Theory of Thermodesorption from Energetically Heterogeneous

Langmuir 2000, 16, 8037-8049

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Theory of Thermodesorption from Energetically Heterogeneous Surfaces: Combined Effects of Surface Heterogeneity, Readsorption, and Interactions between the Adsorbed Molecules Wladyslaw Rudzinski,*,† Tadeusz Borowiecki,‡ Tomasz Panczyk,† and Anna Dominko‡ Department of Theoretical Chemistry and Department of Chemical Technology, Faculty of Chemistry, UMCS, pl. Marii Curie-Sklodowskiej 3, Lublin 20-031, Poland Received December 17, 1999. In Final Form: June 29, 2000 Theoretical analysis of thermodesorption from energetically heterogeneous solid surfaces is commonly based on the absolute rate theory in the form of the Wigner-Polanyi (W-P) equation, which neglects readsorption. As a consequence, the results of such analyses contain errors of unknown type and magnitude. Here we show that the application of the statistical rate theory of interfacial transport, instead of the W-P equations, leads to adsorption/desorption rate equations which simplify taking readsorption effects into account. An extensive model investigation is presented to show how neglecting readsorption can affect the theoretical analysis of experimental TPD peaks. Our investigation shows that readsorption can mimic effects which are actually due to surface energetic heterogeneity and/or to interactions between the adsorbed molecules. In addition to illustrative model calculations, the role of readsorption is also demonstrated by a quantitative analysis of spectra of hydrogen thermodesorption from a nickel catalyst.

Introduction Since the first thermodesorption (commonly know as temperature programmed desorption or TPD) experiments were carried out at the beginning of the seventies, TPD has been used to study the energetic heterogeneity of adsorbents and catalysts surfaces.1-9 Until very recently the interpretation of TPD data was almost exclusively based on the absolute rate theory.10 This assumes that at a fixed temperature T, the rate of adsorption is described by

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

(1)

where θ is the fractional coverage of adsorption sites, s is * To whom correspondence should be adressed. Ph: +48 81 5375633. Fax: +48 81 5375685. E-mail: [email protected]. lublin.pl. † Department of Theoretical Chemistry. ‡ Department of Chemical Technology. (1) Amenomiya, Y.; Cvetanovic, R. J. J. Phys. Chem. 1963, 67, 144. (2) Cvetanovic, R. J.; Amenomiya, Y. Catal. Rev. Sci. Eng. 1972, 6, 21. (3) Falconer, I. L.; Schwarz, I. A. Catal. Rev. Sci. Eng. 1983, 25, 141. (4) Lemaitre, I. L. Temperature-Programmed Methods. In Characterisation of Heterogeneous Catalysts; Delannay, F., Ed.; Marcell Dekker Inc.: New York, 1984; Chapter 2. (5) 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, U.K., 1991; Chapter 6. (6) 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. (7) Bhatia, S.; Beltramini, I.; Do, D. D. Catal. Today 1990, 8, 309. (8) Tovbin, Yu. K. Theory of Physical Chemistry Processes at a GaSolid Interface; Izd. Nauka: Moscow, 1990 (English transl.; CRC Press Inc.: Boca Raton, FL, 1991). (9) 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: Amsterdam, 1997. (10) Clark, A. The Theory of Adsorption and Catalysis; Academic Press: New York, 1970.

the number of sites involved in an elementary adsorption/ desorption process, t is time, p is the partial pressure of the gas-phase adsorbate, T is absolute temperature, a and d are activation energies for adsorption and desorption, and Ka and Kd are the related constants.10 When s ) 1 and dθ/dt ) 0, eq 1 yields the Langmuir adsorption isotherm

( kT ) exp( kT)

Kp(e) exp θ(e)(p, T) ) 1 + Kp(e)

(2)

where K ) Ka/Kd and ) (d - a) and where the superscript (e) refers to equilibrium. Equation 1, with s ) 1, describes what is commonly called “Langmuirian kinetics”. Treatment of TPD data is usually further simplified since almost all papers aimed at a theoretical interpretation of TPD peaks were based on an abbreviated form of eq 1, taking account of the desorption term only:

( )

d dθ ) -Kdθs exp dt kT

(3)

This truncated rate expression is commonly called the “Wigner-Polanyi” equation. One of the reasons for using the truncated rate expression was probably the lack of a fundamental concept required to use both terms in eq 1, though there is little discussion in the literature concerning the reasons for the omission. The missing concept in the case of the energetically heterogeneous solid surfaces is what kind of a relationship exists between a and d on adsorption sites characterized by various values of the adsorption energy ? Does any kind of correlation exists at all? If not, one has to introduce into the theoretical description of TPD two functions, one describing the differential distribution of site activation energy for adsorption, χ(a), and the second for the activation energies for desorption, χ(d). This means that

10.1021/la991651f CCC: $19.00 © 2000 American Chemical Society Published on Web 09/13/2000

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Langmuir, Vol. 16, No. 21, 2000

if one would like to consider desorption and readsorption a two-dimensional energy distribution, χ(a, d), would have to be introduced into the theoretical formulation. This unresolved problem is also the probable reason theoretical studies of isothermal adsorption kinetics usually neglected11-15 the second term in eq 1. At the beginning of the eighties, a new family of theoretical approaches appeared presenting an opportunity of solving the problem of taking into account the simultaneously presence of adsorption and desorption. A common feature of these approaches is that they relate the rate of adsorption/desorption to the chemical potential of the adsorbed molecules. These ideas have been developed in works by Nagai,16-20 Kreuzer and Payne,5,6,21-23 and Ward and co-workers.24-29 In his works Ward et al. outline various difficulties connected with the use of the ART approach.30 Such observations resulted in the introduction of the concept of “sticking coefficient”.31,32 But then, it was also reported that such sticking coefficients may themselves depend on the instantaneous surface coverage.31,33 Ward and co-workers24-29 have launched a new approach, called the “statistical rate theory of interfacial transport” (SRTIT), which aims to eliminate these difficulties. Another objective of that approach is to relate the kinetic behavior of adsorption/desorption to the behavior of the system at equilibrium. One important consequence of this approach is that the activation energies for adsorption and desorption no longer appear in Ward’s SRTIT approach. Instead, the terms related to adsorption and desorption kinetics are functions of the energy of adsorption . This eliminates the problem of establishing a relationship between a and d on a heterogeneous solid surface while allowing one to simultaneously describe the equilibrium isotherm and the kinetics of both desorption and the adsorption. In a recent series of papers, Rudzinski et al.15,34-41 have shown how the SRTIT approach may be further generalized to describe both the kinetics of isothermal adsorption/ (11) Elovich, S. Yu.; Kharakorin, F. F. Probl. Kinet.Catal. 1937, 3, 322. (12) Kharakorin, F. F.; Elovich, S. Ya. Acta Physicochim. U.S.S.R.1936, 5, 325. (13) Elovich, S. Ya.; Zabrova, G. M. Zh. Fiz. Khim. 1939, 13, 1775. (14) Aharoni, C.; Tompkins, F. C. Adv. Catal. 1970, 1, 21. (15) 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: New York, 1999; Chapter 15, pp 355-390. (16) Nagai, K. Phys. Rev. Lett. 1985, 54, 2159. (17) Nagai, K. Surf. Sci. 1986, 176, 193. (18) Nagai, K.; Hirashima, A. Surf. Sci. 1986, L464, 171. (19) Nagai, K. Surf. Sci. 1988, L659, 203. (20) Nagai, K. Surf. Sci. 1991, L147, 244. (21) Kreuzer, H. J.; Payne, S. H. Surf. Sci. 1988, 198, 235; 1988, L433, 200. (22) Payne, S. H.; Kreuzer, H. J. Surf. Sci. 1988, 205, 153. (23) Kreuzer, H. J. Langmuir 1992, 8, 774. (24) Ward, C. A.; Findlay, R. D. J. Chem. Phys. 1982, 76, 5615. (25) Findaly, R. D.; Ward, C. A. J. Chem. Phys. 1982, 76, 5624. (26) Ward, C. A.; Elmoselhi, M. Surf. Sci. 1986, 176, 457. (27) Elliot, J. A.; Ward, C. A. J. Chem. Phys. 1997, 106, 5667. (28) Elliot, J. A.; Ward, C. A. J. Chem. Phys. 1997, 106, 5677. (29) Elliot, J. A.; Ward, C. A. Statistical Rate Theory and Material Properties Controlling Adsorption Kinetics. In Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich G.; Elsevier: New York, 1997; p 285. (30) Rublof, G. W. Surf. Sci. 1979, 89, 566. (31) Kisliuk, P. J. Phys. Chem. Solids 1957, 3, 95. (32) Zangwill, A. Physics at Surfaces; Cambridge University Press: Cambridge, U.K., 1988; p 363. (33) Christman, K.; Schober, O.; Ertl., G. J. Chem. Phys. 1974, 60, 4719. (34) Rudzinski, W.; Aharoni, C. Langmuir 1997, 13, 1089.

