with the basic assumption of applicability of the Langmuir equation to each layer. It is well-known that the original BET isotherm does not describe well the experimental results, especially at the high-pressure range close to p ≈ p0. The criticisms against the BET isotherm and its various proposed modifications have been reviewed in recent papers by Cerofolini and Meda5 and Nikitas.6 (3) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; John Wiley & Sons: New York, 1990. (4) Hansen, R. S.; Fu, Y.; Bartell, F. E. J. Phys. Chem. 1949, 53, 769-785. (5) Cerofolini, G. F.; Meda, L. Surf. Sci. 1998, 416, 403-422. (6) Nikitas, P. J. Phys. Chem. 1996, 100, 15247-15254.
10.1021/la025777f CCC: $22.00 © 2002 American Chemical Society Published on Web 08/08/2002
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The quantity x1 can change in four different ways: by adsorption on the first layer, by desorption from the first layer, by desorption from the second layer, and by adsorption on the bare surface
Figure 1. Multilayer adsorption.
The number of papers dealing with the rate of multilayer formation is much less than in the case of equilibrium. Numerical solutions of the system of equations describing the adsorption-desorption kinetics were presented by Fong et al.7 and Karasova.8 The growth kinetics of multilayer adsorption has been also a subject of various Monte Carlo simulations.9 The complicated form of the desorption-adsorption rate equations with an infinite number of differential equations has not apparently encouraged researchers to solve them. The present authors could find only one attempt to find the analytical solution. Pritzker10 used the method of generating functions to derive an analytical expression for the transient current response to a potential step for multilayer electrochemical adsorption reaction. The advantage of an analytical solution is not only in the shorter computation time required to evaluate the transient adsorption density numerically but also in the expansion of our fundamental knowledge on the mechanism of multilayer adsorption. In addition, the knowledge of the analytical solution allows one to derive different useful approximations and asymptotic expansions. Although the fundamental weaknesses of the Langmuirian multilayer adsorption are well-known, the BET isotherm has remained as a kind of yardstick to which other theoretical approaches are compared. In the same way, as different kinetic schemes of multilayer formation are compared, the corresponding Langmuirian equivalent should be available for comparison. The aim of the present authors is to fill this need of analytical solutions and to derive approximations which are useful in the interpretation of experimental results.
(4)
dx2/dt ) k-x3 + k+Cx1 - k+Cx2 - k-x2
(5)
One can see that for the successive layer with i ) 2, 3, ... the quantity xi(t) can change: by adsorption on the layer i, -k+Cxi; by desorption from the layer i, -k-xi; by desorption from the layer i + 1, +k-xi+1; by adsorption on the layer i - 1: +k+Cxi-1. Hence, we get
dxi/dt ) k-xi+1 + k+Cxi-1 - k+Cxi - k-xi
(6)
i ) 1, 2, ... Here k1+ and k1- are rate constants for adsorption and desorption on the substrate and k+ and k- are constants for adsorption and desorption on molecular layers. And further we use the constants K ) k+/k- and K1 ) k1+/k1and dimensionless parameters κ- ) k1-/k-, κ+ ) k1+/k+, and γ ) K1/K ) κ+/κ-. We also define dimensionless time T ) k-t and dimensionless concentration c ) KC. Using the dimensionless time, we take the Laplace transform
sxj 0 - 1 ) κ- xj1 - κ+cxj0
(7)
sxj1 ) xj2 + κ+cxj0 - cxj1 - κ- xj1
(8)
sxj2 ) xj3 + cxj1 - cxj2 - xj2
(9)
sxji ) xji+1 + cxji-1 - cxji - xji
(10)
i ) 2, 3, ... We make a substitution
xji ) Aσi-1; i ) 1, 2, ...
