Kinetics of Isothermal Adsorption on Energetically Heterogeneous

J. Phys. Chem. B 2000, 104, 9149-9162

9149

Kinetics of Isothermal Adsorption on Energetically Heterogeneous Solid Surfaces: A New Theoretical Description Based on the Statistical Rate Theory of Interfacial Transport Wladyslaw Rudzinski* and Tomasz Panczyk Departament of Theoretical Chemistry, Faculty of Chemistry, Maria Curie-Sklodowska UniVersity, Place Marii Curie-Sklodowskiej 3, Lublin, 20-031, Poland ReceiVed: January 5, 2000; In Final Form: July 20, 2000

Starting from the fundamental equations of the statistical rate theory of interfacial transport (SRTIT), a set of three equations has been developed, resulting in a new description of the kinetics of localized gas adsorption/ desorption on energetically heterogeneous solid surfaces. For the Langmuir model of adsorption, these equations take the following form: dθt/dt ) 2kTχc(c)K ˜ gs[Kpec/kT - 1/(Kp)e-c/kT], θt(c,T) ) ∫0∞χ()[{exp(( - c)/ kT)}/{1 + exp(( - c)kT)}] d, and χc(c) ) ∫0∞χ()[{(1/kT) exp(( - c)/kT)}/{[1 + exp(( - c)/kT)]2}] d, where θt is the average surface coverage, t is time, T and k are the absolute temperature and Boltzmann constant, respectively, p is the nonequilibrium pressure in the experiment, K ˜ gs is an expression depending on the conditions at which the experiment is carried out, K is a temperature-dependent constant, and χ() is the normalized differential distribution of the number of adsorption sites at various values of the adsorption energy . Two adsorption energy distributions, one Gaussian-like and one rectangular, were taken into consideration since they lead to the two well-known isotherm equations, the Langmuir-Freundlich and the Temkin isotherms, commonly used to describe adsorption equilibria. By accepting these energy distributions, one arrives at two kinetic expressions θt(t) which after several simplifying assumptions reduce to the wellknown power-law and Elovich equations. The new equations have been subjected to an exhaustive numerical analysis, and then used to interpret some experimental data reported in the literature. Both the constant K and the adsorption energy distribution χ() can be calculated from equilibrium adsorption isotherms by this analysis. We have therefore shown that by measuring just the adsorption equilibria it is possible to predict the related behavior of adsorption kinetics. In the theoretical procedures available in the literature, only the reverse operation was possible.

Introduction In many adsorption processes important for life, environment, and technology, the time dependence of these adsorption processes is at least as important as their behavior at equilibrium. However, the kinetics of adsorption/desorption on/from nonporous solid surfaces has received most attention in the catalysis literature, not in the literature on adsorption. The reason is the importance of adsorption kinetics in catalytic reactions occurring on solid catalyst surfaces. Until very recently, the theoretical interpretation of adsorption/ desorption kinetics was based on the absolute rate theory (ART) as a representation of the mechanism of adsorption/desorption processes. The simplest application of that strategy led to the Wigner-Polanyi expression for the rate of adsorption (dθ/dt)1

dθ ) Ka p(1 - θ)se-a/kT - Kdθse-d/kT dt

(1)

where θ is the fractional surface coverage, t is time, s is the number of adsorption sites involved in an elementary adsorption/ desorption process, a and d are the activation energies for adsorption and desorption, respectively, Ka and Kd are temperature-dependent constants, k is the Boltzman constant, and T is the absolute temperature. * To whom correspondence should be addressed. Tel: +48 81 5375633. Fax: +48 81 5375685. E-mail: [email protected].

For one-site-occupancy adsorption (s ) 1), at equilibrium, (when dθ/dt ) 0) expression 1 leads to the Langmuir isotherm

(kT ) exp( ) kT

Kp(e) exp θ(e)(p,T) ) 1 + Kp(e)

(2)

where K ) Ka/Kd, ) d - a, and the superscript (e) refers to equilibrium. For that reason, expression 1 has also been called Langmuirian kinetics. The inability of eq 1 to correlate all experimental data for adsorption/desorption kinetics was detected soon after Langmuir published his kinetic derivation of eq 2.2 Instead, the empirical equation, introduced by Roginski and Zeldovich,3,5 but commonly called the Elovich equation, has been widely used:

dθ ) a exp(-bθ) dt

(3)

where a and b are some temperature-dependent constants.2-7 Provided that at t ) 0, there already exists an adsorbed amount θ ) θ0, the integration of eq 3 yields

1 1 1 θt ) ln ba + ln t + ebθ0 b b ba

[

]

(3a)

Integral data for adsorption kinetics are often presented as θt

10.1021/jp000045m CCC: $19.00 © 2000 American Chemical Society Published on Web 09/08/2000

9150 J. Phys. Chem. B, Vol. 104, No. 39, 2000 vs ln(t + t0), where t0 ) 1/ba exp(bθ0) to make the plot linear. Such linear plots are called Elovich integral plots. Attempts to find a rationalization of the Elovich eq 3 concentrated on considering the energetic heterogeneity of the solid surfaces for gas molecules adsorbing on various adsorption sites.6-9 Recently, Rudzinski and Panczyk published an exhaustive analysis10 of existing rationalizations of the Elovich equation found in the literature and based on the Wigner-Polanyi (ART) expression 1 for the kinetics of adsorption on an energetically heterogeneous surface. That analysis showed the fundamental difficulties that one faces when trying to arrive at the Elovich equation by a rigorous generalization of expression 1, based on the ART for the case of an energetically heterogeneous surface. The feeling that the concepts of the absolute rate theory may not be a good representation of the nature of adsorption/ desorption processes started growing as early as the beginning of the 1960s. In 1957 Kisliuk11 proposed an improvement of the ART based on the concept of a “precursor state”, which concept was further modified by King12 and by Gorte and Schmidt.13 Still another theoretical approach to adsorption/desorption kinetics was proposed by Kreuzer and Payne.14,15 Both these concepts gained only a limited popularity in practical analysis of experimental data. Nevertheless, the interpretation of isothermal kinetics data was often carried out in terms of the Elovich equation, while the interpretation of thermodesorption data was carried out in terms of the Wigner-Polanyi eq 1, adapted in a crude way to describe thermodesorption from energetically heterogeneous surfaces. That adaptation led to a well-known difficulty, which has not yet been solved: using this approach, the preexponential constant Kd, determined from an Arrhenius plot often varies over several orders of magnitude. Seebauer et al.16 have reviewed various theoretical representations for the preexponential factor Kd and showed that none of them is able to explain such strong variations in Kd. As we have already mentioned, Rudzinski and Panczyk have recently identified the difficulties in rationalizing the Elovich equation for isothermal kinetics, by using the concepts of the absolute rate theory.10 One fundamental failure of ART and its extensions is that it does not allow us to describe the dependence of adsorption kinetics on surface coverage while it is known that in many adsorption systems the “sticking probability” or “sticking coefficient” depends on the instantaneous coverage. The SRTIT approach used here was therefore developed with the objective of filling this need. Since it was first proposed in an elementary form in 1977,17 the SRTIT approach has been placed on firmer theoretical grounds18,19 and used to examine the rates of gas absorption at a liquid-gas interface,20-22 hydrogen absorption by metals,23 electron exchange between ionic isotopes in solution,18,24 permeation of ionic channels in biological membranes,25 and both nondissociative24,26,27 and dissociative26,27 adsorption kinetics in isothermal isobaric systems. In each case, the SRTIT approach led to improvements in the theoretical description of the rate process. Initially, application of this approach to adsorption kinetics was limited to systems for which both the gas-phase pressure and the temperature could be assumed to be constant during the adsorption process.24,26-28 More recently, Ward and co-workers have applied SRTIT to describe the kinetics of thermodesorption.29,30 Rudzinski and Aharoni31,32 have also presented a new development of the Elovich equation, using the axioms of SRTIT.

