Remarks on the Current State of Adsorption Kinetic Theories for

Langmuir 2002, 18, 439-449

439

Remarks on the Current State of Adsorption Kinetic Theories for Heterogeneous Solid Surfaces: A Comparison of the ART and the SRT Approaches Wladyslaw Rudzinski* Department of Theoretical Chemistry, Faculty of Chemistry UMCS, pl. Marii Curie-Sklodowskiej 3, Lublin, 20-031, Poland

Tomasz Panczyk Laboratory for Theoretical Problems of Adsorption, Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, ul. Niezapominajek 8, 30-239 Krakow, Poland Received June 25, 2001. In Final Form: October 21, 2001 At the end of the 1980s, the classical theories of adsorption/desorption kinetics based on ideas of the ART (absolute rate theory) were challenged by new theories linking the rate of adsorption/desorption with the chemical potentials of free (bulk) and adsorbed molecules. Among the latter theories, the so-called statistical rate theory (SRT) received probably the most advanced theoretical development. It is based on quantum mechanics and statistical thermodynamics. The appearance of the new theories sometimes gives rise to emotional discussions in the literature, but they do not always focus on fundamentals. The present paper is aimed at comparison of the theoretical predictions of the ART and the SRT approaches with the behavior of adsorption/desorption kinetics in real adsorption systems. That comparison is based on accepting the popular Langmuir model of adsorption, next generalized for the case of energetically heterogeneous solid surfaces, typical for real adsorption systems. For the purposes of comparison, the kinetics of CO2 adsorption on scandia has been subjected to quantitative analysis using the expressions offered by both the classical ART approach and the newer SRT approach. For a certain set of parameters, the SRT theoretical expression reproduces very well the behavior of both kinetic and equilibrium adsorption isotherms. Also, the determined parameters exhibit a fully physical meaning. In the case of the ART approach, one can fit the kinetic adsorption isotherms, using many sets of parameters, but some of the determined parameters always exhibit a nonphysical meaning.

Introduction From the beginning of the 20th century, absolute rate theory (ART) has, almost exclusively, been used for theoretical interpretation of adsorption/desorption kinetic data.1 That approach was based on applying some ideas from the field of chemical reactions, even for the case of physisorption systems. However, from the very beginning dramatic discrepancies were reported between theoretical predictions made with the ART approach and the experimentally monitored kinetics of adsorption/desorption. As a result, many scientists started to use empirical equations (e.g., the Elovich, power-law) to correlate experimental data. Others attempted various improvements of ART, introducing the concept of precursor states, for instance.2-4 However, the fundamentals of the classical ART approach remained unchanged. Finally, at the beginning of the 1980s new theoretical approaches to adsorption/desorption kinetics appeared that challenged the fundamentals of the classical ART approach. They related the kinetics of adsorption/desorption to the chemical potentials of the adsorbed and bulk phase molecules. First, it was the rate of desorption Rd that was related to the chemical potential of adsorbed molecules, µa, * Corresponding author. Phone: +48 81 5375633. Fax: +48 81 5375685. E-mail: [email protected]. (1) Clark, C. A. The Theory of Adsorption and Catalysis; Academic Press: New York, 1970. (2) Kisliuk, P. J. Phys. Chem. Solids 1957, 3, 95. (3) King, D. A. Surf. Sci. 1977, 64, 43. (4) Gorte, R.; Schmidt, L. D. Surf. Sci. 1978, 76, 559.

( )

Rd ∼ exp

µa kT

(1)

This assumption was launched first by de Boer5 in 1956. It next received further rationalization in the work of Nagai that was published in 1985,7,8 but Nagai’s work raised immediate criticism from some of the researchers working in this field. After a few years of intense discussion,6-12 Nagai presented finally an impressive proof13 that eq 1 was able to explain the thermal desorption spectrum of the hard hexagon model, whereas the traditional ART approach was not. Also, Kreuzer’s approach14,15 leads to eq 1 for the rate of desorption, when the system is not far from equilibrium. At the beginning of the 1980s, Ward and co-workers16-34 launched another approach of that kind in which the (5) de Boer, J. H. Adv. Catal. 1956, 8, 1. (6) Nagai, K. Phys. Rev. Lett. 1985, 54, 2159. (7) Nagai, K.; Hirashima, A. Chem. Phys. Lett. 1985, 118, 401. (8) Zhdanov, V. P. Surf. Sci. 1986, 171, L461. (9) Nagai, K.; Hirashima, A. Surf. Sci. 1986, 171, L464. (10) Zhdanov, V. P. Surf. Sci. 1986, 171, L469. (11) Cassuto, A. Surf. Sci. 1988, 203, L656. (12) Nagai, K. Surf. Sci. 1988, 203, L659. (13) Nagai, K. Surf. Sci. 1991, 244, L147. (14) Kreuzer, H. J.; Payne, S. H. Thermal Desorption Kinetics. In Dynamics of Gas-Surface Interactions; Rettner, C. T., Ashfold, M. N. R., Eds.; The Royal Society of Chemistry: Cambridge, 1991; p 221. (15) Kreuzer, H. J.; Payne, S. H. Theories of Adsorption-Desorption Kinetics on Homogeneous Surfaces. In Equilibria and Dynamics Of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: Amsterdam, 1997. (16) Ward, C. A. J. Chem. Phys. 1977, 67, 229. (17) Ward, C. A.; Rizk, M.; Tucker, A. S. J. Chem. Phys. 1982, 76, 5606.

10.1021/la0109664 CCC: $22.00 © 2002 American Chemical Society Published on Web 12/27/2001

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Rudzinski and Panczyk

desorption rate Rd was found to depend also on the chemical potential of the molecules in the gas phase µg,

(

)

µa - µg kT

Rd ∼ exp

(2)

whereas the adsorption rate Ra is given by

(

Ra ∼ exp

)

µg - µa kT

(3)