Rudzinski et al.

desorption on energetically heterogeneous solid surfaces and the kinetics of thermodesorption from such surfaces. In particular a paper40 published in Langmuir shows how readsorption may affect the interpretation of experimental TPD peaks. It shows that readsorption simulates the effects of surface energetic heterogeneity. Neglecting readsorption may therefore lead to a serious overestimate of the degree of surface heterogeneity calculated from experimental TPD peaks. In a typical chemisorption system in catalysis, gassolid interactions are believed to be strongly dominant over any possible interactions between adsorbed molecules. As a result, the latter interactions are ignored in the analysis of TPD spectra which are examined in the course of catalysis research. However, it is possible that there are adsorption systems in which the interactions between adsorbed molecules affect the behavior of both adsorption equilibria and adsorption kinetics. All the same, it is thought that these should typically be physisorption systems which play little role in catalysis. There is a large literature treating how the inclusion of interactions between the adsorbed molecules may affect the interpretation of experimental TPD spectra. So far, all the published papers have been based on the WignerPolanyi eq 3. To account for the admolecule-admolecule interactions an additional interaction energy term in was added to d. This configurational energy was calculated using the simple Bragg-Williams (mean-field) approximation, the Bethe-Peierls (quasi-chemical) approximation, a combination of these two approximations, or other analytical approaches. More recently, Monte Carlo simulations have been used for that purpose, and the paper by Meng and Weinberg42 brings an exhaustive review of the relevant literature up to 1994. More recently, papers on this subject have been published by Houle and Hinsberg,43 Weinketz,44 Weinketz and Cabrera,45 and Cortes et al.46 Throughout most of this period, the theoretical studies of the effects of interactions between the adsorbed molecules were based on a model involving an energetically uniform surface, the Wigner-Polanyi eq 3 was accepted, and a function in(θ) calculated in some way. In 1982 Cordoba and Luque47 departed from this norm and used the Bragg-Williams approximation to study the combined effect of adsorbate-adsorbate interactions and surface energetic heterogeneity. Today the most sophisticated examination of the combined effect can be found in papers by Tovbin and co-workers8,9 and by Sales and Zgrablich48 and Cortes et al.,46 who have applied Monte Carlo simulations to the study of the combined effect. However, (35) Rudzinski, W. A New Theoretical Approach to AdsorptionDesorption Kinetics on Energetically Heterogeneous Flat Solid Surfaces, Based on Statistical Rate Theory of Interfacial Transport. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: New York, 1996. (36) Rudzinski, W.; Aharoni, C. Pol. J. Chem. 1995, 69, 1066. (37) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Zientarska, M. Chem. Anal. (Warsaw) 1996, 41, 1057. (38) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Panczyk, T. Langmuir 1997, 13, 3445. (39) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Panczyk, T.; Gryglicki, J. Pol. J. Chem. 1998, 72, 2103. (40) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. Langmuir, in press. (41) Rudzinski, W.; Panczyk, T. Adv. Colloids Interface Sci., in press. (42) Meng, B.; Weinberg, H. J. Chem. Phys. 1994, 100, 5280. (43) Houle, F. A.; Hinsberg, W. D. Surf. Sci. 1995, 338, 329. (44) Weinketz, S. J. Chem. Phys. 1994, 101, 1632. (45) Weinketz, S.; Cabrera, G. G. J. Chem. Phys. 1997, 106, 1620. (46) Cortes, J.; Valencia, E.; Araya, P. J. Chem. Phys. 1994, 100, 7672. (47) Cordoba, A.; Luque, J. J. Phys. Rev. 1982, B26, 4028. (48) Sales, J. I.; Zgrablich, G. Surf. Sci. 1987, 1, 187.

Thermodesorption from Heterogeneous Surfaces

Langmuir, Vol. 16, No. 21, 2000 8039

all the above-described theoretical studies are based on using the Wigner-Polanyi eq 3, i.e., neglecting readsorption. Thus, the current situation can be summed up as follows: There is a lack of papers showing how the three main physical factors acting simultaneously, i.e., surface energetic heterogeneity, readsorption, and interactions between the adsorbed molecules, acting simultaneously, affect the behavior of experimentally recorded TPD peaks. This is the subject matter of the present publication where our considerations of this problem will be based on the SRTIT approach.

1. The Principles of the SRTIT Approach. Consider adsorption on an energetically homogeneous solid surface. The SRTIT approach leads to the following expression for adsorption/desorption kinetics,

)

(

)

dθ µg - µs µs - µg ) K’gs exp - K’gs exp dt kT kT

(4)

where µg is the chemical potential of the molecules in the gas phase, µs is the chemical potential of the adsorbed molecules, and K’gs is a constant related to the exchange rate between the gas phase and the solid surface once the system has reached equilibrium. To a good approximation, K’gs can be written as the product,27,28

K’gs ) Kgsp(e)(1 - θ(e))

µg ) µ0g + kT ln p

(6)

Then for the Langmuir model of adsorption we have

µs ) kT ln

θ q (1 - θ)

(7)

s

where the molecular partition function qs is the product,

Theory

(

when Langmuir model of adsorption is accepted at equilibrium. As in our previous publications we assume ideal gas behavior in the gas phase. So we write

(5)

where the equilibrium state (e) is defined as that to which a system isolated at a surface coverage θ and gas-phase concentration with partial pressure p would evolve. The first term on the rhs of eq 4 describes the rate of adsorption, while the second term on the rhs of eq 4 describes the rate of desorption. It turns out that depending on the conditions at which the experiment is carried out, one may observe somewhat different behavior of the adsorption/desorption kinetics. To see to what extent the experimental conditions (including the technical characteristics of the experimental setup) may affect the observed adsorption/desorption kinetics, we will consider the following three extreme cases: (1) The adsorption process is essentially a nonequilibrium one, and the features of a gas/solid system are “volume dominated”; i.e., the amount of adsorbate in the gas phase above the surface is much larger than the portion adsorbed. In that case, after the system is isolated and equilibrated, the gas pressure p does not change much, so that p(e) ≈ p. (2) The process is a nonequilibrium one, but the features of a gas/solid system are “solid dominated”. In the case of the “solid dominated” system the adsorbed amount is much larger than the amount in the bulk gas phase, so that after isolation of the system and equilibration, θ remains practically unchanged and θ ≈ θ(e). (3) The process is carried out at conditions such that the gas/solid system is close to equilibrium. We will call such systems “equilibrium dominated”; i.e., the process is carried out at such conditions that one may assume θ ≈ θ(e) and p ≈ p(e). This is the case in practice when temperature ramping and gas flow is slow. At equilibrium the ART rate expression (1) yields the Langmuir isotherm (2), and it is interesting to see what rate expressions are obtained using the SRTIT approach