Rudzinski and Panczyk By assuming that transport between two phases at thermal equilibrium results primarily from single molecular events, an equation for the rate of transport between a gas and a solid phase was developed by Ward and co-workers using a first-order perturbation analysis of the Schroedinger equation and the Boltzmann definition of entropy

[ (

)

(

)]

dθ µg - µs µs - µ g ) K′gs exp - exp dt kT kT

(4)

where µg and µs are the chemical potentials of the adsorbate in the gas phase (g) and in the adsorbed (surface) phase (s), respectively, and K′gs represents the rate of the elementary adsorption/desorption processes at equilibrium. Ward and Findlay point out that eq 4 does not apply at extremely low and extremely high values of θ. In their derivation, Ward and Findlay made the assumption that the transient surface configurations of surface-adsorbed molecules are close to these present at equilibrium at the same surface coverage. Rudzinski and co-workers have published a series of papers aimed at applying the SRTIT theory to describe the kinetics of adsorption/desorption on/from energetically heterogeneous solid surfaces. These papers dealt with both isothermal processes,10,31-33 and processes of thermodesorption,32,34-36 commonly used to study the energetic heterogeneity of adsorbents and catalysts surfaces. In applying the SRTIT theory to the kinetics of adsorption/ desorption processes, Rudzinski et al. used the “condensation approximation”37,38 to handle the effects of solid surface heterogeneity. They also limited their theoretical consideration to the rectangular adsorption energy distribution. This approximation is known to present an adequate description of the surface energetic heterogeneity of strongly heterogeneous surfaces. In the case of moderately heterogeneous surfaces, Gaussian-like functions are better at representing the nature of surface energetic heterogeneity. The purpose of this publication is to provide rigorous solutions for the above problem, and to study the errors which result from applying the condensation approximation. We will also consider the kinetics of adsorption on moderately heterogeneous solid surfaces, characterized by Gaussian-like adsorption energy distributions. At the same time, we will limit our discussion to considering only the Langmuir model of adsorption. In this way, we will neglect the interactions between adsorbed molecules while emphasizing the effects due to energetic heterogeneity of solid surfaces. In effect, the present study is oriented more toward chemisorption systems, in which gas-solid interactions play commonly more pronounced role than the interactions between adsorbed molecules. Theory 1. Application of SRTIT to the Langmuir Model of Adsorption on Energetically Homogeneous Solid Surfaces. The way of applying eq 4 will depend on the features of an adsorption system under consideration. When the adsorbed phase is highly mobile, the fast translation of admolecules over the solid surface makes the adsorbed phase act as one thermodynamic entity. In such a case, one has to replace µs in eq 4 by an expression developed for mobile adsorption on a heterogeneous solid surface. For practical applications (chemisorption systems), models of localized adsorption are much more interesting. In this case, adsorbed molecules are trapped in deep local minima of the gas-surface potential, with high energy barriers to lateral

Adsorption on Energetically Heterogeneous Solid Surfaces translation from one local minimum to another. These local minima are commonly called “adsorption sites” or “adsorption centers”. This, for instance, is the case of adsorption accompanying catalytic processes occurring on solid surfaces. The energetic surface heterogeneity in most such cases is a consequence of geometric surface heterogeneity. Distortions in surface structure lead not only to the appearance of different adsorption sites, but also to energy barriers separating these sites. On heterogeneous surfaces, the experimentally monitored overall kinetics is an average of the rates of several elementary adsorption/desorption processes, each taking place on sites with different adsorption features. In order to quantify the overall kinetics, one therefore has to deal with a collection of independent subsystems, each having different adsorption features. However, since the adsorption/desorption is assumed to run at quasi-equilibrium conditions, the local occupancy of the various adsorption sites at a certain average surface coverage θt will be the same as that at equilibrium. One can then define the chemical potential of the adsorbed phase, µs, as a function of average surface coverage θt, but one cannot use it in eq 4. This is because the rate of the elementary adsorption/desorption processes on different sites is different. Equation 4 is used as an expression for the “local” kinetics of adsorption/desorption on a certain subclass of adsorption sites having the same adsorption properties. For models of localized adsorption these adsorption properties are linked to µs through the molecular partition function of adsorbed molecules. Let µg in eq 1 be the chemical potential of the gas in the bulk phase. Then, assuming ideal behavior

µ ) g

µg0

+ kT ln p

θ s q (1 - θ)

() ()

µg0 K′d ) s exp kT q

(7)

(8)

To a good approximation, K′gs can be written as a following product:

K′gs ) Kgs p(e)(1 - θ(e))

(1 - θ) dθ 1 ) Ka p exp - Kd θ exp dt θ kT p kT 2

( )

( )

(11)

where

˜ gsq0s exp Ka ) K

Kd )

K ˜ gs

()

( )

exp s

q0

µg0 kT

µg0 kT

(12)

At equilibrium, when (dθ/dt) ) 0, eq 11 yields the Langmuir isotherm

θ)

Kp(e) exp(/kT) 1 + Kp(e) exp(/kT)

and where

µg0 K′a ) K′gs q exp kT K′gs

(10)

With this notation eq 7 takes the form

(6)

where s

qs ) q0s exp(/kT)

(5)

where qs is the molecular partition function of the adsorbed molecules. Equation 4 takes then the following form:

1 θ dθ 1-θ ) K′a p - K′d dt θ p1-θ

A common arrangement for pressure kinetic adsorption experiments is a beam-dosing experiment in which a solid sample is placed within a large vacuum chamber. This is a “volume dominated” experiment and p(e) ≈ p.29 We will also assume that the gas/solid system is not far from equilibrium, so, θ(e) ≈ θ during the kinetic adsorption experiment. The energetic heterogeneity of the solid surfaces is mainly described by the variation of the adsorption energy, , in the molecular partition function, qs, for molecules adsorbed on different sites37,38

K ˜ gs ) Kgs p(e)

Let µs be the expression for the chemical potential of adsorbed molecules, corresponding to the Langmuir model of adsorption (one-site-occupancy monolayer adsorption, no interactions between the adsorbed molecules). Then for an energetically homogeneous lattice of adsorption sites

µs ) kT ln

J. Phys. Chem. B, Vol. 104, No. 39, 2000 9151

(9)

where the equilibrium state is now defined to be the one to which the system would evolve after it has been isolated at a certain coverage θ, and a fixed amount of the species in the gas phase.29

K ) q0s exp

() µg0 kT

(13)

Notice that Langmuir’s eq 13 is obtained by replacing µs as defined in eq 6, by µg as defined in eq 5. One fundamental feature of the SRTIT approach is that it makes it possible to predict features of adsorption/desorption kinetics from equilibrium adsorption isotherms. For that reason, our extension of SRTIT to the kinetics of adsorption/desorption on/from energetically heterogeneous solid surfaces must be preceded by the development of theoretical expression for equilibrium adsorption isotherms. Therefore, the next two sections will be devoted to developing theoretical isotherms of adsorption for certain models of heterogeneous solid surfaces. Later, theoretical expressions will be developed for the isothermal kinetics of adsorption. 2. Features of Equilibria of Adsorption on Hetreogeneous Surfaces Corresponding to the Langmuir Model of Adsorption. As we have stated, the purpose of the present publication is to provide exact solutions, within the constraints of the Langmuir adsorption model accepted here, and to compare the behavior of the exact solutions to the approximate solutions developed in our previous publications based on the condensation approximation (CA). For convenience we will first summarize the principles of that approach.

9152 J. Phys. Chem. B, Vol. 104, No. 39, 2000

Rudzinski and Panczyk

Figure 1. Effect of temperature on the shape of the function θ() defined in eq 19 when Kp(e) ) 1. The solid line (s) is for T ) 50 K, the line of long dashes corresponds to T ) 200 K whereas the line of short dashes is for 300 K. The dotted line is the condensation function θc() defined in eq 20.