That new approach called SRT (statistical rate theory) leads, thus, to two expressions that can be viewed as a generalization of the expressions proposed by de Boer and Nagai. However, like the first attempts by Nagai to use eq 1 for a quantitative interpretation of experimental TPD data, the more general SRT approach has not received an easy acceptance from some of the researchers using the classical ART approach.35 Meanwhile, the new SRT approach has been successfully used to describe a variety of interfacial phenomena. These include the rates of gas absorption at a liquid/gas interface,16-19 dissociative hydrogen adsorption,20 hydrogen absorption by metals,21 electron exchange between ionic isotopes in solution,22 permeation of ionic channels in biological membranes,23 nondissociative gas adsorption on solid surfaces,24-28 beam dosing adsorption kinetics,29 temperature programmed thermal desorption,30 solid crystal dissolution rate,31 and rates of liquid evaporation.32-34 In a recent series of papers, Rudzinski and coworkers36-43 have generalized the SRT approach to describe the kinetics of adsorption/desorption on/from energetically heterogeneous solid surfaces. These generalizations concern both the isothermal adsorption kinetics (18) Ward, C. A.; Tikuisis, P.; Tucker, A. S. J. Colloid Interface Sci. 1986, 113, 388. (19) Tikuisis P.; Ward, C. A. In Transport Processes in Bubbles, Drops and Particles; Chabra, R., DeKee, D., Eds.; Hemisphere: New York, 1992; p 114. (20) Findlay, R. D.; Ward, C. A. J. Chem. Phys. 1982, 76, 5624. (21) Ward, C. A.; Farabakhsk, B.; Venter, R. D. Z. Phys. Chem., Neue Folge 1986, 147 (S 89-101), 7271. (22) Ward, C. A. J. Chem. Phys. 1983, 79, 5605. (23) Skinner, F. K.; Ward, C. A.; Bardakjian, B. L. Biophys. J. 1993, 65, 618. (24) Ward, C. A.; Findlay, R. D.; Rizk, M. J. Chem. Phys. 1982, 76, 5599. (25) Ward, C. A.; Findlay, R. D. J. Chem. Phys. 1982, 76, 5615. (26) Ward, C. A.; Elmoseli, M. B. Surf. Sci. 1986, 176, 457. (27) Elliott, J. A. W.; Ward, C. A. Langmuir 1997, 13, 951. (28) Elliott, J. A. W.; Ward, C. A. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: New York, 1997. (29) Elliott, J. A. W.; Ward, C. A. J. Chem. Phys. 1997, 106, 5677. (30) Elliott, J. A. W.; Ward, C. A. J. Chem. Phys. 1997, 106, 5667. (31) Dejmek, M.; Ward, C. A. J. Chem. Phys. 1998, 108, 8698. (32) Ward, C. A.; Fang, G. Phys. Rev. E 1999, 59, 429. (33) Fang, G.; Ward, C. A. Phys. Rev. E 1999, 59, 441. (34) Fang, G.; Ward, C. A. Phys. Rev. E 1999, 59, 417. (35) Zhdanov, V. P. J. Chem. Phys. 2001, 114, 4747. (36) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Panczyk, T. Langmuir 1997, 13, 3445. (37) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Panczyk, T.; Gryglicki, J. Pol. J. Chem. 1998, 72, 2103. (38) Rudzinski W.; Panczyk, T. Surface Heterogeneity Effects on Adsorption Equilibria and Kinetics: Rationalisation of Elovich Equation. In Surfaces of Nanoparticles and Porous Materials; Schwarz, J., Contescu, C., Eds.; Marcel Dekker: 1999. (39) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Panczyk, T. Langmuir 1999, 15, 6386. (40) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. Adv. Colloid Interface Sci. 2000, 84, 1. (41) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. J. Phys. Chem. B 2000, 104, 1984. (42) Rudzinski, W.; Panczyk, T. J. Phys. Chem. B 2000, 104, 9149. (43) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. Langmuir 2000, 16, 8037.

and the kinetics of thermal desorption. It has been shown that the well-known empirical kinetic equations, like the Elovich or the power-law expressions, describe the kinetics of isothermal adsorption in systems where the adsorption equilibria are described by some well-known isotherm equations, the Temkin and the Freundlich equations, respectively. Moreover, a more general form of these kinetic equations has been developed, by taking rigorously into account both the adsorption and desorption terms, and next used to fit quantitatively reported experimental data. The success of the SRT approach to quantitatively describe so many different interfacial phenomena must, obviously, raise doubts as to the theoretical backgrounds of some critical comments on SRT published recently.35 One may expect another heated discussion in the literature regarding the fundamentals of the ART and the SRT approaches between the scientists defending the classical ART approach and the scientists who have launched the newer SRT approach. So far, these discussions have been carried out at the level of fundamental assumptions, whose validity must finally be verified by the ability of the related theoretical expressions to describe the experimental findings. Thus, in parallel to the expected further discussion of fundamentals, we have decided to compare the ability of the ART and SRT approaches to describe the kinetics of gas adsorption in some real adsorption systems with energetically heterogeneous solid surfaces. Discussion of Theoretical Principles and Analysis of Experimental Data The attempts to generalize the ART approach for the case of energetically heterogeneous solid surfaces met one large barrier related to the fundamentals of the ART approach. This is the appearance of the “activation energies” for adsorption and desorption, a and d, which are so fundamental to that approach. According to ART, adsorption of an adsorbate molecule is viewed as a chemical reaction between that molecule and a solid surface. In the simplest case of one-siteoccupancy adsorption, and in absence of interactions between the adsorbed molecules, ART offers the following expressions:

( ) ( )

Ra ) Kap(1 - θ) exp Rd ) Kdθ exp -

a kT

d kT

(4) (5)

At equilibrium, when Ra and Rd are equal eqs 4 and 5 yield the Langmuir isotherm

(kT ) exp( ) kT

Kp(e) exp θ

(e)

) 1 + Kp(e)

(6)

where K ) Ka/Kd, ) d - a, θ is the fractional coverage of adsorption sites, and p is the bulk pressure. The superscript (e) is used to denote the equilibrium condition. However, the main difficulty lies in the interpretation of a and d and their interrelations when the surface is energetically heterogeneous. And such is true for the vast majority of adsorption systems of practical importance. By contrast, the dispersion of appearing in the SRT approach has been studied in hundreds of papers treating

Adsorption Kinetic Theories for Solid Surfaces

Langmuir, Vol. 18, No. 2, 2002 441

the equilibria of adsorption on energetically heterogeneous surfaces.44-46 There, the experimentally monitored equilibrium adsorption isotherms have usually been considered as the following averaged functions, θt(e),

θt(e)(p,T) )

∫Ωθ(e)(,p,T) χ() d

(7)

where θ(e)(,p,T) is the “local” adsorption isotherm representing the fractional coverage of adsorption sites characterized by the adsorption energy , χ() is the differential distribution of the number of adsorption sites among the related values, and Ω is the physical domain of . Equation 7 is the so-called “integral equation” approach. It was next followed by the scientists trying to generalize the ART eqs 4 and 5 for the case of energetically heterogeneous surfaces, where both a and d are believed to vary from one adsorption site to another. But here, the scientists trying to apply the ART approach met a fundamental difficulty. To illustrate it, we consider the θ(t) function, found by the integration of the related ART expressions (eqs 4 and 5),

( )