( kT )

qs ) q0s exp

(8)

Equation 4 now takes then the form

1 θ dθ 1-θ ) Kap exp exp - Kd dt θ kT p1-θ kT [1 - θ(e)]p(e) (9)

[

( )

(

)]

where

( )

µ0g Ka ) Kgsq0 exp kT s

Kd )

Kgs q0s

( )

µ0g exp kT

(10)

At equilibrium, when (dθ/dt) ) 0, eq 8 yields the Langmuir isotherm (2), in which K ) xKa/Kd ) q0s exp(µ0g/kT). Thus,

1

1 - θ(e) ) 1 + Kp

(e)

(11)

kT

( )

exp

Thus, for the “volume dominated” systems, where p(e) ) p, we have

1-θ θ dθ ) Kap2 exp exp - Kd 1+ dt θ kT 1-θ kT -1 (12) Kp exp kT

[

( )][ ( )]

( )

whereas for the “solid dominated” systems, where θ(e) ) θ, we obtain

K d 1 θ2 2 dθ Ka ) p(1 - θ) exp dt K K p (1 - θ) kT

( )

(13)

Finally, for the “equilibrium dominated” systems, where both p(e) ) p and θ(e) ) θ, we arrive at the following expression, 2

(1 - θ) dθ ) Kap2 exp - Kdθ exp (14) dt θ kT kT

( )

( )

Equations 12-14 describe the kinetics of adsorption/ desorption on a hypothetical energetically homogeneous solid surface, where all the adsorption sites are characterized by the adsorption energy . Looking at these dependencies of (dθ/dt) on surface coverage θ, we see that none of them is identical with the ART rate expression (1). Moreover, it turns out that various preexponential terms are now functions of coverage, as they are found to be by experiment, but not as they appear in eq 1. Further differences show up when one considers adsorption/ desorption kinetics on the energetically heterogeneous solid surfaces.

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2. Generalization of the Adsorption Kinetics Equations for the Case of Energetically Heterogeneous Solid Surfaces. To bring out these differences consider the processes where desorption is strongly dominant over readsorption so that the first terms in eqs 1 and 12-14 can be neglected. This is the assumption made in thermodesorption studies based on the WignerPolanyi equation. In studies based on the Wigner-Polanyi eq 3, it was assumed that the rate of desorption from sites characterized by a value d depends only on the coverage of these sites and is independent of coverage on other kinds of sites. So the assumed rate of desorption from the total surface (dθt/dt) was described by the expression49-56

dθt ) dt

χ( ) dd ∫0∞ (dθ dt ) d

(15)

where χ(d) is the differential distribution of the number of sites among various values of d, normalized to unity,

∫0∞ χ(d) dd ) 1

(16)

where θ(d) was found by integrating the rate expression (3), to yield

[

( )]

d θ(d) ) exp -Kdt exp kT

(17)

In contrast, in studies of adsorption/desorption kinetics based on the SRTIT approach, the rate of adsorption/ desorption kinetics on sites characterized by an adsorption energy is assumed to be dependent on the coverage of all surface sites present on an energetically heterogeneous surface. This comes about because of the assumption, inherent in the SRTIT approach, that the adsorbed phase is at “quasi-equilibrium”; i.e., all the surface correlation functions are the same as they would be at equilibrium at the same surface coverage. In view of this, it would not be correct to write (dθt/dt) in the form

dθt ) dt

χ() d ∫0∞ (dθ dt )

(18)

where χ() is the differential distribution of the number of sites among corresponding values of , and to find θ() by integrating the appropriate equation from among (12)(14). In the case of SRTIT approach θ() is also function of θt through eq 7 in which µs is dependent on θt. At “quasiequilibrium” conditions the function θ(,θt) is the same as it is at equilibrium. We will express this interdependence in the following way. We rewrite the Langmuir eq 2 in the following form:

( ) ( )

- c kT θ(,c,T) ) - c 1 + exp kT exp

(19)

At equilibrium between the gas and surface phase c ) (49) Carter, G. Vacuum 1962, 12, 245. (50) Witkopf, H. Vacuum 1984, 37, 819. (51) Britten, J. A.; Travis, B. J.; Brown, L. F. Adsorption and Ion Exchange AIChE Symposium; AIChE: New York, 1983; p 7. (52) Carter, G.; Bailey, P.; Armour, D. G. Vacuum 1982, 32, 233. (53) Du, Z.; Sarofim, A. F.; Longwell, J. P. Energy Fuels 1990, 4, 296. (54) Ma, M. C.; Brown, T. C.; Haynes, B. S. Surf. Sci. 1993, 297, 312. (55) Seebauer, E. G. Surf. Sci. 1994, 316, 391. (56) Cerofolini, G. F.; Re, N. J. Colloid Interface Sci. 1995, 174, 428.

kT ln Kp(e), whereas, at the “quasi-equilibrium” conditions assumed by Ward and Findlay, only the following relationship holds (see eq 7):

c ) -µs - kT ln q0s

(20)

Next, θ(,p,T) in eq 19 is related to the experimentally observed overall surface coverage θt(p,T) by the “integral isotherm equation”,57-60

θt(c,T) )

∫0∞ θ(,c,T)χ() d

(21)

We introduce the function c in eq 20 because we will be looking later for approximate and compact analytical solutions for various quantities of interest using methods based on the condensation approximation, where the function c will play a role.58 3. Effects of Interactions between the Adsorbed Molecules. Equation 21 remains valid when the Langmuir model is generalized to take into account interactions between adsorbed molecules though an additional configurational energy term in must be added to the rhs of eq 7 for µs. To find the expression for in we may use the simplest mean-field approach (Bragg-Williams approximation). In this way, for an energetically homogeneous solid surface, we have

4Tc θ θ µs ) kT ln s T q (1 - θ)

(22)

where Tc is the critical temperature of the 2-dimensional lattice gas T dense ordered phase transition. The Tc value is given by -cw/4k, where w is the interaction energy between two molecules adsorbed on two neighboring sites and c is the lattice parameter. According to the notation used here, w takes negative values in the case of attractive interactions between admolecules. The mean-field approximation (MFA) is known to provide a good description of the interaction effects at temperatures far from the critical temperature. Of course, instead of the simple Bragg-Williams approximation we can choose others, for instance the quasi-chemical approximation which is more accurate as critical conditions are aproached. In this paper, the thermodesorption temperatures will in general be far from the critical point at bulk conditions; however, it has been shown that when a surface is energetically heterogeneous, critical temperatures are shifted down to much lower values.58 Thus, we will write eq 22 in the following form:

θ + cwθ µs ) kT ln s q (1 - θ)

(23)

The relation Tc ) -cw/4k is no longer valid for energetically heterogeneous solid surfaces. When one uses the more accurate quasi-chemical (QC) approximation, eq 7 takes the following more complicated but more accurate form: (57) Jaroniec, M.; Madey, E. Physical Adsorption on Heterogeneous Solids; Elsevier: New York, 1989. (58) Rudzinski, W.; Everett, D. M. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (59) Cerofolini, G. F.; Re, N. Riv. Nuovo Cimento 1993, No. 7, 16. (60) Cerofolini, G. F.; Rudzinski, W. Theoretical Principles of Singleand Mixed- Gas Adsorption Equilibria on Heterogeneous Solid Surfaces. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: New York, 1997; p 1.