Figure 2. Derivatives of the functions shown in Figure 1.

function, and θt in eq 17 is given by

limθt ) Tf0

We rewrite eq 6 in the following form:

( ) ( )

- c kT θ) - c 1 + exp kT exp

(14)

(15)

For both equilibrium and nonequilibrium conditions, from eq 6 we have

c ) -µs - kT ln q0s

(16)

The average surface coverage θt measured by experiment, is given by

θt )

∫Ω θ(,c,T)χ() d

(18)

Figure 1 shows the temperature dependence of the function θ(,c,T)

θ() )

1 + exp(r/τ)

Tf0

∂θ 1 ) θ(1 - θ) ∂ kT

{

0 for < c 1 for g c

(23)

and the variance of (∂θ/∂) is given by πkT/x3. In the limit T f 0, (∂θ/∂) tends to the Dirac delta function δ( - c). At finite temperatures (∂θ/∂) reaches its maximum at the point c, at which (∂2θ/∂2) ) 0

∂ 2θ 1 ) θ(1 - θ)(1 - 2θ) ) 0 2 ∂ (kT)2

(24)

i.e. θ(c) ) 1/2. To analyze errors inherent in CA, we use the RJ approach.38 Thus, the integral (17) is evaluated by parts

θt( p,T) ) θ(, p,T)X ()|+∞ 0 -

X () d ∫0+∞(∂θ ∂ )

(25)

(19)

where r ) ( - c)/kT0, and τ ) T/T0, are dimensionless quantities. We see that the kernel θ() in eq 13 is a steplike function of , which becomes the step function θc()

limθ() ) θc() )

(22)

and the integration constant is chosen so that X (∞) ) 0. Calculating θt in this way is a long-established procedure in theories of adsorption equilibria, and constitutes the “condensation approximation” (CA). However, the condition T f 0 is not the critical one for the CA approach to be applicable. The essential condition, as we will soon show, is that the variance of the function χ() be larger than the variance of the derivative (∂θ/∂). To ensure this we first consider the derivatives (∂θ/∂) of the functions θ() shown in Figure 1. These derivatives are shown in Figure 2. Note that

(17)

where χ() is the differential distribution of the number of adsorption sites, among corresponding values of , and Ω is the physical domain of . For mathematical convenience it is common practice to assume Ω to be the interval (0,+∞). This assumption does not have a significant effect on the calculated values of θt, except for very high, (θt f 1), or very low (θt f 0), surface coverages.25 So, we assume that

exp(r/τ)

∫χ() d

X (c) )

∫0+∞χ() d ) 1

(21)

c

where

where at equilibrium

c ) -kT ln Kp(e)

∫+∞χ() d ) -X (c)

(20)

in the limit T f 0. This means that at moderate temperatures, adsorption and desorption will proceed in a fairly steplike fashion in the sequence of decreasing (adsorption) or increasing (desorption) energies . In the limit T f 0, θ() becomes an ideal step

From this it can be shown that except for very small and very high values of θt the first term on the rhs of eq 25 is negligible compared to the second. Next, the second term is evaluated by expanding X () into its Taylor series around the point ) c, at which (∂θ/∂) reaches its maximum

-

()

∫0+∞ ∂θ ∂

X () d )

( )[

∫0+∞ ∂θ ∂

X (c) +

( )

1 ∂2X 2 ∂2 Thus

c

(∂X ∂ )

c

( - c) +

]

( - c)2 + ... d (26)

Adsorption on Energetically Heterogeneous Solid Surfaces

θt ) -X (c) -


( )∫ ( ) ( )∫ ( ) ∂X ∂

+∞

c

0

1 ∂2X 2 ∂2

∂θ ( - ) d ∂ c +∞

c

0

∂θ ( - )2 d + ‚‚‚ (27) ∂ c

The first leading term on the rhs of eq 27 is the result obtained by applying the condensation approximation. Because of the symmetry of the function (∂θ/∂), the second term on the rhs of eq 27 almost vanishes and the first nonvanishing correction to -X (c) is the third term on the rhs of eq 27. For the Langmuir model of adsorption, to a good approximation 2

π (c - )2 d ) (kT)2 ∫0+∞(∂θ ) ∂ 3

(28)

Thus, χ() is not the only function which can appear in the theoretical formulation of adsorption on heterogeneous solid surfaces. Every time one assumes

φ() d ) φ(c) ∫0+∞(∂θ ∂ )

(29)

one accepts the accuracy level corresponding to the condensation approximation. As was true in the case of θt, φ(c) will be the leading term, approximating (usually pretty well) the exact value of the integral. It can also be shown that at finite temperatures, φ(c) yields a fairly exact value of integral (25) if the variance of φ() is somewhat larger than the variance of (∂θ/∂).23 The function χc(c), when it is calculated from the relation (17)

χc(c) ) -

∂θt ∂c

(30)

and related to θ() and χ() by the following expression

χc(c) )

χ() d ∫0+∞(∂θ ∂ )

(31)

is usually called the “condensation function” and corresponds to the actual adsorption energy distribution χ(). In the limit T f 0, when the derivative (∂θ/∂) becomes the Dirac delta function [δ( - c)], χc(c) becomes the exactly the function χ(). 3. Adsorption on Energetically Heterogeneous Surfaces Characterized by the Gaussian-like and the Rectangular Adsorption Energy Distributions. It has been shown that certain empirical isotherms describe adsorption in many real systems very well.37,38 It has also been shown that the applicability of these equations has its source in the energetic heterogeneity of solid surfaces and each of these isotherms is related to a specific form of adsorption energy distribution χ(). One of the goals of this publication is to see whether these isotherms have their counterparts in equations for adsorption kinetics. An interesting starting point would be to study adsorption kinetics in systems whose equilibrium coverage is described by such isotherms. Let us assume that χ() is represented by the following Gaussian-like function

( ) ( )]

- 0 1 exp R R χ() ) - 0 1 + exp R

[

2

(32)

Figure 3. Function θ(,θt) defined in eq 37, drawn for three values of the heterogeneity parameter: R/kT ) 2 (- - -), R/kT ) 5 (- - -), R/kT ) 10 (s),when T ) 273 K, and ) 0.

centered at ) 0, the variance of which is πR/x3. Then, χ() is given by

[

χ() ) - 1 + exp

( )] - 0 R

-1

(33)

Accepting the CA level of accuracy θt, defined by eq 21, takes the following form:

[

θt ) 1 + exp

)]

(

c - 0 R

-1

(34)

At equilibrium, θt can also be expressed as a function of the equilibrium bulk pressure p(e), through relation 15:

[

Kp(e) exp

θ(e) t )

[

( )] ( )] 0 kT

kT/R

0 1 + Kp exp kT (e)

kT/R

(35)

Equation 35 is the well-known Langmuir-Freundlich isotherm, which is probably the most frequently used isotherm for correlating experimental data on equilibrium adsorption on heterogeneous solid surfaces. Thus, by accepting function (32) to represent χ(), we may hope to arrive at expressions which will adequately describe the kinetics of adsorption/desorption in a number of real adsorption systems. Since adsorption is assumed to run at quasi-equilibrium conditions, we will have only one value of the chemical potential µs for all the adsorbed molecules. This value will be a function of the average surface coverage θt. The function µs(θt) can therefore be found from eqs 15 and 21, as long as we are satisfied with the CA level of accuracy. For the particular case of χ() in eq 32, the relation µs(θt) takes the following form:

µs(θt) ) - kT ln qs0 - 0 + R ln

θt 1 - θt

(36)

The “local” surface coverage of sites characterized by an adsorption energy , θ(), is dependent on the average surface coverage θt. By expressing c in eq 14 by c calculated from eq 34, we have

( ) ( ) ( ) ( )

θt R/kT - 0 exp 1 - θt kT θ(,θt) ) θt R/kT - 0 1+ exp 1 - θt kT

(37)

Figure 3 shows the θ(,θt) function when ) 0, for three values of the parameter R/kT.