(11)
into eq 10 giving
Theory We shall derive the differential equation for multimolecular adsorption by a method that is a generalization of Langmuir’s treatment of the unimolecular layer. The solution of this equation should provide the total quantity of adsorbed species per unit area as a function of time. Let x0, x1, x2, ..., xi, ... represent the fractions of surface area covered by only 0, 1, 2, ..., i, ... layers of adsorbed molecules (see Figure 1). The bulk concentration of the adsorbate is C. We assume that the concentration at the immediate vicinity of the surface is continuously replenished and the concentration gradient is negligible. During the process of adsorption, the change of the bare surface x0 is caused by both the adsorption on the bare surface and desorption from the first layer
dx0/dt ) k1-x1 - k1+Cx0
dx1/dt ) k-x2 + k1+Cx0 - k+Cx1 - k1-x1
sAσi-1 ) Aσi + cAσi-2 - cAσi-1 - Aσi-1
(12)
Both sides are divided by Aσi-2 giving
sσ ) σ2 + c - cσ - σ
(13)
σ2 - (s + c + 1)σ + c ) 0
(14)
or
Let the roots of this equation be σ1 and σ2. Apparently the general solution for eq 10 is then
xj i ) Aσ1i-1 + Bσ2i-1
(15)
To be valid for i ) 1, we have
A + B ) xj1
(3)
(16)
Due to the definition of the surface fractions: (7) Fong, M. C.; Liu, C. K.; Classford, A. P. M. Eur. Space Agency, [Spec. Publ.], SP 1982, (ESA SP-178, Spacecraft Mater. Space Environ.), 71-80. (8) Karasova, I. Czech. J. Phys. 1989, B 39, 1378-1391. (9) Grabowski, K.; Patrykiejew, A.; Sokolowski, S. Thin Solid Films 1999, 352, 259-268; 2000, 379, 279-307. (10) Pritzker, M. D. J. Electroanal. Chem. 1989, 262, 15-33.
∞
1
xji ) ∑ s i)0 or
(17)
Kinetics of Multilayer Langmuirian Adsorption ∞
∞
Langmuir, Vol. 18, No. 18, 2002 6791
∞
xji ) A ∑ σ1i + B ∑ σ2i ) ∑ i)1 i)0 i)0 A 1 - σ1
x1,eq ) +
B 1 - σ2
)
1 s
(19a)
σ2 ) 1/2(s + c + 1) + 1/2((s + c + 1)2 - 4c)1/2
(19b)
Apparently the sum of geometric progression 18 is valid only if
(20)
In the case of equilibrium we have the same eqs 8 to 10 but with s ) 0 and xi without the Laplace transformation. The first equation comes from eq 3 by putting the derivative to zero
The total amount of the adsorbed species on the surface as the fraction from the total monolayer is ∞
Ξ)
∑ kxk
(29)
k)1
When applied to eqs 28 we have ∞
Ξeq )
∑ kxk,eq k)1 ∞
γc(1 - c)
kσk-1 ∑ 1,eq ) 1 + (γ - 1)c k)1
) Aeq
(21a)
1
)
(1 - c)2 γc
(1 - c)(1 + (γ - 1)c)
x2 - (c + κ-)x1 + κ+cx0 ) 0
(21b)
xi+2 - (c + 1)xi+1 + cxi ) 0
(21c)
Ξeq ) (22)
1 1 1 - c 1 + (γ - 1)c
(31)
In the general case we go back to the condition 20. At very low concentration we have
i ) 1, 2, ... and the roots are
cf0
σ1,eq ) c, σ2,eq ) 1
(30)
Thus we have come to the expression for the well-known isotherm derived by Brunauer, Emmett, and Teller.2,10 This can be also written
The general solution is of the same form i-1 xi ) Aeqσi-1 1,eq + Beqσ2,eq
(28c)
i ) 1, 2, ...
σ1 ) 1/2(s + c + 1) - 1/2((s + c + 1)2 - 4c)1/2
x1 - γcx0 ) 0
(28b)
1-c ci xi,eq ) γ 1 + (γ - 1)c
- xj0 (18)
The explicit expressions of the roots are
|σ1| < 1 and |σ2| < 1
γc(1 - c) 1 + (γ - 1)c
σ2 98 s + 1
(23)
(32)
Only the first root can be accepted due to the condition 20, and in addition we have the condition
This is greater than 1 because the Laplace variable s cannot be negative. Consequently we have to reject the root σ2 and put eq 15 to the form
c 0
κ+cσ1
) )
(39)
Here we have the same result as in eq 30, i.e., the BET equation. Now we shall take the inverse Laplace transform of Ξ h (s). At any value of Laplace domain s, we have (see also condition 24)
Ξ h)
{
F2 - F1 κ+c c 2 (1 F )(1 F ) (1 - κ )(F1 - F2) 1 2 s 1 1 c 1 c 1 + + r2F2(1 - F2) s - r2 s r1F1(1 - F1) s - r1 s
Ξ h )-
σ1 s(1 - σ1)
)
c - σ1 s2
(51)
The inverse Laplace transform is easily taken by using eq 49 and convolution
Ξ(T) )
∫0T (Tτ - 1) exp(-(1 + c)τ)I1(2c1/2τ) dτ
cT - c1/2
(52)