( )

a d dθ ) Kap(1 - θ) exp - Kdθ exp dt kT kT

(8)

for a hypothetical, homogeneous solid surface. While assuming the boundary condition θ(t)0) ) 0, we have

Ka (d-a)/kT pe Kd 1θ(t) ) Ka 1 + pe(d-a)/kT Kd

{

(

(

) )}

Ka (d-a)/kT pe +1 t Kd

( )) a kT

(12)

The function θ(a, p,T,t) was next used as the kernel in the equation for θt(p,T),38,48

θt(p,T) )

∫Ωθ(a,p,T) χ(a) da

(13)

to describe the kinetics of isothermal adsorption. Of course, the obtained expression could then be used only to represent the initial adsorption kinetics at not too high surface coverages (or long adsorption times). Similarly, scientists using ART to investigate thermal desorption neglected, as a rule, the (re)adsorption term and used the integrated form of eq 5 to represent the kinetics of thermal desorption,

(

( ))

θ(d,p,T,t) ) exp -Kdt exp -

θt(p,T,t) ) (9)

Then, a correct generalization for an energetically heterogeneous solid surface reads

θt(t,p,T) )

(

θ(a,p,T,t) ) 1 - exp -Kapt exp -

d kT

(14)

Then again, θ(d, p,T,t) is a function of only the one variable, d, and θt(p,T,t) is given by

-d/kT

exp -Kde

The lack of an answer for that fundamental question has overshadowed for decades the studies both of isothermal adsorption kinetics and of the kinetics of thermal desorption. The scientists investigating the isothermal adsorption kinetics focused usually on the initial rates of adsorption at lower surface coverages, that is, neglected the simultaneously occurring desorption, so the integrated form of eq 4 becomes then a function of only one variable, the activation energy for adsorption, a:

∫θ(t,p,T,a,d) χ(a,d) da dd

(10)

where χ(a,d) is a two-dimensional differential distribution of the activation energies, a and d, normalized to unity,

∫Ω∫χ(a,d) da dd ) 1

(11)

and Ω is the related two-dimensional physical domain. However, such a rigorous generalization of the ART expressions has not yet been attempted so far for one fundamental reason. Not one fundamental paper was published that reported how a and d may change from one adsorption site to another. Do they vary independently, or does some correlation exist between a and d when going from one site to another? Only certain guesses were made in the literature.47 (44) Jaroniec, M.; Madey, E. Physical Adsorption on Heterogeneous Solids; Elsevier: New York, 1989. (45) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Solid Surfaces; Academic Press: San Diego, 1992. (46) Do, D. D. Adsorption Anaysis: Equilibria and Kinetics; Imperial College Press: London, 1998. (47) Tovbin, Yu. Theory of Adsorption-Desorption Kinetics on Flat Heterogeneous Surfaces. In Equilibria and Dynamics of Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: New York, 1997.

∫Ωθ(d,p,T) χ(d) dd

(15)

Equation 15 was frequently treated as an integral equation for the unknown function χ(d), assumed to characterize the surface energetic heterogeneity.49-55 However, no single attempt has been reported so far in the world literature to treat the formally correct eq 10 as an integral equation for the unknown two-dimensional function χ(a,d) and to recover such a function from experimental data. No useful procedure has been developed so far on the grounds of the ART approach that would allow one to describe the kinetics of adsorption/desorption on/from an energetically heterogeneous surface, when the full form of the ART rate eq 9 is taken into consideration. It is very important that such a modification would be applicable in the whole range of adsorption times, that is, be applicable when both adsorption and desorption must be taken into account. Even in the case of the one-dimensional integral eq 7, solving it with respect to χ() is not easy. This is the socalled “ill-posed” solution, that is very sensitive to the errors in the experimental (discrete) values of θt(e)(p,T).45 It is to be expected that this problem would be much more serious in the case of the two-dimensional integral eq 10. (48) Aharoni, C.; Tompkins, F. C. Adv. Catal. 1970, 21, 1. (49) Carter, G. Vacuum 1962, 12, 245. (50) Carter, G.; Bailey, P.; Armour, D. G. Vacuum 1982, 32, 233. (51) Du, A. F.; Sarofim, J. P.; Longwell, J. P. Energy Fuels 1990, 4, 296. (52) Brown, T. C.; Leary, A. E.; Ma, M. C.; Haynes, B. S. In Fundamental Issues in the Control of Carbon Gasification Reactivity; Lahaye, J., Ehrburger, P., Eds.; Kluwer: Norwell, MA, 1990; p 307. (53) Ma, M. C.; Brown, T. C.; Haynes, B. S. Surf. Sci. 1993, 297, 312. (54) Seebauer, E. G. Surf. Sci. 1994, 316, 391. (55) Cerofolini, G. F.; Re, N. J. Colloid Interface Sci. 1995, 174, 428.

442



This shows the barriers which one would face trying a formally correct generalization of the ART approach for the case of energetically heterogeneous solid surfaces, represented by eq 10. By contrast, the SRT approach offers an easy way to handle in a formally correct way the kinetics of adsorption/ desorption on/from the actual energetically heterogeneous solid surfaces.36-46 That approach assumes that the adsorbed phase is at “quasi-equilibrium”, that is, all the surface correlation functions are practically the same as they would be in equilibrium at the same surface coverage. For a hypothetical, energetically homogeneous solid surface, the rate equation (dθ/dt) takes the form

[ (

)

)]

(

µa - µs µs - µa dθ ) K′gs exp - exp dt kT kT

(16)

where K′gs is the exchange rate between the gas phase and the solid surface once an isolated system has reached equilibrium. To a good approximation, it can be expressed as follows:

K′gs ) Kgsp(e)(1 - θ(e))

(17)

For the Langmuir model of adsorption,

θ - kT ln qs µs ) kTln 1-θ

(18)

where qs is the molecular partition function of the adsorbed molecules, commonly written as the following product:

(kT )

qs ) qs0 exp

(19)

Then the SRT rate equation (dθ/dt) takes the form

1-θ dθ 1 θ ) Kap exp exp - Kd × dt θ kT p1-θ kT

[

( )

)]

(

[1 - θ(e)]p(e) (20) where

Ka )

Kgsqs0

( )

µg0 exp kT

( )

µg0 Kd ) s exp kT q0 Kgs

(21)