( ) ( )]

- 0 1 exp 1 R kT χ() ) - 0 FN 1 + exp kT

[

2

1e e m

(41)

)]

(42)

where the normalization factor FN is

[

FN ) 1 + exp Figure 4. Effect of the heterogeneity parameter R on the shape of the adsorption energy distribution χ(), given in eq 41. Here EU is an energy unit comparable to kT. In this figure, l ) 2EU, 0 ) 5EU, and m ) 8EU. Values of the parameter R are: 0.5EU (- - -), 1.0EU (- - -), and 5.0EU (s). One can see that as the heterogeneity (parameter R) increases, χ() tends to assume the shape of the rectangular adsorption energy distribution.

One can see that when R/kT becomes large θ(,θt) becomes a step function. This means that, even at finite temperatures, adsorption and/or desorption proceeds in a fairly stepwise fashion, if the surface is strongly heterogeneous. As can be seen in Figure 3 it is not only in the limit T f 0 that the CA approach is applicable. CA is also applicable at finite temperatures if the surface is strongly heterogeneous, i.e., when the derivative (dχ/d) takes on small values. This can be illustrated by considering the first nonvanishing correction to -χ(c) in eq 27

( ) ()

1 ∂ 2X 2 ∂2

2

( )

∞ ∂θ ∂X π (c - )2 d ) (kT)2 ∫ 0 ∂ 6 ∂ c

(

)] [

1 - 0 kT

-1

- 1 + exp

(

m - 0 kT

-1

Figure 4 shows that when R f ∝, the function in eq 41 approaches a rectangular energy distribution. As in the case of the Gaussian-like adsorption energy distribution (32), we first apply the condensation approximation in considering adsorption in these systems. Let us consider the adsorption equilibrium first: (e) θ(e) t (p ,T) )

m kT + ln Kp(e) m - 1 m - 1

(43)

The above equation is the well-known Temkin’s isotherm. This isotherm has frequently been used to describe adsorption in catalytic systems. Figure 5 shows an example of χ() defined in eq 40, along with the function θ() χ(), when the variance of χ(), {1/2x3}(m - 1) is roughly 3 times larger than the variance of (∂θ/∂), i.e., when (m - l) ≈ 6πkT.

(38)

c

We will evaluate an approximate value of this correction for the Gaussian-like function χ(), defined in eq 32. For that purpose we accept the relation between c and θt in eq 34, developed by neglecting the correction given in (38). Then we obtain

dX π2 (kT)2 6 d

( )

c

)

π2 kT 2 θ (1 - θt) 6 R t

( )

(39)

The maximum value of this correction is when θt ) 1/2, i.e., π2/24(kT/R)2 ≈ 0.4(kT/R)2. This correction will decrease not only if T f 0, but when the ratio kT/R decreases. In the following sections such estimates of the correction term in eq 39 will be very important in our estimation of the accuracy of the analytical expressions for adsorption kinetics. The surface energetic heterogeneity of strongly heterogeneous surfaces is frequently described by using the rectangular function:

{

1 for ∈ (1, m) χ() ) m - 1 0 elsewhere

(40)

One may argue that this function does not look like a realistic representation of physical behavior; nevertheless, the above function is often found to be a useful representation of strongly heterogeneous surfaces. The reason for that is as follows: For physical reasons, there must exist a minimum l, and a maximum m, of the adsorption energy . Thus, the Gaussianlike function (32) should be viewed in the way shown in Figure 4. All the functions shown in Figure 4 are normalized to unity and given by the equation

Figure 5. Behavior of χθ (s) when χ() is the rectangular function (40), and l ) 5 kJ/mol, m ) 50 kJ/mol.

4. Using SRTIT To Describe the Kinetics of Adsorption on Energetically Heterogeneous Surfaces and the Langmuir Model of Adsorption, and the Condensation Approximation. The shadowed area in Figure 5 represents the fraction of the surface which is covered, i.e., θt. The rate of adsorption (∂θt/ ∂t) will simply depend on the rate with which the adsorption/ desorption “front” will move with time on the energy scale. This will be true for any adsorption energy distribution, because it follows from eq 17.

dc dθt ∂θt dc ) ) -χc(c) dt ∂c dt dt

(44)

We note that c is the value of for sites whose local coverage θ ) 1/2, so, we define the function F(c,t,T):

1 F(c,t,T) ) θ(c,t,T) - ) 0 2

(45)

Then

[ ]

(∂F/∂t)c (∂θ/∂t) dc ))dt (∂F/∂c)t (∂θ/∂)t

(46)

θ)1/2,)c

To represent (∂θ/∂t)c we use SRTIT as defined in eq 10 with

Adsorption on Energetically Heterogeneous Solid Surfaces ) c, and θ ) 1/2. This gives

(∂θ∂t )

c,θ)1/2

1 1 1 ) Ka p ec/kT - Kd e-c/kT 2 2 p

(47)

Now, because the adsorption/desorption runs at quasi-equilibrium, we can calculate (∂θ/∂) from the related isotherm eq 14.

(∂θ∂)

) )c,θ)1/2

1 4kT

(48)

In this way we arrive at the following general and accurate rate expression for the Langmuir model of adsorption:

dθt 1 ) 2kTχc(c) Ka p ec/kT - Kd e-c/kT dt p

[

]

[

]

(50)

and c(θt) is then given by the simple eq 21. The above equation corresponds to the condensation approximation and was developed in our previous publications where it was used to study adsorption/desorption kinetics.10,31-36 The two above equations can be understood intuitively. We note that the experimentally observed rate of adsorption is controlled by the local rate of adsorption on sites having adsorption energy close to c. The term within the square brackets of eqs 49 and 50 is the SRTIT expression for the rate of adsorption on these sites. The observed overall rate of adsorption dθt/dt should be proportional to that local rate of adsorption, and to the population of these sites on a solid surface, represented by χ(c) giving eq 50. The more or less exact form of χ() can be extracted from equilibrium adsorption data. Usually, it is done by solving the eq 17, with respect to χ(), when θt(p,T) is known from experiment. Equation 17 is a first kind of the Fredholm integral equation and its solution has been the topic of dozens of papers.38 Once we determine χ(), the function c(θt) is calculated from eq 21. In that way we arrive at an explicite form of the kinetic expression (dθt/dt) as a function of the experimentally monitored average surface coverage θt. We would like to stress the important feature of this new approach to adsorption/desorption kinetics: namely, knowledge of the behavior of an adsorption system at equilibrium (the adsorption isotherm equation), makes it possible to predict the behavior of its adsorption kinetics (the expression for dθt/dt). This feature of the new approach needs to be emphasized, since theoretical approaches previously available in the literature do not offer such a possibility. Usually, the kinetic equations are postulated on some rational basis, and then the equilibrium isotherm equation is deduced by putting dθt/dt ) 0.39 It might appear that eq 49 has two unknown constants, Ka and Kd. In fact it has only one constant, K ˜ gs which multiplies ∂θt/∂t, without changing its behavior. To show this we substitute (Ka/Kd) ) K2 and KaKd ) (K ˜ gs)2, and write (49) in the following form:

dθt 1 ˜ gs Kpec/kT - e-c/kT ) 2kTχ(c)K dt Kp

[

]