Langmuir, Vol. 16, No. 21, 2000 8041

cw θ + + µs ) kT ln s 2 q (1 - θ) c kT (f - 1 + 2θ)(1 - θ) (24) ln 2 (f + 1 - 2θ)θ Here

{

[

f ) 1 - 4θ(1 - θ) 1 - exp

( )]} -w kT

1/2

cw ckT (f - 1 + 2v)(1 - v) in ) - ln v ) θ, θt (28) 2 2 (f + 1 - 2v)v where

(24a)

{

[

f ) 1 - 4v(1 - v) 1 - exp

Equations 23 and 24 are valid for energetically homogeneous solid surfaces. In the case of energetically heterogeneous surfaces, the effects of the interactions between the adsorbed molecules will also depend on surface topography. In the theoretical studies of adsorption equilibria two limiting kinds of surface topography can be considered: 1. The “patchwise” topography can be considered, when identical adsorption sites appear in the form of large surface domainss“patches”. The boundary effects due to situations when two neighboring admolecules are adsorbed on different patches are neglected. Then, the additional force field acting on an adsorbed molecule, due to the interactions between the adsorbed molecules, i.e., the second terms in eqs 23 and 24, is a function of the local surface coverage θ() of the patch. So, in the case of patchwise topography, eqs 23 and 24 remain unchanged. 2. The “random” surface topography can also be considered, when adsorption sites of differing energy are randomly distributed on a heterogeneous surface. As a consequence, the local surface coverage at any small area of the surface is the same across the surface and equal to the average surface coverage θt. Thus, the additional force field due to the interactions between the adsorbed molecules will be a function of θt. As a result, eq 7 takes the following more general form. For the simplest MFA approach we have

µs ) kT ln

where v ) θ for patchwise topography and v ) θt for random topography. In the case of the more accurate QC approach we have

θ + cwθt qs(1 - θ)

(25)

whereas for the more accurate QC approach we obtain

θ cw + + µs ) kT ln s 2 q (1 - θ)

(-w kT )]}

1/2

(28a)

Finally, to establish the relationship between θ() and θt from eqs 19-21, one introduces the appropriate expressions for µs form eq 20. 4. Application of SRTIT to the Study of Thermodesorption from Energetically Heterogeneous Surfaces. In TPD experiments at every temperature θt is found from the relation

θt(T) )

N0 F Nm βNm

∫TT c(T) dT 0

(29)

where θt(T) is the fraction of the surface covered at time t when the temperature is T, β is the heating rate in the “ramping” function T ) T0 + βt, F is the volumetric flow rate of the carrier gas, N0 is the amount of preadsorbed species in a given experiment, and Nm is the monolayer capacity of this sample, which has been determined in a separate experiment, and the instantaneous gas-phase concentration, c(T) of the desorbing species is given by

βNm dθt c(T) ) F dT

(29a)

Because the thermodesorption is assumed to run at quasiequilibrium conditions, defined as the conditions at which the surface correlation functions are practically the same as those at equilibrium, (dθt/dT) is evaluated from eq 21.

dθt ∂θt ∂θt dc ) + dT ∂T ∂c dT

(30)

So, from eqs 29 and 30 we have

c kT (f - 1 + 2θt)(1 - θt) ln (26) 2 (f + 1 - 2θt)θt

dc ∂θt Fc(T) + ) χc(c) βNm dT ∂T

(31)

where χc(c) is the “condensation distribution function”, defined as follows:

where

{

[

f ) 1 - 4θt(1 - θt) 1 - exp

(-w kT )]}

1/2

(26a)

Other models of surface topography have been considered in the adsorption literature, but we will limit our interest to these two kinds of surface topography to see how surface topography affects experimental TPD spectra. We complete our considerations in this section by developing rate equations for local kinetics of adsorption, dθ/dt, on heterogeneous solid surfaces in the presence of interactions between adsorbed molecules. According to eqs 4 and 23-26 these generalized local kinetics equations are obtained from eqs 12-14, by adding to an interaction energy term in, which for the MFA approach takes the form

in ) -cwv v ) θ, θt

(27)

χc(c) )

χ() d ∫0∞(∂θ ∂ )

(32)

Note that, at a given surface coverage θt, the local coverage θ defined in eq 19 is a function of time t, because both T and θt in c (eq 21) change with time. Thus, from eq 19 we have

( - c)β dθ ∂θ dT ∂θ dc ) + ) θ(1 - θ) dt ∂T dt ∂c dt kT2 dc 1 θ(1 - θ) (33) kT dt Now, consider the value of (dθ/dt) on adsorption sites whose adsorption energy ) c. On these sites θ ) 1/2, and from eq 33 we have

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1 dc ))c,θ)1/2 4kT dt

(dθdt )

(34)

The rhs of eq 34 is now compared with the value of (dθ/ dt)θ)1/2,)c, evaluated from eqs 12-14 in which c is replaced by c + in. For the “volume dominated” systems we have

( ) dθ dt

)c,θ)1/2

[

( )][

) Kap2 exp

(

c + in kT

Kd exp -

)

c + in kT 1 + Kp exp

)]

(

c + in kT

-1

(35)

whereas for the “solid dominated” systems we obtain,

( ) dθ dt

)

)c,θ)1/2

(

)

Kd 1 Ka 2(c + in) pexp 2K 2K p kT

(36)

dθ dt

)c,θ)1/2

(

)

c + in 1 ) Kap2 exp 2 kT

(

)

c + in 1 K exp (37) 2 d kT For patchwise topography in in eqs 35-37 is given by eq 27 or 28, in which v ) 1/2, whereas for random topography v ) θt. After replacing p by Pc(T) in eqs 35-37, and comparing them with eq 34, we arrive at the following three expressions for (dc/dT) ) (1/β)(dc/dt). For the “volume dominated” systems, we have

[

(

)

c + in dc 4kT KaP2c2(T) exp - Kd )dT β kT c + in c + in exp 1 + KPc(T) exp kT kT

(

)][

)]

(

-1

(38)

whereas, for the “solid dominated” systems, we obtain

(

)

2kTKa 2kTKd 1 2(c + in) dc )Pc(T) + exp dT βK βK Pc(T) kT (39) Then, for “equilibrium dominated” systems,

(

)

c + in dc 2kT 2kT )K P2c2(T) exp K + dT β a kT β d c + in exp (40) kT

(

c2 ∂2χc 2 χ(c) ) χc(c) - (kT) 2 ∂ 2

(41)

c

where the value of the coefficient c2 depends on surface topography. For surfaces with random topography, c2 is equal to -3.29, whereas, for surfaces characterized by patchwise topography,58

Finally, for “equilibrium dominated” systems

( )

normalized to unity. As discussed in previous work, that boundary condition will be the value of c at a certain temperature T. This issue was examined in detail in a previous publication.41 Having determined the CA function χc(c), one can calculate the true distribution function, χ(), by solving eq 32. Solving such equations is usually not a trivial problem. Fortunately, for the special case of eq 32, compact approximate solutions have been developed in works published by Rudzinski and Jagiello.58,61-64 Using these methods the adsorption energy distribution χ() can be calculated, to a good approximation, as follows:

)

Equations 38-40 can be treated as first-order differential equations for the function c(T), because the function c(T) is known from experiment, while θt(T), appearing in in (in the case of random topography), is calculated from eq 28. For physical reasons, c(T) should be a one-to-one function of T. Having determined the function c(T), one can find the condensation approximation χc(c) of the adsorption energy distribution χ() from eq 31 provided that ∂θt/∂T is known. However, as we have shown previously,40 ∂θt/∂T pops-up as only a correction to the first term on the rhs of eq 31 and can usually be safely neglected. We have also shown in previous publications that by correctly determining χc(c) we also find a boundary condition for the solution of any of the eqs 38-40. A properly chosen boundary condition in those solutions causes the function χc(c), determined from experimental TPD peaks c(T), to be

(kTw ) ]

[

c2 ) -3.29 exp -1.64

1.24

(42)

5. Predictive Features of the SRTIT Approach. According to the above considerations, the shape of experimentally monitored TPD peaks can be influenced by the following factors: (1) surface energetic heterogeneity, i.e., the adsorption energy dispersion χ(); (2) topography of surface adsorption sites; (3) interactions between the adsorbed molecules; (4) readsorption kinetics; (5) the technical features of the experimental setup, i.e., is the sample in a “volume” or a “solid” dominated setting; (6) the speed of thermodesorptionsthe transition from the “quasi-equilibrium” conditions to conditions at which the system can be considered to be “equilibrium dominated”. Of course, longitudinal diffusion could also affect TPD peaks to a some extent, but we will omit discussing of that effect for now. Thus, there is a longish list of factors influencing TPD peaks. Since they act simultaneously, their combined effect can easily result in TPD peaks which are not easily understood. On the other hand, such understanding is necessary if we are to draw quantitative, or even correct qualitative conclusions, from an analysis of experimental TPD data. For this reason, we have decided to carry out numerical calculations to see how various factors affect TPD peaks in the presence of the other accompanying factors. To do this we now develop appropriate equations to calculate the c(T) function for a given adsorption energy distribution χ(). The first step in the theoretical calculation of c(T), when χ() is known, is the evaluation of the CF (condensation function), χc(c), from eq 32. According to the notation in eq 20, the kernel (∂θ/∂) will always be the same,

( ) ( )]

c - 1 kT ∂θ ) ∂ kT c - 1 + exp kT exp

[

2

(43)

(61) Rudzinski, W.; Narkiewicz, J.; Patrykiejew, A. Z. Phys. Chem. (Leipzig) 1979, 260, 1097. (62) Rudzinski, W.; Jagiello, J. J. Low Temp. Phys. 1981, 1, 45. (63) Rudzinski, W.; Jagiello, J.; Grillet, Y. J. Colloid Interface Sci. 1982, 87, 478. (64) Jagiello, J.; Schwarz, J. A. J. Colloid Interface Sci. 1991, 146, 415.


Langmuir, Vol. 16, No. 21, 2000 8043

Replacing c(T) in eq 38 with its value calculated from eq 31, we arrive at the following quadratic equation for dc/dT:

dc )dT dc ∂θt 2 β2 + Kae(c+in)/ kTNm2P2 χc - Kde-(c+in)/ kT 2 dT ∂T 4kT F β dc ∂θt β + 1 + Ke(c+in)/ kTNmP χc F dT ∂T (44)

(

(

)

)

Of its two roots only the following one has physical meaning:

dc ) - [F2 + βe(c+in)/kTNmP(FK + 8χcKakTNmP) × dT (∂θt/∂T)]/[2βχce(c+in)/kTNmP(FK + 4χcKakTNmP)] + (45) {F[F2 + NmP(16χcFKKdkT + 64χc2KaKd(kT)2NmP + (∂θt/∂T)(2βFK + 16βχcKakTNmP + β2e(c+in)/kTK2NmP(∂θt/∂T))e(c+in)/kT)]1/2}/ [2βχce(c+in)/kTNmP(FK + 4χcKakTNmP)] The above equation serves for the “volume dominated” systems. For the “solid dominated” system, the expression for dc/dT is obtained by replacing c(T) in eq 39 by its value calculated from eq 31. After solving the resulting quadratic equation for dc/dT, we arrive at the following expression for its physically meaningful root:

dc ) -[(FK + 4χcKakTNmP)(∂θt/∂T)]/[2χc(FK + dT 2χcKakTNmP)] + (46) {F[NmP(8χcFKKdkT + 16χc2KaKd(kT)2NmP + β2e2(c+in)/kTK2NmP(∂θt/∂T)2)]1/2}/[2βχce(c+in)/kTNmP (FK + 2χcKakTNmP)] Finally for the “equilibrium dominated” systems the expression for dc/dT is obtained by replacing c(T) in eq 40 by its value calculated from eq 31. After solving the resulting quadratic equation, we obtain the following expression for the physically meaningful root:

dc ) -[F2 + 4βχce(c+in)/kTKakTNm2P2(∂θt/∂T)]/ dT [4βχc2e(c+in)/kTKakTNm2P2] + (47) {F[F2 + 16χc2KaKd(kT)2Nm2P2 + 8βχce(c+in)/kTKakTNm2P2(∂θt/∂T)]1/2}/ [4βχc2e(c+in)/kTKakTNm2P2] As for the function ∂θt/∂T, to a good approximation, it can be represented by the following expressions developed in ref 58. For random topography and the QC approximation,

∂θt ) -3.29k2Tχ’(c) ∂T

(48)

Figure 1. The behavior of χ() defined in eq 50 for 0 ) 80 kJ/mol and three values of the heterogeneity parameter R: 8kJ/ mol (s), 10 kJ/mol (- - -), 12 kJ/mol (---).

whereas, for patchwise topography, we obtain

∂θt w ) -3.29k2Tχ’(c) exp -1.52 ∂T kT

[

1.2

( ) ]

(49)

After solving equations 45-47, we obtain three functions for each of c(T) and T(c)sfor the “volume”, “solid”, and “equilibrium dominated” systems, respectively. In the solution of these equations, the boundary condition is dictated by the requirement that the resulting functions χc(c) must be normalized to unity. We now take χ() to be the following quasi-Gaussian function:

χ() )

(

)[

(

)]

- 0 - 0 1 exp / 1 + exp R R R

2

(50)

centered at ) 0, its variance being πRx3. The behavior of χ() defined in eq 50 is shown in Figure 1. If one accepts the CA level of accuracy and the Langmuir model of adsorption, function (50) yields the LangmuirFreundlich isotherm at equilibrium.35,57,58 This isotherm has frequently been used to correlate experimental adsorption isotherm data. This leads us to believe that the real χ() function of many surfaces can be approximated by eq 50. 6. Illustrative Model Calculations and Their Discussion. At the beginning of section 5, we listed six physical factors which may influence the shape of observed TPD peaks. These factors can act in various combinations, so that in order to fully understand their influence we would need to carry out a very extensive set of model calculations. The strategy of our model investigation was simplified to avoid this problem by proceeding as follows: in each simulation a chosen set of the six parameters was kept constant while other parameters were varied. We start our investigation with the case w ) 0, i.e., when the effects due to interactions between the adsorbed molecules can be ignored. The results are shown in Figures 2-5. The aim of presenting the data in Figure 2 is to show how experimental conditions may affect TPD peaks in systems characterized by various adsorption and desorption constants, Ka and Kd. It is interesting to see that for some sets of parameters the “solid”, “volume”, and the “equilibrium” dominated systems behave in a similar way. This is the case for TPD peaks shown in Figure 2C,D. As they correspond to very different values of K, we look for the source of this phenomenon to the corresponding Ka and Kd values. We note that Figure 2C,D corresponds to the situation when either Kd or Ka is large. Figure 2 also shows that changing the heating rate will not affect the similarity in behavior of the S (“solid dominated”), V (“volume dominated”), and E (“equilibrium dominated”) systems.