At equilibrium, when (dθ/dt) ) 0, one arrives at the Langmuir isotherm, in which

( )

desorption rate is not inversely proportional to the pressure, as has been incorrectly argued in a recent article criticizing the SRT approach.35 The above equation, being one of the possible equations describing the “Langmuirian kinetics”, has to be compared with eq 8 offered by the classical ART approach. So, let us remark that for the kinetic processes which are not far from equilibrium (the assumption in eq 17), both the ART and the SRT approaches lead to the same dependence on the desorption term Rd, for the Langmuirian model of adsorption considered here. Looking to the existing interpretations of the activation energy of desorption, one can see that d should be close to interpreted usually as the local minimum in the gas-solid potential function taken with a reverse sign. As we have already mentioned, that form of the desorption term has frequently been used to carry out theoretical interpretation of the spectra of thermal desorption. It means that accepting the SRT approach will not invalidate the results already published on thermal desorption kinetics, which were obtained by neglecting readsorption. But there is, however, a dramatic difference in the interpretation of the adsorption terms Ra corresponding to the ART and SRT approaches. The Ra term offered by the SRT approach has different pressure and coverage dependence than the Ra term corresponding to the ART approach. In view of that dramatic discrepancy in the related theoretical predictions, studying that problem might provide the answer to whether the fundamentals of ART or SRT are more correct. As the SRT and ART desorption terms have identical coverage dependence, the so frequently carried out studies of thermal desorption kinetics could not provide arguments for a better applicability of the ART approach or the SRT approach. One can arrive at reliable conclusions only if the full form of the rate equation dθ/dt ) Ra - Rd is applied to analyze experimental data. As the experimental data are usually reported as the integral adsorption kinetics isotherms θ(p,T,t), we will use further the integral forms of the ART rate expression8 and of the SRT rate expression,24 to fit experimental data. The corresponding ART integral function θ(p,T,t) is then given in eq 9. Now, let us consider the integral form of the SRT function θ(p,t,T). The integral form of the eq 23, obtained with the boundary condition θ(t)0) ) 0, reads

t(θ) )

µg0 K ) xKa/Kd ) qs0 exp kT

Depending on the conditions at which the kinetic experiment is carried out, (dθ/dt) can take various forms. In their recent works, Rudzinski et al.37-41 considered the case when the experimental conditions are not far from equilibrium. Then,

1 - θ(e) ≈ 1 - θ and p(e) ≈ p

(22)

At such conditions, the rate equation takes the form

(1 - θ)2 dθ ) Kap2 exp - Kdθ exp (23) dt θ kT kT

( )

(

)

This example shows that the SRT expression for the

[

(

x (

Kde-/kT

(x

)

(

Ka /kT pe Kd ar tanh Ka 2 2/kT pe -1 Kd

)

)

]]

Ka /kT pe Kde-/kT Kd Ka Ka 2 2/kT pe - 1 θ - p2e2/kT Kd Kd

[(

Kde-/kT

where

[ )]

Ka 2 2/kT 1 ln|Kde-/kT pe K K a 2 2/kT d -/kT 2Kde pe -1 Kd Ka 2 2/kT Ka pe - 1 θ2 | 2 p2e2/kTθ + Kd Kd

)

+ C (24)


x (

C)

Kde-/kT

Ka /kT pe Kd ar tanh Ka 2 2/kT pe -1 Kd

)

[x ] (

1

-

Ka /kT pe Kd

)

Ka 2 2/kT 1 ln|Kde-/kT pe | (24a) Ka 2 2/kT Kd pe -1 Kd

(

2Kde-/kT

Langmuir, Vol. 18, No. 2, 2002 443

)

As the surface energetic heterogeneity is the main factor affecting the kinetics and adsorption equilibrium on the actual solid surfaces, one can arrive at reliable conclusions only if one uses the averaged forms θt(p,T,t) taking into account the surface energetic heterogeneity to fit experimental data. For the ART approach, θt(t,p,T) is given in eq 10. To arrive at the kinetic isotherm θt(p,T,t) corresponding to the SRT approach, one has to calculate the following one-dimensional integral,

θt(p,T,t) )

∫

θ(,p,T,t) χ() d Ω

(25)

where the local kinetic isotherm θ(,p,T,t) is that one defined in eq 24. In the SRT eq 25, the averaging is carried out with the χ() function. This function is also used to arrive at the equilibrium isotherm θt(p,T). And because χ() can be determined from the equilibrium experimental data for θt, studies of adsorption equilibria on heterogeneous solid surfaces make it possible to predict the features of the adsorption kinetics in an adsorption system just by assuming a certain value of the desorption constant Kd. The integration in eq 25 can easily be performed numerically, but Rudzinski et al.42 have shown that in the case of typical heterogeneous surfaces, compact analytical solutions are also available, for some important functions χ(), which when inserted in eq 7 lead to well-known equilibrium isotherm equations (Temkin, Freundlich). The result of integration in eq 25 is then one of the well-known empirical equations for adsorption kinetics (Elovich, power-law). Thus, the SRT approach makes it extremely easy to establish rigorous relations between the features of adsorption equilibria and adsorption kinetics. On the contrary, the ART approach makes these relations obscure. However, before considering it in more detail, let us compare the two kinetic equations, the ART eq 9 and the SRT eq 24, both leading to the Langmuir isotherm at equilibrium. Both of them contain the physical parameter KL,

KL )

(

)

( )

Ka d - a p exp ) Kp exp in eq 9 Kd kT kT

(26a)

and

KL )

x

Ka p exp ) Kp exp in eq 24 (26b) Kd kT kT

( )

( )

where the subscript L refers to the Langmuir model of adsorption. ) Then, eq 9 contains yet the second parameter KART d Kd exp(-d/kT), whereas eq 24 has the second parameter KSRT ) Kd exp (-/kT). Then, d

Figure 1. The comparison of the SRT θt(t) function, calculated from eq 24 by assuming that KL ) 10 and KdSRT ) 1, with the ART θt(t) functions calculated from eq 9 by assuming KL ) 10 and KdSRT ) 1, for three various values of a: -5kT, 0kT, and 5kT.