The Langmuir constant K in eq 51 can be calculated from an appropriate analysis of the equilibrium adsorption isotherm θt. 5. Kinetics of Adsorption on Heterogeneous Solid Surfaces Characterized by Gaussian-like and Rectangular Adsorption Energy Distributions. Equations Developed by Applying the Condensation Approximation. We start our considerations using the Condensation Approximation. Let us first consider the case of the Gaussian-like adsorption energy distribution. The coverage dependence of (dθt/dt) is obtained by expressing c in eq 49 by θt(c) defined in eq 21. For the particular case of the Gaussian-like adsorption energy distribution (32), c(θt) is given by eq 34. Then (dθt/dt) in eq 49 takes an explicit form obtained using

c ) 0 + R ln

(49)

In the limit T f 0, or for a strongly heterogeneous solid surface, χc(c) f χ(c). In that case, expression 49 can also be written in the following form:

dθt 1 ) 2kTχ(c) Ka p ec/kT - Kd e-c/kT dt p


(51)

1 - θt θt

(52)

and

1 χ(c) ) θt(1 - θt) R

(53)

dθt 2kTKap 0/kT (1 - θt)(R/kT)+1 ) e dt R θ(R/kT)-1 t

2kTKd -0/kT θ(R/kT)+1 t e (54) Rp (1 - θ )(R/kT)-1 t

Integration of eq 54, with the boundary condition θt(t)0) ) 0, yields:

θt )

[x [x

Ka 0/kT e p tanh{2xKaKdt} Kd

1+

]

kT/R

Ka 0/kT e p tanh{2xKaKd t} Kd

]

kT/R

(55)

The above equation can also be written in the following form:

θt )

[Ke0/kTp tanh{2K ˜ gst}]kT/R 1 + [Ke0/kTp tanh{2K ˜ gst}]kT/R

(56)

Note the striking similarity of expression 56 to the LangmuirFreundlich isotherm (35) which in fact is the limit of expression 56 when t f ∞. eq 56 is thus the sought-after counterpart for adsorption kinetics to the Langmuir-Freundlich isotherm for adsorption equilibria. The kinetic eq 56 is published here for the first time, so, we decided to carry out an analysis of its characteristic features. Figure 6 shows the time dependence of surface coverage θt(t), predicted by eq 56 for various values of the nonequilibrium pressure p. Part B is the “Elovich representation” commonly used to display experimental data for adsorption kinetics. The behavior of θt(t) shown in Figure 6 is often seen in reported experimental data. In Figure 7 we show two examples of such behavior from the literature. Figure 8 shows the effect of surface energetic heterogeneity on the time dependence of θt. The isothermal kinetics of desorption has rarely been studied; on the other hand, there is a large body of experimental data on the isothermal kinetics of adsorption.42 Moreover, the experimental studies of adsorption were often focused on low coverages in order to test various expressions for adsorption



ln θt )

Figure 6. θt(t) in eq 56 drawn as a function of the dimensionless time 2K ˜ gs (A), and of log 2K ˜ gs (B), for various values of the dimensionless pressure Kp exp(0/kT); 0.1 (- - -), 1.0 (- - -), and 10 (s), when the dimensionless heterogeneity parameter kT/R ) 0.5.

[

{

|

( ) |}] | ( ) |}]

θt0 R/kT 1 0/kT 1 - θ Kpe t0 θt0 R/kT 1 0/kT 1 + Kpe tanh 2K ˜ gst + ar tanh Kpe0/kT 1 - θt0 Kpe0/kT tanh 2K ˜ gst + ar tanh

[

{

kT/R

kT/R

(60)

whereas for the simplified eq 59, we have

θt )

Figure 8. Function θt(t) given in eq 56 drawn for three values of the heterogeneity parameter kT/R; 0.2 (- - -), 0.5 (- - -), and 0.8 (s), when Kp exp(0/kT) ) 1.

(59)

Thus, we have arrived at the empirical power law, which has been used to correlate experimental data for adsorption kinetics. We can now see eq 56 to be a generalized form of the powerlaw equation, valid for the whole region of surface coverages, and taking desorption into account. As for the accuracy of eq 56, the errors introduced by applying the CA approach will be the same as in the case of adsorption equilibria. This means that the main correction term will have the same form as in eq 39. This in turn means that errors will be largest when θt ) 1/2, and that eq 56 will become invalid at low (θt f 0) and high (θt f 1) surface coverages. Experimental kinetic data measured at low coverage should be treated with caution, when subjected to quantitative theoretical interpretation using eq 56. The boundary condition θt(t)0) ) 0 may not be valid in many kinetic experiments. Some preadsorbed amount may be present and the only question is whether, for a particular case, it can be ignored in a quantitative analysis of experimental data. In cases where at t ) 0, a certain amount θt0 is already present on the surface and eqs 56 and 59 take the following form. For eq 56 θt )

Figure 7. (A) Effect of pressure on the kinetics of CO2 adsorption on Sc2O3, shown in Figure 4 of the paper by Pajares at. al.40 (B) Effect of pressure on the kinetics of O2 adsorption on pollycrystalline tungsten, reported by Lopez-Sancho and de Segovia.41

0 kT kT + ln 2Kap + ln t R R R

[

2kTKap 0/kT e t + θR/kT t0 R

]

kT/R

(61)

Changing experimental conditions (pressure, temperature) at which a kinetic experiment is carried out may cause changes in the preadsorbed amount θt0, and these changes may in turn lead to serious misinterpretation of the measured kinetic data. To show the risk of such misinterpretation, we have carried out the following model investigation. We have drawn the quantity (θt - θt0) calculated from eq 60 as a function of K ˜ gst. Such functions are in fact the “coverage” determined in experiments, because θt0 is not monitored as a rule. The result of our calculations is presented in Figure 9.

kinetics. Let us therefore neglect for the moment the second term on the rhs of eq 54 and examine the resulting differential equation. Consider initial adsorption in the region of small surface coverages, i.e., when θt f 0. Then eq 54 can, to a good approximation, be rewritten as follows:

θ(R/kT)-1 dθt ) t

2kTKap 0/kT dt e R

(57)

Integration with the boundary condition θt(t)0) ) 0, yields

θt(t) ) (2Kape0/kT)kT/RtkT/R

(58)

This would suggest that, at small surface coverages, ln θt should be the following linear function of ln t:

Figure 9. Experimentally monitored function (θt - θt0), drawn for three values of θt0: θt0 ) 0 (s), θt0 ) 0.2 (- - -), θt0 ) 0.4 (- - -) when kT/R ) 0.5, and Kp exp(0/kT) ) 1.

Let us now consider the case of a strongly heterogeneous solid surface for which the rectangular adsorption energy distribution (40) is appropriate as an approximation for the actual

Adsorption on Energetically Heterogeneous Solid Surfaces energy distribution. Accepting the CA level of accuracy, c(θt) is evaluated from the equation

θt )

m c m - 1 m - 1

(62)

so that

c ) m - (m - 1)θt

J. Phys. Chem. B, Vol. 104, No. 39, 2000 9157 Because the Elovich integral eq 3a has been widely used, to correlate experimental data on adsorption kinetics, it seems worth examining what misinterpretation could arise from its improper application. To do this the Elovich eq 3a was used to fit the theoretical θt(t) curve generated by eq 56, which uses the Gaussian-like adsorption energy distribution (32). Figure 10 shows the result of these computer exercises.