Short Time Limit. In possible analytical applications of the results, received here, it would be practical to have a simple equation for describing the first moments of the adsorption process which may allow the estimation of heterogeneous rate constant of the adsorbate or its bulk concentration. Such an expression can be obtained for the
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Langmuir, Vol. 18, No. 18, 2002 6793
very beginning of the process when the coverage is low allowing the linearization of the general solution 50. To receive the short-time approximation we use only the first terms in the series expansions of the exponential functions
er1T - 1 ≈ r1T, er2T - 1 ≈ r2T
(53)
Substituting these to eq 50 we have
{
c
F1 - F2
∫0
T
]
(1 - F1)(1 - F2)
[
1 1 + (1 - F2) (1 - F1) -(c+1)τ
(T - τ)e
-1
1/2
τ I1(2c τ) dτ
}
Ξ h (s′) ≈
(54)
]
Hence we have the following expression for the very beginning of the adsorption process:
κ +c cT ) κ+cT ) k1+Ct (1 - κ-)F1F2
To reduce the constant term of the square root in the expression of σ1 to zero, we apply translation s f s′ - (1 - c1/2)2. This is permissible if we multiply the result of inverse Laplace transformation by exp[- (1 - c1/2)2T]. Equation 59 becomes then
(60)
Omitting the terms with zero and positive integer exponents for s′, we obtain as the first term
F1 - F2 1 1 ) 0 (55) + (1 - F2) (1 - F1) (1 - F1)(1 - F2)
Ξ(T) )
(59)
s(1 - σ1)
2[s′ - (1 - c1/2)2]2
One can see that in this approximation the factor in brackets is equal to zero
[
σ1
Ξ h (s′) ) Ξ h (s + (1 - c1/2)2) ) 2c - 2c1/2 - s′ + s′ 1/2(s′ + 4c1/2)1/2
F2 - F1 κ+c cT Ξ(T) ) (1 - κ )(F1 - F2) (1 - F1)(1 - F2) c c T+ T+ F2(1 - F2) F1(1 - F1) 1/2
Ξ h (s) )
(56)
This is the result, which can be also obtained just by inspecting the first differential eq 3. Long Time Limit. Among of the most important information needed from an analytical method based on adsorption is the time needed to accomplish the process up to the required accuracy. However, to be able to visualize the influence of different parameters, one needs an approximate and reasonably simple formula for estimating the adsorption time. For that purpose we need the asymptotic expansion of function Ξ(T) for large values of T. Most easily the asymptotic expansion is obtained from the Laplace transform Ξ h (s) of eq 38. We may use a general theorem:11 If we can write a power expansion in the Laplace domain where exponents λν are not generally integers
c1/4 s′ 1/2 (1 - c1/2)4
(61)
Taking into account the limit from eq 39 and Γ(-1/2) ) - 2π1/2, we obtain
Ξ(T) ≈ c1/4 c - 1/2 T-3/2 exp[-(1 - c1/2)2T] (62) 1 - c 2π (1 - c1/2)4 Reduced Concentration ) 1. An interesting limiting value is obtained when in addition to the previous assumptions we assume that c ) 1. The approximation is relevant to the case when adsorbate is weakly soluble and the saturation concentration is low. For example, the solubility of various thiols in aqueous solutions is typically low and the adsorption process at reduced concentration tending to unit may approach its limit given by the following theoretical consideration. At c ) 1, using eqs 46 we have eq 59 in form
Ξ h (s) )
σ1 s(1 - σ1)
)
-s + (s(s + 4))1/2
(63)
2s2
In this case there is no upper limit, i.e., the adsorption continues as a deposition process. The series expansion can be done without transformation
Ξ h (s) ≈ s-3/2 - 1/2s-1 + 1/8s-1/2 - 1/128s1/2 + ... (64) We obtain
∞
f(s) )
∑gν(s - F0)λ
(57)
ν
ν)0
Ξ(T) ≈
-N < λ0 < λ1 < ... f ∞ the following asymptotic expansion is valid ∞
gν
t-λ -1 ∑ ν)0Γ(-λ )
L-1{f(s)} ) F(t) ≈ eF0t
ν
) (58)
ν
It should be noted that zero and positive integer values of λν ) n are automatically dropped off from the expansion because Γ(0) ) Γ(- n) ) ∞. All Layers Similar. We take first the simple case κ ) 1 and K ) K1; i.e., we assume that the first layer is similar to the other layers. These assumptions simplify considerably eq 42 giving
1 1 1 T1/2 - + T-1/2 3 2 Γ( /2) 8Γ(1/2) 1 T-3/2 + ... (65) 128Γ(-1/2) 1 1 2 1/2 1 T - + 1/2 T-1/2 + T-3/2 + ... 1/2 2 π 8π 256π1/2
General Case. In the general case we proceed in an analogous way. Starting from eq 38 we get