( )

KART ) KSRT exp d d

a kT

(27)

Thus, while comparing the features of the ART kinetic isotherm (9) and the SRT kinetic isotherm (24) we will choose various pairs of (KL, KSRT d ), and for every pair of those parameters we will try in addition various values of a. Figure 1 shows the results of our calculations. Now, let us consider the behavior of the ART eq 9 and of the SRT eq 24 for the case of energetically heterogeneous solid surfaces. In the case of the SRT approach, such a generalization is straightforward and is given in eq 25. On the contrary, no such straightforward generalization seems to be possible in the case of ART eq 9. Below, we list the obscure problems which one faces when trying to generalize the ART eq 9 for the whole range of surface coverages 0 e θt e 1, when both adsorption and simultaneously occurring desorption must be taken into account. Studies of adsorption equilibria show that the value ) d - a varies from one adsorption site to another. The related dispersion χ() can be deduced from equilibrium adsorption isotherm θt(e)(p,T). To arrive from eq 12 at the Elovich or power-law empirical equations, one has to assume that there is a dispersion of a, described by either a rectangular (Elovich) or exponentially decreasing (power-law) distribution χa(a).38,48 Theoretical interpretations of thermal desorption spectra are always based on the assumption that there is a certain dispersion of d on various adsorption sites, described by a χd(d) function deduced from experimental desorption spectra. Thus, while staying on the grounds of the classical ART approach we have to assume that both a and d vary from one site to another. Nothing can be stated a priori about the possible correlations between a and d, except that the highest possible correlation d - a ) const does not exist, because values are found to vary, when one analyzes the adsorption equilibria. So, various speculations are possible, but we believe that a final judgment should always come from appropriate analysis of available experimental data. Thus, trying various generalizations of the ART eq 9 for the case of heterogeneous solid surfaces, we will again take into consideration the data describing the kinetics of CO2 adsorption on scandia,49 analyzed in one of our previous works.42 There, we have shown that in the case of typically heterogeneous solid surfaces, the condensation approxi-

444



mation (CA) that is based on the SRT approach can be used to arrive at compact analytical expressions, representing the integral in eq 25. In the case of the system CO2/scandia, it was found that the adsorption energy distribution χ() describing the energetic surface heterogeneity can be well represented by the following Gaussianlike function, χ(),

1 χ() ) c

(

)

- 0 c - 0 1 + exp c exp

[

(

)]

2

(28)

that is centered at ) 0, and the variance of χ() is equal to cπ/x3. Then, the result of integration in eq 25 takes the following simple compact form:42

θt(p,T,t) )

[Kpe0/kT tanh{2pKgst}]kT/c 1 + [Kpe0/kT tanh{2pKgst}]kT/c

(29)

In the limit t f ∝, when the kinetic isotherm θt(t) reaches the plateau, eq 29 reduces to the well-known LangmuirFreundlich isotherm, (e)

θt (p,T) )

[Kpe0/kT]kT/c 1 + [Kpe0/kT]kT/c

(30)

which has been used by so many researchers to successfully describe the adsorption equilibria, in various adsorption systems.44,45 A suitable choice of the three equilibrium constants K exp(0/kT), Kgs, and kT/c has led Rudzinski and Panczyk to an excellent fit of the experimental kinetic isotherms measured at a temperature of 250 °C, but for four different pressures p, ranging from 2.5 to 8.0 Torr.42 Figure 2 shows the best-fit obtained by assuming K exp(0/kT) ) 0.265 Torr-1 and kT/c ) 0.74. Now, we will try to fit these kinetic isotherms using a generalized form of the ART expression (9). Unfortunately, no such simple compact analytical equation for θt(t) can be developed on the basis of the ART approach. Because the experimental kinetic isotherms of CO2 adsorption on scandia reach plateau, it means that the equilibrium isotherm at 250 °C is described by the Langmuir-Freundlich eq 30, with the parameters K exp(0/kT) ) 0.265 Torr-1 and kT/c ) 0.74; that is, the surface energetic heterogeneity of scandia for CO2 adsorption is very well represented by the Gaussian-like function (28). Thus, we will accept the values of the parameters that we have already determined for this system, c ) 0.74 kT ) 5.88 kJ/mol and K exp(0/kT) ) 0.265 Torr-1 at 250 °C, in our coming analysis of that adsorption kinetics data based on a generalized form of the ART isotherm (9). What we surely know is that the dispersion of the difference (d - a) in eq 9 should be represented by the function (28), in which c ) 5.88 kJ/mol. But now we face the fundamental problem of how should we treat the term Kd exp(-d/kT) appearing in eq 9. Various speculations are possible. Figure 3 shows some of the results of our numerical best-fit exercises. There we present two best-fits, obtained by assuming that a is a linearly decreasing function of , as has been suggested in the theoretical works published by Tovbin.47 Thus, we assumed that

Figure 2. The best-fit of the experimental kinetic isotherms of CO2 adsorption on scandia, reported by Pajares et al. (ref 56) for T ) 250 °C, obtained by the authors (ref 42) using eq 20 in which K exp(0/kT) ) 0.265 Torr-1 and kT/c ) 0.74. (Other details can be found in the author’s previous publication (ref 42).)

Figure 3. (A) The comparison between the theoretical values of θt(t,p) (s) calculated from the ART expression (9), averaged with the Gaussian-like adsorption energy distribution (28) by assuming that γ ) 0, with the experimental data for the kinetics of CO2 adsorption on scandia monitored at 250 °C and at the four pressures 2.5, 3.8, 4.9, and 8.0 Torr. The other parameters are listed in Table 1. (B) The comparison between the theoretical values of θt(t,p) (s) and experimental data, when γ ) -1/2.

a ) 0a + γ

(31)

d ) + a ) + 0a + γ

(32)

Then,

While accepting this assumption, we attempted to fit quantitatively the kinetic isotherms of CO2 adsorption on scandia, by choosing properly the two unknown constants, that is, 0 and Kd. The other constant K ) Ka/Kd is


Langmuir, Vol. 18, No. 2, 2002 445

Table 1. Values of the Parameters Used to Arrive at the Fit of the Experimental Data Presented in Figure 3A,Ba best-fit monitored p [Torr]

Nm [µg/g] (best-fit)

Kdeff [s-1] (best-fit)

0 [kJ/mol] (best-fit)

K [Torr-1] (calculated)

2.5 3.8 4.9 8.0

When the Assumed γ ) 0 1110 (1160) 2.33 × 107 86.8 1032 (1038) 3.18 × 107 86.8 1064 (1047) 2.68 × 107 86.8 980 (942) 2.90 × 107 86.8

5.72 × 10-10 5.72 × 10-10 5.72 × 10-10 5.72 × 10-10

2.5 3.8 4.9 8.0

When the Assumed γ ) -1/2 1105 (1160) 5.41 × 104 128 1035 (1038) 8.08 × 104 128 1068 (1047) 7.39 × 104 128 987 (942) 8.98 × 104 128

4.40 × 10-14 4.40 × 10-14 4.40 × 10-14 4.40 × 10-14

a In the parentheses, the values of N are listed, found by fitting m the kinetic data by eq 29 in our previous publication (ref 40).

then calculated from the already known parameter K exp(0/kT) ) 0.265 Torr-1. We put 0a ) 0 in our practical calculations, which means that instead of the actual value of Kd, we estimated its effective value, Kdeff,

( )

Kdeff ) Kd exp -

0a kT

(33)