(63)

After inserting (61) into eq 50, we have

{

[

}

˜ gs dθt 2kT K m - (m - 1)θt ) Kp exp dt m - 1 kT -m + (m - 1)θt 1 exp Kp kT

{

}]

(64)

When desorption, represented by the second term within the square bracket, is neglected, one arrives at the well-known Elovich equation. That equation is probably the most frequently used expression to correlate experimental data for adsorption kinetics. This could mean that many real solid surfaces are strongly heterogeneous. The reported deviations from the “Elovich behavior” at higher coverages may be due to the fact that the Elovich equation clearly neglects the effects of the simultaneously occurring desorption. Another reason for such deviations may be the lack of an appreciation that even at low surface coverages the Elovich equation can be applied only in cases of strongly heterogeneous surfaces. In the case of moderately heterogeneous surfaces, the Gaussian-like adsorption energy distribution (32) might better represent the nature of surface energetic heterogeneity. As we can see from the discussion above, in such cases, instead of using Temkin’s isotherm, the Langmuir-Freundlich isotherm should be used to describe adsorption equilibria, while the generalized power law (60) is appropriate for describing adsorption kinetics. We have shown in a previous publication10 that the integration of eq 64 with the boundary condition θt(t)0) ) 0, leads to the following equation for θt(t):

θt )

[

kT × m - 1

{

|

1 1 + Kp exp(m/kT) ln Kem/kTp tanh 2K ˜ gs t + ln 2 Kp exp(m/kT) - 1

|}]

(65)

At small surface coverages, when the second (desorption) term on the rhs of eq 64 can be neglected, the integral expression θt(t) takes the following forms: For the conditions when t ) 0, θt ) 0, we have

θt )

kT ln[2Ka pem/kT + 1] m - 1

(66)

When at t ) 0, θt ) θt0, we obtain

θt )

kT ln[2Ka pem/kTt + e((m-1)/kT)θt0] m - 1

(67)

Notice that eq 67 is essentially the integral Elovich equation, eq 3a, in which

a)

2kTKa p m/kT e m - 1

b)

m - 1 kT

(67a)

Figure 10. (Solid line) Function θt(t) calculated from eq 56 when Kp exp(0/kT) ) 10, and kT/R ) 0.5. (Broken line) the Elovich eq 3a approximating best the solid line, when a ) 21.5, b ) 5.54, θt0 ) 0.018.

Let us assume that the solid line represents the adsorption kinetics on a real, moderately heterogeneous, solid surface. In trying to fit such experimental data, one will discover a good fit using the integral Elovich equation over a large region of surface coverages. At the same time, one will discover serious departures from the Elovich equation in the initial adsorption kinetics. This may easily lead one to the improper conclusion that the kinetics of adsorption obey a different law at low coverages.43 6. Exact Solutions for the Adsorption/Desorption Kinetics, on Heterogeneous Surfaces Characterized by the Rectangular Adsorption Energy Distribution. To study the differences due to using either the approximate expression 50 or the exact expression 49 for dθt/dt, we will use the rectangular function (40) to carry out the appropriate model calculations. This approach makes both θt(c,T) and χc(c) into simple analytical expressions thereby making the presentation clearer. For the rectangular function (40), χc(c) takes the following form:

χc(c) )

{[

(

)]

c - m 1 1 + exp m - 1 kT

[

-1

-

1 + exp

)] }

(

c - 1 kT

-1

(68)

and θt(c,T) in eq 17 can also be evaluated analytically to yield

( (

) )

m - c kT kT ln θt(c,T) ) m - 1 1 - c 1 + exp kT 1 + exp

(69)

After solving eq 69 with respect to c, we arrive at the following expression

(

)

(m - 1)θt -1 kT c(θt) ) -kT ln m (m - 1)θt + 1 exp - exp kT kT exp

()

(

)

(70)


Figure 11. Comparison of χ() defined in eq 40 (s), and the related CA function χc(c) defined in eq 64 (- - -) when l ) 40 kJ/mol, m ) 60 kJ/mol, and T ) 273 K.


Figure 13. Kinetic data (9) for NO adsorption on platinum black at 3-4 Torr and 0 °C, reported by Otto and Shelef.

Figure 14. Equilibrium isotherm of NO adsorption on platinum black at 0 °C (9), reported by Otto and Shelef.44

Figure 12. Comparison of θt(t) calculated from eq 63 (- - -), with the exact θt(t) calculated from eqs 49, 68, and 70 (s), for three pairs of values: (A) l ) 20 kJ/mol; m ) 80 kJ/mol; (Β) l ) 35 kJ/mol; m ) 65 kJ/mol; (C) l ) 45 kJ/mol; m ) 55 kJ/mol. All at Kp ) 10-6, T ) 273 K. The functions θt(t) are drawn as functions of the dimensionless, time 2K ˜ gst.

The function c(θt) from eq 70 is next inserted into eq 49, in which χc(c) is given by eq 68. Figure 11 shows a comparison of an actual rectangular adsorption energy distribution, (40) with its condensation approximation χc(c) given by eq 68. Figure 12 shows a comparison of the function θt(t) calculated from the less accurate expression (65), with the function θt(t) calculated rigorously by a numerical integration of eq 49 in which; χc(c) is given by eq 68; c(θt) is given by eq 70. Figure 12 shows what is to be expected from the general analysis of the applicability of CA. The more heterogeneous the solid surface (that is, the larger the value of (m - l)), the closer is the approximate CA function to the correct θt(t) function. At the same time we can observe some special properties of the rectangular distribution function. For example, the agreement between the CA solution for θt(t) and the true values of that function, calculated from eqs 49, 68, and 70, is generally very good up to surface coverages θt less than one-half. The deviations of the CA solution (65) from the true function are very large at higher surface coverages. Furthermore, a plateau may appear in (65) at surface coverages θt > 1, which is obviously a nonphysical result for the Langmuir model of adsorption being considered here. We face a different situation in the case of the CA function (56), corresponding to the Gaussian-like adsorption energy distribution (32). An analysis of the correction term, presented

in eq 39, suggests that the largest deviations from the true solution θt(t) may appear at half surface coverage, and that these deviations should disappear at both smaller and larger coverages. The calculations presented in Figure 12 suggest certain procedures which should be followed in applying the kinetic isotherms developed here when these are used to correlate experimental data. These are: (a) One should avoid using eq 65 unless the measured kinetic isotherm θt(t) exhibits a sharp transition from one straight line, observed over a wide region of coverages, to another straight line, parallel to the t-axis, at high coverages, as shown in Figure 12A. The lack of such a sharp transition suggests that the surface is not heterogeneous enough for eq 65 to be applicable. In such instances eq 60, developed for the Gaussian-like energy distribution, should be applied. (b) On the other hand, eq 60 cannot be used for systems whose θt(t) kinetics show behavior like that in Figure 12A. One cannot simply use small values of kT/R, because then both the equilibrium and kinetic theoretical isotherms (35) and (60) are greatly influenced by the unrealistic limits (-∞, +∞) inherent to the Gaussian-like distribution (32). Nonetheless, the behavior shown in Figure 12A has been observed in experimental data reported in the literature. This is the case in the study of the kinetics of NO adsorption on platinum black reported by Otto and Shelef. Figure 13 shows the experimental θt(t) curve redrawn from Figure 3 of the paper by Otto and Shelef.44 The isothermal kinetics of adsorption shown in Figure 13 suggests that platinum black has a strongly heterogeneous surface. It is therefore to be expected that the corresponding equilibrium isotherm (θt(e)) should very well be represented by the Temkin isotherm (43). The equilibrium measurements confirmed that expectation. Figure 14 shows the corresponding Temkin equilibrium isotherm, redrawn from Figure 1 of the paper by Otto and Shelef.44 Although, as in the case of adsorption equilibria, the second central moment (the variance) of χ() will be the factor affecting the behavior of θt most strongly, higher central moments (the


Figure 15. (Solid line) Exact function θt(t) corresponding to the rectangular distribution, and calculated from eqs 49, 68, and 70 when Kp ) 10-8, T ) 273 K, l ) 40 kJ/mol, m ) 60 kJ/mol. (Dotted line) Function θt(t) calculated from eq 56 for the same values of Kp and T, the same mean adsorption energy 0 ) 50 kJ/mol, and the same variance 5.77, i.e., R ) 0.71.