8044


Figure 2. Ideal systems, w ) 0. Effects due to construction of an experimental set and experimental conditions. The solid line (s) is for “solid dominated” systems and the slightly broken line (- - -) is for the “volume dominated” systems, whereas the strongly broken line (---) is for the “equilibrium dominated” systems. Key: (A-C) Ka ) 10-2 min-1 atm-2 (constant) and with changing Kd (A) Kd ) 105 min-1, (B) Kd ) 106 min-1, (C) Kd ) 107 min-1; (D-F) Kd ) 106 min-1 (constant) and with changing Ka (D) Ka ) 10-1 min-1 atm-2, (E) Ka ) 10-2 min-1 atm-2, (F) Ka ) 10-3 min-1 atm-2. Other parameters: R ) 10 kJ/mol; β ) 40 K/min; F ) 0.03 dm3/min; 0 ) 80 kJ/mol.

Figure 3. Effect of the heating rate on the behavior of the “solid”, “volume”, and “equilibrium” dominated systems. Key: (A-C) the TPD curves from Figure 2C plotted for different heating rates β, (A) β ) 20 K/min, (B) β ) 40 K/min, (C) β ) 60 K/min; (D-F) the TPD curves from Figure 2D plotted for different heating rates β, (D) β ) 20 K/min, (E) β ) 40 K/min, (F) β ) 60 K/min.

In general, however, when the behavior of the S, V, and E systems is different, the V and E systems will show a similar behavior, which behavior is significantly different from the behavior of S systems. This is shown in Figure 4. Figure 5 shows that readsorption simulates increased surface heterogeneity. As R or Ka increases, the variance of the TPD peak increases. However, increases in R and Ka affect the detailed shape of TPD peaks in somewhat different way. To distinguish between the effects due to surface energetic heterogeneity and those due to read-

Rudzinski et al.

Figure 4. Ideal systems, w ) 0. Effect of Kd and Ka on the behavior of the “solid”, “volume”, and “equilibrium” dominated systems. Key: (A-C) Ka ) 10-3 min-1 atm-2 (constant) and with changing Kd 108 min-1 (s), 106 min-1 (- - -), 104 min-1 (---); (D-F) Kd ) 106 min-1 (constant) and with changing Ka 10° min-1 atm-2 (s), 10-2 min-1 atm-2 (- - -), 10-5 min-1 atm-2 (---). Other parameters: β ) 40 K/min; F ) 0.03 dm3/min; R ) 10 kJ/mol; 0 ) 80 kJ/mol.

Figure 5. Ideal systems, w ) 0. Combined effect of readsorption and heterogeneity. Key: (A-C) Ka ) 10-3 min-1 atm-2 and with changing R 12 kJ/mol (s), 10 kJ/mol (- - -), 8 kJ/mol (---); (D-F) R ) 10 kJ/mol and with changing Ka 100 min-1 atm-2 (s), 10-2 min-1 atm-2 (- - -), 10-5 min-1 atm-2 (---). Other parameters: Kd ) 106 min-1; β ) 40 K/min; F ) 0.03 dm3/min; 0 ) 80 kJ/mol.

sorption one has to consider the higher moments of the function c(T). Figure 6 shows how interactions between adsorbed molecules and the topography of a heterogeneous solid surface affect TPD peaks when readsorption plays a significant role. One can see that interactions between adsorbed molecules have a greater effect on TPD peak shape in systems with random surface topography. This applies to both the second and higher moments of the function c(T). This should not surprise us. Such difference between the surfaces having random and having patchwise topography has, since a long time, was known in theories of equilibria of adsorption.58


Figure 6. Nonideal systems, w * 0. Combined effect of w and surface topography. Key: (A-C) random topography and changing w: -2 kJ/mol (s), 0 (- - -), 2 kJ/mol (---); (D-F) patchwise topography. Other parameters: Ka ) 10-3 min-1 atm-2; Kd ) 106 min-1; F ) 0.03 dm3/min; β ) 40 K/mol; R ) 10 kJ/mol; 0 ) 80 kJ/mol.

Figure 7. Nonideal systems, w*0. Combined effect of w and Ka (readsorption). Key for random topography systems: (A-C) Ka ) 10-2 min-1 atm-2 and changing w: -2 kJ/mol (s), 0 (- -), 2 kJ/mol (---); (D-F) w ) -2 kJ/mol and changing Ka 10-1 min-1 atm-2 (s), 10-3 min-1 atm-2 (- - -), 10-5 min-1 atm-2 (---). Other parameters: Kd ) 106 min-1; R ) 10 kJ/mol; 0 ) 80 kJ/mol; F ) 0.03 dm3/min; β ) 40 K/min.

We set out in the Introduction to show how readsorption affects the behavior of TPD peaks in the real adsorption systems, where readsorption, surface energetic heterogeneity, and interactions between the adsorbed molecules contribute to the observed behavior. Figures 7 and 8 show to what extent readsorption can mimic the effects of interactions between the adsorbed molecules. As we investigated the effects of readsorption by changing Ka, we also examined the likely behavior of attractive forces between the adsorbed molecules. It seems obvious that changes in Ka will affect the interactions between two molecules adsorbed on neighboring sites. These interactions should still be represented well by a Lennard-Joneslike function, with somewhat different parameters than in the bulk phase.

Langmuir, Vol. 16, No. 21, 2000 8045

Figure 8. Nonideal systems, w * 0. Combined effect of w and Ka (readsorption). Key for patchwise topography systems: (AC) Ka ) 10-2 min-1 atm-2 and changing w: -2 kJ/mol (s), 0 (- -), 2 kJ/mol (---); (D-F) w ) -2 kJ/mol and changing Ka 10-1 min-1 atm-2 (s), 10-3 min-1 atm-2 (- - -), 10-5 min-1 atm-2 (---). Other parameters: Kd ) 106 min-1; R ) 10 kJ/mol; 0 ) 80 kJ/mol; F ) 0.03 dm3/min; β ) 40 K/min.

Figures 7 and 8 suggest that in the most probable case of attractive interactions between the adsorbed molecules, TPD peaks of S systems are the ones most strongly affected by readsorption for both random and patchwise topography. The sensitivity to readsorption decreases in the sequence: S w V w E. In all cases the combination of readsorption and the interactions between admolecules results in complicated effect of peak shape. We see that increasing readsorption mimics not only increased energetic heterogeneity but also the behavior caused by increasing attractive forces between admolecules. We see this in the influence of the two causes, readsorption and lateral interactions, on the shape of c(T) and on the temperature at peak maximum. At the same time increased readsorption has an opposite effect, to that of increased attractive forces between admolecules, on the second moment of c(T). Finally, Figures 9 and 10 show to what extent interactions between adsorbed molecules can mimic surface energetic heterogeneity in the systems where readsorption plays a significant role. Looking at Figure 9 one can see that as far as the second moment of c(T) is concerned, attractive interactions mimic a decrease in surface heterogeneity. We conclude that interactions between the adsorbed molecules affect not only the width of the peak (the first moment) but also the higher moments (the shape) of c(T). Other selections of variables could be investigated, but we believe that the figures shown give an adequate picture of the interactions between the various physical factors influencing experimental TPD peaks. It should be noted that for the case w * 0 calculations were carried out using the quasi-chemical approximation, which is considered to be fairly accurate for this case. This may not be true in the case of strongly repulsive forces acting between the adsorbed molecules, as has been shown by Sales and Zgrablich.48 Their computer simulations, based on the Wigner-Polanyi rate equation, suggest that strong repulsive forces may result in the appearance of two desorption peaks instead of one, as would be expected for an unimodal adsorption energy distribution.