We attempted various values of γ in our numerical calculations. For instance, our numerical exercises showed that when γ ) 0 the best agreement with experimental data is achieved by assuming 0 ) 86.8 kJ/mol and Kd and Nm values that were slightly different for different pressures. They are listed in Table 1. The agreement between theory and experiment presented in Figure 3 is essentially as good as that obtained by using the SRT eq 29 and shown in Figure 2. To arrive at the good fit presented in Figure 3, we had to assume some slightly different values of Kdeff and of the monolayer capacity Nm for every pressure p, but this can be attributed, of course, to experimental errors. We observed a similar effect while fitting the kinetic data by eq 29, in our previous publication. There, the estimated values of Kgs and Nm were also slightly different for the different nonequilibrium pressures. Table 1 shows the comparison of the estimated Nm values. Parts A and B of Figure 3 show almost equally good fits of experimental data, obtained by assuming that γ is either zero or equal to -1/2. Our numerous computer exercises showed that similar good fits can be obtained for values of γ that ranging from zero to about -0.4. The uncertainty in choosing the γ value must be viewed as an embarrassing feature of the ART approach. Our computer exercises also showed that no computer fit of these experimental data can be achieved by assuming γ > 0. Meanwhile, if we assume that the elementary act of physisorption is a chemical reaction which can be considered in terms of ART, then a should be positive, because the activation energies of various chemical reactions have positive values. However, one can argue that the above result is only a proof that the assumption (31) is correct. Provided that 0a would take a sufficiently high positive value, the sum (0a + γ) could be positive in the whole range of existing (physical) values of , and then a > 0, as required in the ART approach. The final judgment could come from the studies of the temperature dependence of Kdeff, plotted as ln Kdeff versus 1/T. The tangent of that Arrhenius plot should give the

Table 2. Values of the Parameters Used to Arrive at the Fit of the Experimental Data Presented in Figures 4 and 5 best-fit monitored T [°C]

Nm [µg/g] (best-fit)

Kdeff [s-1] (best-fit)

0 [kJ/mol]

K [Torr-1] (calculated)

149 185 250 293

When the Assumed γ ) 0 736 3.90 × 107 86.8 758 4.28 × 107 86.8 1139 3.11 × 107 86.8 1897 2.60 × 107 86.8

5.67 × 10-10 5.67 × 10-10 5.67 × 10-10 5.67 × 10-10

149 185 250 293

When the Assumed γ ) -1/2 763 5.95 × 103 128 711 2.51 × 104 128 1138 7.43 × 104 128 2670 8.45 × 104 128

4.35 × 10-14 4.35 × 10-14 4.35 × 10-14 4.35 × 10-14

value -0a/k. A nice circumstance here is that the data measured by Pajares et al.56 offer such a possibility. In addition to the kinetic isotherms θt(t) measured at one temperature 250 °C and at the four pressures p, Pajares et al.56 have also reported a number of kinetic isotherms θt(t) measured at one pressure 2.8 Torr but at five different temperatures ranging from 105 up to 293 °C. (We have extended our investigation to four of them that were measured at temperatures closest to the temperature 250 °C, for which the four isotherms were reported, measured at different pressures.) First, we used the ART expression (9) to fit the experimental kinetic isotherms, measured at a pressure of 2.8 Torr and in different temperatures. Now, we have carried out our computer fitting by assuming the two cases γ ) 0 and γ ) -1/2. In every case (γ ) 0, γ ) -1/2) and for every studied temperature T, we assumed that c, 0, and K must have the values determined already for that case (γ ) 0 or γ ) -1/2), from fitting the experimental kinetic isotherms, measured at different pressures, p, and at the one temperature 250 °C. For the two cases γ ) 0 and γ ) -1/2, these parameters are collected in Table 1. Taking these parameters, we next fitted the kinetic isotherms measured at different temperatures, just by adjusting one parameter Kdeff at every studied temperature. Our computer exercises also showed that to achieve some agreement between ART theory and experiment, we were forced to assume Nm values that were much different for different temperatures. The values of Kdeff(T) and Nm(T) found in that way are collected in Table 2. Figures 4 and 5 show the agreement between experiment and ART theory, obtained in that way by using the parameters collected in Table 2. Of course, one may argue that the value of K may too be temperature dependent, but it has been generally assumed in adsorption literature (and theoretically proved) that K is a slowly varying function of temperature. Usually it varies with a low power of T.1,44,45 Figures 4 and 5 show some moderate agreement between the ART theory and experiment, obtained by using the parameter values collected in Tables 1 and 2. However, the theoretical interpretation of the determined values of these parameters must raise serious doubts regarding the consistency of the ART approach. We begin with the determination of the 0a from the Arrhenius plot shown in Figure 6. When we assume γ ) 0, the determined 0a value is negative, and the a() function takes only negative values. (56) Pajares, J. A.; Garcia Fierro, J. L.; Weller, S. W. J. Catal. 1978, 52, 521.

446



Figure 6. The Arrhenius plots for the Kdeff values determined by assuming γ ) 0 (]) and by assuming γ ) -1/2 (O), collected in Table 2.

Figure 4. The agreement between the kinetic isotherms of CO2 adsorption on scandia, reported by Pajares et al. (ref 56) for different temperatures, and the ART function (9) averaged with the adsorption energy distribution (28) when assuming γ ) 0, using the related best-fit parameters collected in Table 2.

Figure 7. The agreement between the experimental kinetic isotherms of CO2 adsorption on scandia, measured at p ) 2.8 Torr and at different temperatures, with the theoretical kinetic isotherms calculated by using the SRT expression (29) and the parameters collected in Table 3. Figure 5. The agreement between the kinetic isotherms of CO2 adsorption on scandia, reported by Pajares et al. (ref 56) for different temperatures, and the ART function (9) averaged with the adsorption energy distribution (28) when assuming γ ) -1/2, using the related best-fit parameters collected in Table 2.