Figure 16. Effect of the nonequilibrium pressure p on the behavior of θt(t) calculated from eqs 49, 68, and 70 when l ) 40 kJ/mol, m ) 60 kJ/mol, T ) 273 K. The solid, slightly broken, and strongly broken lines correspond to Kp values equal to: 10-4 (s), 10-6 (- - -), and 10-8 (- - -), respectively.


Figure 18. Effect of the preadsorbed amount θt0 on the experimentally monitored “Elovich plot” when Kp ) 10-6, T ) 273 K, l ) 20 kJ/ mol, m ) 80 kJ/mol, and θt0 ) 0 (s), θt0 ) 0.1 (- - -), θt0 ) 0.3 (- - -).

and is taken to be zero. This type of (experimental) Elovich curve is generally the one reported in experimental works. 7. Kinetics of Adsorption at Low Surface Coverages (Short Adsorption Times). As a final point we would like to raise the issue of the use of the integral Elovich eq 3a rather than its differential form (3). The fact that the classical Elovich eq 3 does not take desorption into account and is the result of an approximation at the CA level of accuracy leads to inaccuracies when one is using that equation to correlate experimental data. As a result, various deviations from the Elovich behavior have been reported and discussed in the literature.42 The integral form (3a) contains one more adjustable parameter, θt0, which should help it give a better fit to experimental data. In spite of that, serious difficulties have been reported in fitting experimental data at low surface coverages. To explain this Aharoni and Ungarish43 proposed a hypothesis invoking the existence of a “pre-Elovich kinetics” which follow behavior other than that described by the Elovich equations. This hypothesis gained considerable popularity in the adsorption literature. It seems, however, that another explanation is possible on the basis of the equations developed in this work. We have shown in Figure 10 that attempts to correlate experimental data using the integral Elovich equation for the systems characterized by a Gaussian-like adsorption energy distribution may lead to a false hypothesis involving “activated” kinetics at low surface coverages. According to eq 59

ln θt ) Figure 17. Effect of surface heterogeneity on the behavior of θt(t) calculated from eqs 49, 68, and 70, when Kp ) 10-6, T ) 273 K, and for three pairs of l and m values: l ) 20, m ) 80 kJ/mol (s); l ) 35, m ) 65 kJ/mol (- - -); l ) 45, m ) 55 kJ/mol (- - -).

shape) of χ() affect it too. To study that effect, the solid line from Figure 12 corresponding to the rectangular adsorption energy distribution is compared in Figure 15 with the function θt(t) calculated from eq 56 corresponding to the Gaussian-like adsorption energy distribution, when all the adsorption param˜ gs and the variance are the same. eters: Kp, T, 0, K We end our analysis in this section by showing in Figures16 and 17, how pressure, p, and surface heterogeneity, (m - l), affect the exact solution, θt(t), calculated from eqs 49, 68, and 70. Finally, Figure 18 shows how the preadsorbed amount, θt0, affects the integral Elovich representation of experimental data if the preadsorbed amount is not determined in the experiment,

[

]

0 kT R kT ln + ln[t + t0] R 2kTKap R R

(71)

where

( )

0 R exp t0 ) θR/kT t0 2kTKap kT

(72)

Thus, it is the logarithm of θt(t), plotted against ln[t + t0], which should give linear behavior and not θt(t) versus ln[t + t0], as suggested by the integral Elovich eq 3a. Figure 19 shows one danger of misinterpretation of experimental data. When some preadsorbed amount, θt0 * 0, exists at t ) 0, the log[θt - θt0] vs log t plots will be linear, as expected. However, whenever θt0 * 0 the slope of such linear plots will be larger than the kT/R expected on the basis of eq 71. As a consequence, such plots will underestimate the degree of surface energetic heterogeneity. In the case of the rectangular adsorption energy distribution, the analytical consideration is not as straightforward, but the


Figure 19. Behavior of log(θt - θt0) calculated from eq 60 when Kp exp(0/kT) ) 1; kT/R ) 0.5 and for three values of θt0: 0 (s); 0.1 (- - -); 0.4 (- - -).

Figure 20. Three θt(t) functions from Figure 15 shown now in loglog coordinates (θt0 ) 0).

Figure 21. Effect of θt0 on the kinetics calculated from eqs 49, 68, and 70, when Kp ) 10-6, T ) 273 K, l ) 40 kJ/mol, and m ) 60 kJ/mol. The calculations were performed for θt0 ) 0 (s), θt0 ) 0.1 (- - -), and θt0 ) 0.3 (- - -).

effect is the same. Figure 20 shows the three functions θt(t) shown in Figure 17, calculated from eqs 49, 68, and 70, this time plotted using the log-log coordinates of ln θt vs ln t. One can see in Figure 20 that when θt f 0, the log-log plots become linear. That means that the classical integral Elovich eq 3a will not apply at low surface coverages. We see, therefore, that the inapplicability of Elovich equation at low surface coverages does not mean that the kinetics of adsorption at low surface coverages is governed by a different law than that applicable at higher surface coverages. Figure 21 shows how the unobserved preadsorbed amount θt0 can affect the experimentally observed kinetics for systems characterized by the rectangular adsorption energy distribution. We note that real adsorption energy distributions χ() will, in general, be much more complicated than the Gaussian-like function (32) or the rectangular function (40). The two functions we have considered may be thought of as “smoothed” forms of the actual distributions of surface energetic heterogeneity in real

Rudzinski and Panczyk gas-solid adsorption systems. Thus, the equations developed here must be treated as approximate. This understanding is essential to an appreciation of our attempts, presented in the next section, to apply these equations to experimental data. It seems likely that, as long as the surface heterogeneity of a real adsorption system is described by a unimodal function, the equations developed here should be more or less applicable. Serious difficulties may appear in cases, when the true function χ() is composed of two or more separated peaks. Fortunately, the general eq 51 still makes it possible to handle such cases using numerical methods, as long as the shape of the distribution can be described. 8. Discussion of Experimental Results. We have noted in the discussion above that none of the approaches to adsorption/ desorption kinetics available in the existing literature makes it possible to predict adsorption kinetics from the behavior of an adsorption system at equilibrium, as represented by the adsorption isotherms. This explains why experimental studies of adsorption kinetics and equilibria developed along two separate paths with little interaction between them. Simultaneous measurements of kinetic and equilibrium adsorption isotherms were rarely reported in the literature. Most of the papers on adsorption kinetics were published in catalytic journals. There, due to the strong energetic heterogeneity of most catalysts surfaces, the measured experimental isotherms could usually be correlated by using the Temkin equation, which had long been associated with a rectangular adsorption energy distribution. At the same time it was noted that the adsorption kinetics in these systems could often be represented by the Elovich equation, which was also associated with a rectangular adsorption energy distribution, though this derivation of the Elovich equation was criticized by Rudzinski and Panczyk.10 Nonetheless, what was essential in this was the understanding that the Temkin equilibrium isotherm, and the Elovich kinetic expression are related by assuming the same energetic properties of the solid surfaces. That idea encouraged some experimentalists to carry out simultaneous measurements of adsorption equilibria and adsorption kinetics, in order to examine the relationship between Temkin’s and Elovich’s parameters. Unfortunately, a lack of understanding of just how the adsorption energy distribution simultaneously affects adsorption equilibria and kinetics led many authors to an erroneous analysis of the experimental data. The data for adsorption kinetics in such works were normally correlated using the Elovich equation, whereas the adsorption data were correlated using plots of logarithm of adsorbed amount vs logarithm of pressure. As we now know, such plots should only be used in the case of the Freundlich isotherm. We also know that the Freundlich equation is the low-pressure limit of the more general Langmuir-Freundlich equation, eq 35, corresponding to the Gaussian-like adsorption energy distribution (32) whereas Elovich kinetics arise from a rectangular energy distribution. For example, Figure 7A shows the kinetics (θt(t)) of CO2 adsorption on scandia (Sc2O3) measured by Pajares et al.40 at 250 °C. In another paper,45 Pajares et al. reported on corresponding adsorption measurements carried out at equilibrium. To correlate their kinetic data, they initially introduced an empirical equation, which they later abandoned and returned to using the Elovich equation. At the same time, they used loglog Freundlich plots to correlate their experimental isotherms measured at equilibrium.