Ξ h (s′) ≈ (c)5/4κ+[(κ+ - 1)c + 2(1 - κ-)c1/2 + 2κ- - 1] 1/2 4
+
-
(1 - c ) [(κ - 1)c + (2 - κ )c
1/2
-
+ κ - 1]
2
s′ 1/2 (66)
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and the final result is Ξ(T) ≈
γc (1 - c)(1 + (γ - 1)c)
κ+c5/4 exp[- (1 - c1/2)2T][(κ+ - 1)c + 2(1 - κ-)c1/2 + 2κ- - 1] 2π1/2(1 - c1/2)4[(κ+ - 1)c + (2 - κ-)c1/2 + κ- - 1]2
×
T-3/2 (67)
Although this expression is valid for most values of parameters, one should be cautious when applying it. At certain special values of parameters the asymptotic approach becomes exceedingly slow and even divergent. Discussion The general analytical solution for multilayer coverage, eq 50, may be used to compute the time dependence of coverage in those cases, when the interfacial kinetics is relatively slow compared to the rate of diffusive transport. The only essential assumption here is the “Langmurian character” of the adsorption process. Although the Langmuirian adsorption involves the assumption of negligible interaction of adsorbate molecules which may not be realistic in various cases, one should take into account the limiting and ubiquitous role of the BET equation which actually demands the availability of the corresponding kinetic equation, at least for comparison with more realistic adsorption processes. The mathematical method derived in this work allows estimating the influence of the finite adsorption rate on the functional time dependence of coverage given a set of suitable constants. The analytical solution is admittedly rather complicated and consequently the influence of different parameters on the rate is not easily seen. However, the main value of the analytical solution is in the possibility to derive approximations for different cases. The solution is obtained as a function of dimensionless time T ) k-t. Thus plotting the functional time dependence of coverage vs T, one can compare the shapes of the adsorption curves for kinetically different cases. As an example Figure 2 provides the calculated dependence of coverage Ξ on time T for values of κ- < 1 (a) as well as κ- > 1 (b) by using eq 50. In this figure one can see a significant influence of the variation of κ- on the shape of the curves. The same limiting equilibrium value of coverage Ξeq is approached faster when κ- increases. If κ- tends to increase, the adsorption rate especially in the very beginning of the process increases rapidly. The problems in the experimental methods based on the adsorption usually concern the very beginning or the very end of the process. As a rule the analyst, developing analytical methods based on kinetics, is interested in the rate of the process in the very beginning, as the adsorbate and adsorbent have been brought into mutual contact. The measurement of the initial rate is usually done graphically by plotting the coverage (or a physical quantity proportional to it) as the function of time. It is obvious that the procedure is relevant only if the rate law of adsorption is known. Equation 56 provides such an approximation derived for the initial stages of the multilayer adsorption. As one can see in Figure 3, the very beginning of the process can be accurately approximated by the formula 56 giving a linear dependence on time with slope k1+C in the real time scale. In other words, the initial rate of multilayer adsorption is directly proportional to the rate constant k1+ and concentration, meaning that the values
Figure 2. Calculated dependence of coverage Ξ on time T for various values of κ-: (a) κ- < 1 and (b) κ- > 1. The other parameters are γ ) 2 and c ) 0.5, corresponding to the limiting coverage 1.333.
Figure 3. Comparison of different approximations for solving the coverage Ξ vs time T: curve 1, dependence of coverage Ξ on time T calculated from eq 50; curve 2, approximation by eq 56 at initial stages of the adsorption; curve 3, approximation by eq 67 for high values of time. Parameters are γc ) 6, c ) 0.1, and κ- ) 0.15, corresponding to the limiting coverage 0.966.