This contradicts the fundamental assumption of the ART approach that a must be positive. When one assumes that γ ) -1/2, from the Arrhenius plot in Figure 4, we determine a positive value of 0a ) 36.48 kJ/mol, but the a ) 0a -1/2 is positive only in a limited region of small values. The maximum in the χ() function is now located at ) 128 kJ/mol (see Table 1 or 2); that is, positive values of a exist only in the region < 73 kJ/mol. Also, highly nonphysical behavior shows the Nm values determined for different temperatures. Such strong changes of Nm with temperature cannot, by any means,

be interpreted as being due to experimental errors. Moreover, the estimated Nm values grow regularly with temperature, which must immediately raise a suspicion that such strong changes forced by best-fit calculations must compensate the nonphysical behavior of some of the terms in the ART expression. Now, let us consider the results of fitting the kinetic isotherms measured in different temperatures by the SRT eq 29. As previously, we accepted for every temperature the value c ) 5.88 kJ/mol, whereas the parameter Kgs was adjusted at every temperature. The parameters K and 0 were also adjusted but were assumed to be the same for every temperature. Figure 7 shows the results of our best-fit exercises. The related parameters are collected in Table 3. We can observe in Figure 7 a much better agreement with experiment of the theoretical SRT functions than

Adsorption Kinetic Theories for Solid Surfaces Table 3. Parameters Used to Fit the Kinetic Isotherms Measured at Different Temperatures by the SRT Equation 29a temp [°C]

Nm [µg/g]

2Kgs [Torr-1 s-1]

0 [kJ/mol]

K [Torr-1]

149 185 250 293

1217 1175 1172 1202

0.00598 0.00807 0.0109 0.0129

13.7 13.7 13.7 13.7

1.13 × 10-2 1.13 × 10-2 1.13 × 10-2 1.13 × 10-2

a

The related fit is shown in Figure 7.

the agreements presented in Figures 4 and 5 and corresponding to the ART approach. While comparing Figures 4 and 5 with Figure 7, we must conclude that the SRT approach leads to a much better agreement between theory and experiment. Also, the analysis of the determined parameters advocates strongly for a better physical reliability of the SRT theoretical expressions. Contrary to the nonphysical behavior of some of the parameters and the functions calculated by using ART, we see in Table 3 a highly physical behavior of the estimated Nm values, for instance, which are only slightly different for different temperatures. Now, let us point out a nonphysical behavior of the 0 values determined by fitting the kinetic isotherms by the ART theoretical expressions. Because the kinetic isotherm θt(t) measured at different temperatures reaches a plateau in every case, the parameters K and 0 in Table 3 can be considered as determined from equilibrium adsorption isotherms measured at different temperatures. Therefore, the values 0 ) 13.7 kJ/mol and K ) 1.13 × 10-2 Torr-1 must be considered as true physical values of 0 and K for this adsorption system. Their comparison with the 0 and K values found by fitting the kinetic isotherms by the ART eq 9 shows dramatic discrepancies between the ART values and the true ones. All that, of course, must put into question the physical meaning of the ART eq 9 and the applicability of the ART approach itself. Although while using ART we obtain an infinite number of 0 values leading to a moderate agreement with experiment, the lowest 0 value determined by assuming γ ) 0 is 87 kJ/mol, that is, is roughly 7 times higher than the actual value of 0 ) 13.7 kJ/mol. While accepting the only possible values of γ < 0 (leading still to some agreement with experiment), the ART equation leads us to still higher and more unrealistic values of 0. However, the dramatic conflict between the ART equation for adsorption kinetics and the equations describing the adsorption equilibria can only be seen when both the adsorption and the desorption terms, Ra and Rd, are taken into account to correlate the experimental data for adsorption kinetics for large time intervals of time and surface coverage. Taking into account either only the adsorption term Ra or only the desorption term Rd does not make it possible to reveal that dramatic conflict. But, of course, such a truncated expression would not make it possible to fit kinetic data in the whole interval of times, starting from very short times to those corresponding practically to equilibrium. Then, as we have already mentioned, there is one circumstance which screened these differences so far. Namely, the vast majority of the kinetic experimental studies concerned the kinetics of thermodesorption. And as can be seen in eq 23, the desorption terms Rd offered by both the ART and the SRT approaches have identical coverage dependence. Also, taking into account the possible interactions between the adsorbed molecules will lead to identical

Langmuir, Vol. 18, No. 2, 2002 447

generalizations of the desorption term Rd. An interaction term in will have to be added to either d in the ART eq 9 or to in the SRT eq 23. This interaction term has been calculated in the literature using a variety of methods (mean-field approximation, quasi-chemical approximation, computer simulations).57-64 Of course, in the case of heterogeneous surfaces, calculating that interaction term in is not a trivial problem at all. It involves, for instance, considering the surface topography. However, an impressive body of theoretical papers has already been published, proposing various approaches to that problem. It has raised a great interest of both the scientists investigating the adsorption equilibria and of the scientists studying the kinetics of thermodesorption. The scientists studying the adsorption (commonly thermodesorption) kinetics usually made the same fundamental assumption as their colleagues studying the adsorption equilibria. Namely, calculating that additional interaction term has usually been done by assuming that the adsorbed phase is in quasi-equilibrium, that is, all the surface correlation functions are the same as those at equilibrium, at the same total surface coverage θt. Thus, the scientists using the ART approach commonly made the assumption which is fundamental for the new SRT approach. But here again, another dramatic difference between the ART and the SRT approaches is encountered: While considering the interactions between the adsorbed molecules in the ART approach, the additional term in is added only to d in the desorption term Rd. It cannot be added to or subtracted from a, for the fundamental reasons. When in is added to a, for instance, at equilibrium the in terms cancel, which would mean the existence of interactions in the course of adsorption progressing with time and the lack of these interactions at equilibrium. When, on the contrary, in would be abstracted from a, at equilibrium f + 2in, which would mean that at equilibrium the interaction term is 2 times higher. Thus, it is assumed in the ART approach that the interactions between the adsorbed molecules do not affect the kinetics of adsorption. This fundamental but highly questionable feature of the ART approach has not yet sufficiently been brought into light in the previous studies of the adsorption/desorption kinetics based on applying the classical ART approach. Thus, again we can see that only by simultaneously taking into account the adsorption term Ra will one reveal fully the differences between the features of the ART and the SRT approaches. Let us consider now the meaning of Kgs in more detail. Also the SRT approach assumes that when a molecule coming from a gas phase adsorbs on a solid surface, it must both cross a gas-phase boundary and form an adsorptive bond as it encounters the solid. Similarly, when a molecule desorbs it must break the adsorptive bond and enter the gas phase. Although the collisions of the molecule with the solid surface can be treated using either classical or quantum mechanics, the prediction of the rate of bond formation requires a quantum mechanical description because this process involves the probability of the event (57) Meng, B.; Weinberg, H. J. Chem. Phys. 1994, 100, 5280. (58) Clark, A. The Theory of Adsorption and Catalysis; Academic Press: New York, 1970. (59) Houle, F. A.; Hinsberg, W. D. Surf. Sci. 1995, 338, 329. (60) Weinketz, S. J. Chem. Phys. 1994, 101, 1632. (61) Weinketz, S.; Cabrera, G. G. J. Chem. Phys. 1997, 106, 1620. (62) Cortes, J.; Valencia, E.; Araya, P. J. Chem. Phys. 1994, 100, 7672. (63) Sales, J. I.; Zgrablich, G. Surf. Sci. 1987, 187, 1. (64) Cordoba, A.; Luque, J. J. Phys. Rev. B 1982, 26, 4028.