TABLE 1: Values of the Parameters Found by Fitting Eq 56 to the Kinetic Data for CO2 Adsorption on Scandia, at Various Nonequilibrium Pressures, As Reported by Pajares et al.40 a best fit

monitored P (Torr)

Nm (µg/g)

2K ˜ gs

K exp(0/kT)

kT/R

calculated θt(e)

2.5 3.8 4.9 8.0

1160.0 1038.3 1047.4 942.8

0.02080 0.03698 0.03785 0.05412

0.265 0.265 0.265 0.265

0.74 0.74 0.74 0.74

0.424 0.501 0.548 0.635

a The parameter N is the value of the monolayer capacity, i.e. θ m t ) Nt/Nm.

Figure 22. (Solid lines) Theoretical values, calculated from eq 56, by using the best-fit parameters reported in Table 1.

Apart from the fact that the Elovich and Freundlich equations are not theoretically compatible for adsorption kinetics and equilibria, use of the log-log plot in this instance was inappropriate for another reason. Recall that the Freundlich equation is the low-coverage limit of the Langmuir-Freundlich isotherm (35), so that the log-log correlation, when applied to the Freundlich equation, can only be used for low surface coverages while Pajares et al.44 used the log-log plots for data, corresponding to high surface coverages. This led Pajares et al. to estimate unrealistically small values of kT/R. Such small kT/R values, when used in the kinetic rate expression (56), made it impossible to fit the experimental kinetic curves of θt(t) for any values of the other parameters. To see what might be going on, we decided to fit the kinetic data shown in Figure 7A, by eq 56, in which kT/R was treated as a best-fit parameter. Figure 20 shows the results of this fitting. Table 1 reports the corresponding parameter values. The strategy used in our bestfit calculations needs to be understood. It was impossible to fit the θt(t,p) data for all the nonequilibrium pressures p using just one set of the parameters, Nm, 2K ˜ gs, K exp(0/kT), and kT/R. That was not a surprise, in view of our experience with other studies of equilibrium isotherms of adsorption on energetically heterogeneous solid surfaces. The reason for this is as follows.

Figure 23. Dependence on the pressure p of the best-fit K ˜ gs values collected in Table 1.

It is generally known, and frequently reported in the literature on adsorption equilibria, that real adsorption energy distributions, in real adsorption systems are complicated functions. Even their “smoothed” forms cannot, as a rule, be approximated by Gaussian-like functions. A variety of more complex analytical functions have been used in the literature to approximate these complex forms of the function χ(), always with limited success.37,38 The Gaussian-like function (32), can approximate the real function χ() in a limited range of adsorption energies (1,2), with a constant set of parameters R and 0. If the isotherm is studied at conditions such that 1 < c < 2, the experimental data will then be fairly well approximated by eq 56 using constant values of the parameters R and 0. At the same time the parameter Nm will only be a best-fit parameter, not the true value of monolayer capacity. This feature of adsorption isotherms on energetically heterogeneous solid surfaces has long been recognized. As a consequence it is a common practice to fit adsorption at different surface coverages using different isotherms37,38 (Dubinin-Astakhov, Langmuir-Freundlich, BET, etc.) On the basis of the development presented above we tried to fix, the same values of the parameters K ˜ gs, 0, and R, for all the nonequilibrium pressures, and let Nm take different values for the various pressures. This, however, still led us to a poor fit of the experimental kinetic data. Finally, we fixed only the 0 and R parameters as the same value for all the nonequilibrium pressures, and let Nm and K ˜ gs vary for the several pressures. In this way we arrived at the fit of experimental data presented in Figure 20, using the values of the best-fit parameters shown in Table 1. It seems from our fitting that the values of K ˜ gs are not very sensitive to the values of the other parameters, indicating a small correlation between K ˜ gs, and the other parameters. Figure 23 shows that there is an approximately linear relationship between p and the fitted values of K ˜ gs, in accordance with its interpretation in eq 12. Summary Starting from the fundamental equation of SRTIT (eq 4) for the kinetics of adsorption/desorption on energetically homogeneous solid surfaces, a set of three equations, listed in the abstract, has been developed to describe the kinetics of localized monolayer adsorption on energetically heterogeneous solid surfaces. In the case of strongly heterogeneous surfaces, when χc ≈ χ, that equation system reduces to a set of just two of the equations listed in the Abstract. These equations make it possible to predict the kinetics of an adsorption system from knowledge of its pressure and

9162 J. Phys. Chem. B, Vol. 104, No. 39, 2000 temperature dependence at equilibrium. The developed equations take into account both adsorption and desorption occurring simultaneously. Kinetic equations have been developed which correspond to the isotherm equations of Langmuir-Freundlich and Temkin. Both the isotherms and the corresponding rate expressions thus developed can be used to correlate experimental data on adsorption on energetically heterogeneous solid surfaces. An examination of the behavior of the kinetic equations developed here was carried out by comparing their predictions to the kinetic behavior of real adsorption systems. The parameters found in this fitting were shown to exhibit physically meaningful behavior. Special attention was given to features of adsorption/ desorption kinetics at low surface coverages and short adsorption times. That analysis showed that the classical Elovich equation may erroneously suggest that data obtained during the initial stage of adsorption obey different rate laws than do the data obtained at longer times and higher surface coverages. Acknowledgment. The authors would like to express their thanks to Prof. B. W. Wojciechowski from the Department of Chemical Engineering, Queen’s University in Kingston, Canada, for his assistance during the preparation of this manuscript. References and Notes (1) Clark, A. The Theory of Adsorption and Catalysis; Academic Press: New York, 1970. (2) Langmuir, I. J. Am. Chem. Soc. 1918, 40, 1361. (3) Roginsky, S. Z.; Zeldovich, Ya. B. Acta Phys. Chem. USSR, 1934, 1, 554. (4) Zeldovich, Ya. B. Acta Phys. Chem. USSR, 1934, 1, 449. (5) Zeldovich, Ya. B. Ph.D. Thesis, Leningrad, 1936. (6) Elovich, S. Yu.; Kharakorin, F. F. Probl. Kinet. Catal. 1937, 3, 322. (7) Kharakorin, F. F.; Elovich, S. Ya. Acta Physicochim. USSR 1936, 5, 325. (8) Elovich, S. Ya.; Zabrova, G. M. Zh. Fiz. Khim. 1939, 13, 1775. (9) Aharoni, C.; Tompkins F. C. AdV. Catal. 1970, 21, 1. (10) Rudzinski, W.; Panczyk, T. Surface Heterogeneity Effects on Adsorption Equilibria and Kinetics: Rationalizations of the Elovich Equation. In Surfaces of Nanoparticles and Porous Materials; Schwarz, J., Contescu, C., Eds.; Marcel Dekker: Amsterdam, 1999; pp 355-390. (11) Kisliuk, P. J. Phys. Chem. Solids 1957, 3, 95. (12) King, D. A. Surf. Sci. 1977, 64, 43. (13) Gorte, R.; Schmidt, L. D. Surf. Sci. 1978, 76, 559. (14) Kreuzer, H. J.; Payne, S. H. Thermal Desorption Kinetics. In Dynamics of Gas-Surface Interactions; Rettner, C. T., Ashfold, M. N. R., Eds.; Royal Society of Chemistry: Cambridge, UK, 1991.

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Kinetics of Isothermal Adsorption on Energetically Heterogeneous

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