of the rate constant k1+ can be experimentally obtained from the initial slope of the coverage curve, at least in principle. One should be aware, however, that especially in the very beginning of the adsorption process the loss
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of material in the vicinity of the surface is so fast that the influence of finite diffusion rate may not be neglected. The other end of the adsorption process may be important for various analytical methods focusing on heterogeneous processes followed up close to the equilibrium state. For those cases when rapid assay is essential, one should be able to predict approximately how long it takes until the process is sufficiently quantitative. The present theoretical results provide useful approximations for estimating the time needed for the adsorption process. Thus, Figure 3 illustrates the approximation by eq 67 for high values of time. One can see that at given set of parameters a practically perfect long-time approximation was achieved. The simple formulas derived for the long-time limit allow one to estimate the influence of different variables on the adsorption rate. For practical purposes it is desirable to have a simple formula for easy estimation of time needed to reach a certain fraction of full coverage. The simplest case is when the first layer is similar to the other layers. This is not as crude an approximation as it first may appear. Often the first monolayer is adsorbed very rapidly and its influence on the total rate may be negligible. The monolayer produced in this fast process forms now actually the first surface on which the closely similar adsorbate molecules are adsorbed forming the multilayer system. For this variant the expression 62 received above can be written in the following form
Ξ(T) ≈ Ξeq - ΛT-3/2 exp[-(1 - c1/2)2T]
(68)
where the parameters Λ and Ξeq are
Λ)
c1/4 2π1/2(1 - c1/2)4
Ξeq )
(69)
c 1-c
If we want to estimate the time needed to reach the state close to the equilibrium value, e.g., (1 - h)Ξeq where h is a small fraction of the total coverage, we have to solve the following equation for Th
Th-3/2 exp[-(1 - c1/2)2Th] ) h(Ξeq/Λ)
(70)
Taking the inverse and raising to power 2/3 we obtain
[
] ( )
[
]
2 Λ Th exp (1 - c1/2)2Th ) 3 hΞeq
2/3
(71)
and further
2 2 (1 - c1/2)2Th exp (1 - c1/2)2Th ) 3 3 Λ 2 (1 - c1/2)2 3 hΞeq
( )
2/3
(72)
This can be solved by using the Lambert W function12
Th )
[
( )]
3 2 Λ W (1 - c1/2)2 1/2 2 3 hΞ 2(1 - c ) eq
2/3
Lambert W function can be calculated by using the major mathematical program packages, e.g., Mathematica, Maple, or Mathcad. In the present case the argument of W function is large and we may use its asymptotic expansion
W(z) ) ln z - ln ln z +
ln ln z ln ln z(ln ln z - 2) + + ... (74) ln z 2 ln2 z
Taking only the first term gives
Th ≈ )
[
( )]
3 2 Λ ln (1 - c1/2)2 1/2 2 3 hΞ 2(1 - c ) eq
2/3
(75)
(3/2) ln(2/3) + 3 ln(1 - c1/2) + ln(Λ/Ξeq) - ln h (1 - c1/2)2
This equation allows estimation of the order of magnitude of time needed for the adsorption process to reach a certain percentage of full coverage, say (1 - h) × 100% where h , 1. In the same way it is possible to obtain a similar equation for the general case, when the first layer differs from the other layers. Numerous calculations have shown that the equation above predicts a maximal estimate for time values. This practically means that the time needed to achieve (1 - h) × 100% of the equilibrium coverage is always less than Th obtained from eq 75. Conclusions The original BET equation was derived for the gas phase adsorption and assuming negligible interaction of adsorbate molecules. The equation has been strongly criticized for its simplistic character but still it has surprisingly well withstood time and increased sophistication of adsorption models. Its kinetic counterpart has not made its appearance in the literature, presumably due to the uncommon mathematical problem of solving the infinite set of differential equations and the very complicated nature of the solution. Admittedly, the usefulness of eq 50 may be doubtful because the influence of different parameters cannot be directly seen. However, the availability of the full equation allows a graphical estimation of parameters, and above all, it allows the derivation of different asymptotic approximations. The influence of finite diffusion rate has not been taken into account. At first sight this might seem to be a serious limitation. As a matter of fact, in our previous publication we considered the finite diffusion rate in a simple monolayer formation obeying the Langmuirian adsorption law.1 A simple analytical solution was not obtained even in the case of planar geometry, and one can just imagine the expressions obtained when the Fick laws and infinite set of rate equations are fitted together in the case of multilayer adsorption. However, this may be needless in many cases, because the diffusion limitation may be restricted in a short period in the very beginning of the adsorption process and the later stages are governed by the adsorption rate. In these, probably very common cases, the asymptotic rate laws of eqs 62 and 67 may help the experimenters to deduce the influence of different factors on the rate of reaching the coverage reasonably close to the equilibrium state.
(73)
(12) Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jeffrey, D. J.; Knuth, D. E. Adv. Comput. Math. 1996, 5, 329-359.
Acknowledgment. Financial aid from the Academy of Finland is gratefully acknowledged (Grant No. 50537). LA025777F