8046


Figure 9. Nonideal systems, w * 0. Combined effect of w and heterogeneity. Key for random topography systems: (A-C) R ) 10 kJ/mol and changing w: -2 kJ/mol (s), 0 (- - -), 2 kJ/mol (---); (D-F) w ) -2 kJ/mol and changing R 8 kJ/mol (s), 10 kJ/mol (- - -), 12 kJ/mol (---). Other parameters: Ka ) 10-4 min-1 atm-2; Kd ) 106 min-1; β ) 40 K/min; F ) 0.03 dm3/min; 0 ) 80 kJ/mol.

Figure 10. Nonideal systems, w * 0. Combined effect of w and heterogeneity. Key for patchwise topography systems: (A-C) R ) 10 kJ/mol and changing w -2 kJ/mol (s), 0 (- - -), 2 kJ/mol(---); (D-F) w ) -2 kJ/mol and changing R 8 kJ/mol (s), 10 kJ/mol (- - -), 12 kJ/mol (---). Other parameters: Ka ) 10-4 min-1 atm-2; Kd ) 106 min-1; β ) 40 K/min; 0 ) 80 kJ/mol.

However, the existence of strong repulsive forces is highly improbable. In carrying out our model calculations, we considered only extreme physical situations like “patchwise-random” distributions, “solid-volume” dominated settings, etc. The nature of real systems will lie somewhere in between. Such limiting models are meant to help us understand how the various factors could affect the behavior of experimentally recorded TPD peaks. Finally, we neglected the temperature dependence of Ka and Kd. Both Ka and Kd are expected to have a weak dependence on the temperature. Their temperature dependence will be governed by the temperature dependence of the molecular partition functions q0s and µ0g which are related to the internal degrees of freedom of the

Rudzinski et al.

adsorbed and bulk molecules. In view of this, the temperature dependence of Ka and Kd will in general be a function of low powers of T.10 We have neglected this influence in view of the strong exponential dependence introduced by the terms exp(/kT) and the high values of 0 ) 80 kJ/mol used in our model calculations. Our model investigation shows that even in the case of unimodal c(T) functions their quantitative analysis will present a very complex problem due to the many factors which have to be taken into consideration. It is therefore important to design a TPD experiment which will make available the most information about the features of adsorption/desorption kinetics and adsorption equilibria. The classical TPD experiment will not provide enough information to elucidate the kinetic parameters Ka and Kd, the parameters characterizing the surface energetic heterogeneity, the parameters characterizing the interactions between the adsorbed molecules, and the topography of an energetically heterogeneous solid surface. This conclusion can be drawn from Figure 3, presenting the classical type of thermodesorption experiment. A single peak can only yield the position of the maximum and a series of moments, with decreasing confidence and therefore utility, around that maximum. It seems reasonable therefore to obtain several peaks whose parameters vary due to the experimental conditions used. A classical TPD study might yield a few peaks recorded for the same preadsorbed amount but obtained at different heating rates β. Alternatively one could measure several peaks recorded at the same heating rate β but at for different initially preadsorbed amounts. This type of data set could contain sufficient information to unravel the six or so effects which influence the TPD peak. Recently Wojciechowski proposed a new type of thermodesorption experiment which, roughly speaking, is a combination of the two above-mentioned classical experimentssbut not only that. His method provides a means of determining equilibria and the corresponding rates of adsorption and desorption over a range of temperatures. Having such a large amount of information, one might have a chance of quantifying the many parameters appearing in the theoretical description of adsorption kinetics and equilibria on an energetically heterogeneous solid surface. Although Wojciechowski’s new type of thermodesorption experiment could be carried out in the commercially available TPD apparata, a specially designed instrument should make the experiment easier. Such an instrument has just been designed and is called the “temperature scanning-stream swept reactor” (TS-SSR). A description of its operation can be found elsewhere.65 Meanwhile, having at disposal only typical TPD data, we have been looking for a way to extract from such data more information than it is usually extracted. Our efforts toward this direction are described in the following section. 7. Numerical Analysis of Experimental TPD Data. According to our discussion in the previous sections, TPD spectra may be affected simultaneously by a variety of physical factors. Elucidating the role of each of them separately would require having a set of data much more complete and sophisticated than the traditional TPD measurements. So, having at disposal only the typical TPD measurements one can extract only a part of the information which, (65) Wojciechowski, B. W. The Application of Temperature Scanning to Adsorption Studies. Annales UMCS 1999, Section AA, in press (the preprints are available on request from SE REACTORS INC., 1810 Canal Drive, Kingston, Ontario, Canada, K7L 4V3).


Langmuir, Vol. 16, No. 21, 2000 8047

theoretically, might be extracted. Moreover, even to extract that incomplete information, a special procedure will have to be applied. Then, the TPD measurements should be done for systems, where some of the physical factors affecting TPD spectra do not play an important role, so the analysis can substantially be simplified. With such a strategy in mind we have carried out thermodesorption in a typical chemisorption system where gas-solid interactions predominantly affect the behavior of TPD spectra. So, we could eliminate from our analysis complications due to interactions between the adsorbed molecules. However, even then, several physical factors are to be considered. First a decision is to be made whether a considered TPD spectrum should be classified to correspond to an S-, V-, or E-type of experiment. Next, a way is to be found to determine the Ka and Kd constants. We have solved these problems in the following way. Namely, we have recorded TPD spectra for various heating rates. Next, for every temperature we found the value of θt(T) extrapolated to the zero heating rate, β ) 0. The so obtained extrapolated θt(T) was considered to correspond to full equilibrium between the gas and the adsorbed phase. Then, the function c can also be calculated from the relation

c ) -kT ln Kp(e)

(51)

Thus, for every temperature T, the corresponding value of c can be found. (We assume for the moment that the value of K is known.) Next the function θt(c) is found from the obvious (experimental) condition

θt(T(c)) ) 1 -

∫TT c(T) dT

F Nmβ

0

(52)

where T0 is the temperature at which preadsorption was made. When T0 is sufficiently low θt(T0) ≈ 1. Finally, the CA function χc(c) is calculated from the equation

∂θt(c) χc(c) ) ∂c

Figure 11. The three spectra of hydrogen thermodesorption from the nickel catalyst used in our experiment. (See Appendix.)