448



Figure 8. The Arrhenius plot for the Kgs values used to fit by the SRT eq 29, the kinetic isotherms of CO2 adsorption on scandia measured at different temperatures. The solid line is the best-fit of experimental points obtained by eq 34 in which ln K0gs ) -2.82 and gs ) 10.4 kJ/mol.

occurring. The SRT approach leads to the conclusion that Kgs should be proportional to the probability that a molecule which strikes an adsorption site actually forms an adsorptive bond. Therefore, in Figure 8 we show the Arrhenius plot for the Kgs values determined for different temperatures and collected in Table 3. The Arrhenius plot in Figure 8 reveals the existence of a certain energy barrier gs for adsorptive bond formation,

ln Kgs ) ln K0gs -

gs kT

(34)

Conclusions One can observe a continuing controversy between some scientists using (and defending) the classical ART approach and the scientists linking the features of adsorption/ desorption kinetics to the chemical potentials of the free (bulk) and the adsorbed molecules. This controversy started two decades ago, when Cassuto11 and independently Zhdanov8 criticized first Nagai,6,7 using the new approach. More recently, Zhdanov35 has criticized again the newest and the most theoretically advanced version of that new approach, called the SRT. As previously, the criticism focuses on fundamentals of that new appraoch. On the other side, Ward and co-workers28 who have launched the SRT approach have criticized the classical ART approach by both questioning its fundamentals and showing a poorer applicability of ART to reproduce adsorption/desorption kinetics in some adsorption systems with fairly homogeneous solid surfaces. So far, the criticism of the SRT approach has focused on fundamentals, without presenting proof that the classical ART approach is superior in representing the behavior of the kinetics of adsorption in some real adsorption systems. Because the discussion has reached the level of mutually controversial fundamental assumptions, the most promising seems to be the strategy of verifying the fundamental assumptions through comparison of their related theoretical predictions with available experimental data. In other words, there is an urgent necessity to carry out comparative theoretical studies of the ART and the SRT approaches based on comparison with experiment. The present paper is aimed at presenting such a comparative study by considering the kinetics of isothermal adsorption. To draw reliable and clear conclusions, both the considered experimental data and their theoretical analysis must fulfill certain conditions. The experimental data must be recorded at various pressures and temperatures, and in time intervals starting

from very short times up to times corresponding practically to equilibrium. Also, we have shown in one of our recent papers65 that a simultaneous analysis of experimental heats of adsorption provides very valuable auxiliary information. Then, their theoretical analysis must be correctly done by using the full form of the rate expression including both the term describing the rate of adsorption and the term describing the rate of desorption. Next, a proper physical model must be accepted representing well the physical features of the investigated adsorption systems. First of all, one has to take into account the energetic heterogeneity of the vast majority of real solid surfaces. If none of the ART or SRT approaches could still well describe the features of adsorption/ desorption kinetics, it might be a check that interactions between the adsorbed molecules play a remarkable role. In the case of CO2 adsorbed on scandia, the SRT eq 29 yields an excellent fit of all the kinetic isotherms measured at various pressures and temperatures, so there was no reason to assume that the interactions between the adsorbed molecules play a significant role in this adsorption system. The failure of the ART approach to correlate these data at least as well as does the SRT approach must, therefore, be seen as an argument in favor of the fundamentals of the SRT compared to the fundamentals of the ART. However, a much more serious problem related to the use of the ART approach is that the determined parameter values contradict the fundamental assumptions of ART. It would be strange to assume that the expressions developed from incorrect fundamentals might better describe the behavior of real adsorption systems than the expressions obtained by accepting correct fundamentals. This, of course, must raise doubts concerning the theoretical background of the criticism of the SRT approach, attempted in one of recently published papers.35 We finish our summary by showing why the ART approach has commonly led to good interpretation of TPD spectra in the papers published so far. That interpretation has, almost exclusively, been carried out by using the abbreviated form of the rate equation including only the desorption term. Below, we are going to show that in the case of thermodesorption, neglecting the (re)adsorption term has far smaller negative consequences than in the case of the isothermal adsorption kinetics. In the case of the isothermal kinetics, the key is the dramatic difference between the ART and SRT adsorption terms, arising from their different dependences on pressure p and coverage θ. And at constant T, the p and θ determine the rate of adsorption. In the case of thermodesorption, the key is the temperature dependence mostly governed by the exponential terms. So, let us consider the temperature dependence of (dθ/dt) in a hypothetical TPD experiment in which system CO2/ scandia would be subjected to investigation. From the SRT equations, we have determined the parameters gs and 0. So, let us consider the rate of thermodesorption dθt/dt at the TPD peak maximum, that is, when the rate of desorption is predominantly determined by the desorption from the sites characterized by the most probable adsorption energy 0 (the condensation approximation). Then, to a good approximation the temperature dependence of Ra and Rd will be governed by the following exponential terms: (65) Rudzinski, W.; Panczyk, T. J. Phys. Chem. B 2001, 105, 6858.


(

Ra ∼ exp

(

Rd ∼ exp

)

(

Langmuir, Vol. 18, No. 2, 2002 449

)

0 - gs 3.3 kJ/mol ) exp kT RT

)

(

(35)

)

-0 - gs 24.1 kJ/mol ) exp kT RT

(36)

Thus, in the course of thermodesorption the Ra term would behave like a slowly varying function of T compared to the rapid changes of Rd with temperature. In other words, because the adsorption term does not practically affect the temperature dependence of dθt/dt neglecting this term should not overshadow much the results of a theoretical analysis of TPD spectra based on the ART approach. And, as we have emphasized already, the ART and SRT desorption terms are identical.66 In the hypothetical TPD experiment, we would discover, of course, that the most probable activation energy for CO2 desorption from scandia is about 24.1 kJ/mol. (66) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. Appl. Catal., A 2001, 5815, 1.

One may pose the question of why did we not assume a certain variation of gs on various adsorption sites. This is because we could arrive at an excellent agreement with experiment without making such an assumption. Let us, however, remark that while using the ART and putting γ ) 0, we made a similar assumption that there is no dispersion of the activation energies of adsorption on different sites. But there, this assumption led to a poor prediction of the isothermal adsorption kinetics in different temperatures and led us to physically unacceptable values of parameters. Thus, while concluding the results of our theoretical/ numerical studies of the isothermal kinetics of CO2 adsorption on scandia, we would say that the obtained results strongly favor using the new SRT approach rather than the classical ART approach. Acknowledgment. This work was supported by the Polish State Committee for Scientific Research (KBN) Grant No. 3 T09A 066 18. LA0109664

Remarks on the Current State of Adsorption Kinetic Theories for

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