(53)

The function χc(c) calculated in such a way should be identical with the corresponding function χc(c) calculated from one of the recorded TPD spectra, provided that Kd and Ka have correctly been chosen, and a proper decision was made whether that particular TPD spectrum should be classified as an S-, V-, or E-dominated spectrum. In our investigation strategy, the TPD spectrum obtained for the lowest experimental heating rate β ) 12.2 K/min was assumed to be E-dominated and used to calculate χc(c) by applying eqs 40 and 31. In practice the numerical analysis was carried out as follows. For a chosen pair of Ka and Kd parameters, the corresponding χc(c) function was calculated from the TPD spectrum corresponding to the lowest heating rate, from eqs 40 and 31. Then, this χc(c) function was compared to the χc(c) function calculated from eqs 51-53 by taking K ) (Ka/Kd)1/2. While c was calculated in eq 51, the value of p(e) was found from c(T) defined in eq 29a in which β was put equal to 12.2 K/min, and dθt/dT was found from the extrapolated to β ) 0 function θt(T). The pair Kd and Ka values, for which best overlapping of the two χc functions was obtained, we accepted as the true (real) values of the Kd and Ka parameters.

Figure 12. Dependence of θt on β for fixed temperatures. Every set of three circles (ooo) connected by lines (s) are θt values corresponding to a fixed temperature T, changed by 50 K when going from one to another set of circles. The set of data corresponding to the highest values of θt was recorded at 50 °C.

The experimental system which was investigated by us was hydrogen thermodesorption from an alumina-supported Ni catalyst. Such systems are known as typical chemisorption systems in which gas-solid interactions play a strongly predominant role and as systems with strongly heterogeneous surfaces. The details of our experiment have been described in Appendix. Figure 11 shows the three spectra of hydrogen thermodesorption from the Ni catalyst, recorded at β ) 12.2, 25, and 50 K/min. Looking for a convenient way to arrive at the θt(T) extrapolated to β ) 0, we have observed that, for a fixed temperature T, the θt value defined in eq 52 was a linear function of β. The next Figure 12 shows that this linearlike extrapolation can successfully be applied. The obtained by extrapolation to β ) 0 function θt(T) is shown in Figure 13. The next Figure 14 shows the comparison of the χc(c) functions calculated from the extrapolated function θt(T) by using eqs 51-53, with the χc(c) functions calculated from the TPD spectrum recorded at 12.2 K/min, and using eqs 40 and 31. One can see that there is only one pair of the parameters Ka and Kd, for which the χc(c) functions calculated in the two different ways overlap. This pair of parameters, and the corresponding function χc(c), are believed to be the true ones, characterizing our H2-Ni catalyst chemisorption system. Such a good overlapping also means that, indeed, the effects due to interactions between the adsorbed molecules can safely be ignored in the investigated chemisorption system. Although χc(c) functions are not exact χ() functions, they mimic the most essential features of the true χ()

8048


Rudzinski et al.

Figure 13. Function c(T) defined in eq 29a calculated by taking β ) 12.2 K/min and the dθt/dT derivative calculated from the function θt(T) extrapolated to β ) 0. The concentration c(T) is expressed as the mole fraction of hydrogen in the gaseous phase.

Figure 15. (A) Comparison of the χc(c) functions calculated by neglecting readsorption, i.e., taking Ka ) 0 (---) and taking readsorption into account, i.e., putting the correct value Ka ) 10-2 min-1 atm-2 (s). (B) Comparison of the χ() functions calculated by using eq 41, from the χc(c) function in (A) neglecting readsorption (---) and the χc(c) function in (A) correctly taking readsorption into account (s). In all the calculations the value of Kd is the same as in Figure 14, i.e., is equal to 107 min-1.

Figure 15 clearly shows that neglecting readsorption in quantitative analysis of our TPD spectra would results into a serious overestimation of the degree of surface heterogeneity of our Ni catalyst. Both the χc(c) and χ() functions calculated by neglecting readsorption are much more diffuse, suggesting, thus, a higher degree of surface energetic heterogeneity. Of course, such common overestimation of the degree of surface energetic heterogeneity is made in all the analyses based on using the WignerPolanyi equation neglecting readsorption. However, the comparison of the nearly exact χ() functions in Figure 15B shows a more dramatic change in the picture of surface energetic heterogeneity obtained by ignoring readsorption and taking it correctly into account. Figure 14. Broken lines (---) are the χc(c) functions calculated for a certain pair Ka and Kd values from the TPD spectrum recorded at β ) 12.2 K/min, using eqs 40 and 41. The solid lines (s) are the χc(c) functions calulated for K ) (Ka/Kd)1/2, from the θt(T) function corresponding to β f 0, using eqs 51-53.

functions. So, looking at Figure 14A, we can observe the effect which has already been predicted by our model calculations in the previous section. Namely, that neglecting readsorption (putting Ka ) 0) yields the χc(c) function which is more diffuse than the actual χc(c) function in Figure 14C. That means, neglecting readsorption leads to strong overestimation of the degree of surface energetic heterogeneity. This effect can more clearly be seen by comparing the χ() functions, calculated using eq 41 from the χc(c) function (---) shown in Figure 14A (readsorption ignored) and the χc(c) function (s) in Figure 14C (readsorption taken into account). For that purpose we used the χc(c) functions calculated from the TPD spectrum recorded at β ) 12.2 K/min, using eqs 40 and 31. This is because the θt(T) function obtained by extrapolation to β f 0 leads to slightly nonphysical values of χc(c) below certain value of c, which is a side effect of the extrapolation to β ) 0 at the lowest desorption temperatures.

Conclusions The simplified ART adsorption/desorption rate expressions called “Wigner-Polanyi kinetics” are not adequate for interpreting the kinetics of adsorption/desorption processes in systems with energetically heterogeneous surfaces. This is because a number of factors which influence simultaneously the TPD process are not considered in this formulation. The most important of these is the presence of both adsorption and desorption, both of which can occur simultaneously but are not taken into consideration in Wigner-Polanyi kinetics. We believe that conclusions drawn from a Wigner-Polanyi analysis are potentially in error in ways which cannot be foreseen. Applying the statistical rate theory of interfacial transport leads to adsorption/desorption rate expressions which are functions of all the important physical factors which play a role in adsorption equilibria. This makes it possible to relate the features of adsorption/desorption kinetics to corresponding features of adsorption equilibria. Most importantly for our present purposes, this method provides a way in which readsorption can be taken into consideration in the theoretical analysis of experimental TPD peaks. Application of the SRTIT approach has now shown that the shape of experimental TPD peaks is affected by the physical setting of the sample, an aspect which was not


taken into consideration in previous work: the “solid”, “volume”, and “equilibrium” dominated settings. Simulations using the SRTIT approach show that in all these settings readsorption can mimic effects which are actually due to surface heterogeneity or to interactions between the adsorbed molecules. Thus, neglecting readsorption, as is commonly done in the analysis of TPD peaks, may lead to false conclusionssand not only at the quantitative level of interpretation. A good illustration for that is the quantitative analysis of the TPD spectra recorded by us for hydrogen desorption from the nickel catalyst. Appendix The carrier gas was argon (99.99% pure) additionally purified over the same OXICLEAR deoxidant, next with

Langmuir, Vol. 16, No. 21, 2000 8049

molecular sieves, and finally with MnO/γ-Al2O3. The TPD measurements were conducted with an AMI1 TPD apparatus (Altamira Instruments Inc.). A thermal conductivity detector was used as the detector. The experimental procedure was following. A sample of 0.0200 g was heated to 800 °C at a 10 K/min heating rate in a stream of a mixture containing 5% vol H2 in argon flowing at a rate of 30 cm3/min. Then the reduction process lasted for 30 min at 800 °C. Next, the sample was cooled to 20 °C in the same mixture at a rate of 50 K/min. After that, the mixture was replaced with argon for 20 min. The TPD spectra were measured at heating rates 12.2, 25, and 50 K/min in the temperature range from 20 to 700 °C. LA991651F

Theory of Thermodesorption from Energetically Heterogeneous

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