Langmuir 1997, 13, 3445-3453
3445
New Method of Estimating the Solid Surface Energetic Heterogeneity from TPD Spectra Based on the Statistical Rate Theory of Interfacial Transport Wladyslaw Rudzinski,*,† Tadeusz Borowiecki,‡ Anna Dominko,‡ and Tomasz Panczyk† Department of Theoretical Chemistry and Department of Chemical Technology, Faculty of Chemistry UMCS, Pl. Marii Curie-Sklodowskiej 3, 20-031 Lublin, Poland Received September 23, 1996. In Final Form: March 31, 1997X Recently a new theoretical method of a quantitative analysis of surface energetic heterogeneity from the experimental TPD spectra, based on the statistical rate theory of interfacial transport has been proposed by Rudzinski and co-workers. The purpose of the present publication is to show the predictive features of that new method. Also prospects of a further generalization of that method toward taking into account readsorption effects are discussed.
Introduction The temperature programmed desorption (TPD) is nowadays one of the most frequently used methods to characterize the energetic properties of adsorbent and catalyst surfaces. Among dozens of the papers published, there are also exhaustive reviews like those by Falconer and Schwarz,1 Kreuzer,2 Bhatia et al.,3 Cerofolini4 and Tovbin.5 The principles of that method were published by Amenomiya and Cvetanovic in 1963,6 and 9 years later the first theoretical paper on the application of that experiment to study energetic surface heterogeneity was published by Cvetanovic and Amenomiya.7 In fact, the TPD technique has been used, first of all, to study the surface energetic heterogeneity of catalysts and adsorbents. The various peaks observed on TPD diagrams were ascribed to various kinds of surface adsorption sites characterized by different activation energies for desorption. The interpretation of TPD spectra has been performed mostly on a qualitative level. Only a few papers reported the attempts to draw some quantitative information about surface energetic heterogeneity from the experimental TPD spectra. Czanderna et al.8 Cvetanovic and Amenomiya,7 Carter,9 Grant and Carter,10 Dawson and Peng,11 and Tokoro et al.12 assumed that the surface energetic heterogeneity manifests itself as a linear variation (decrease) of the * Author to whom correspondence should be addressed. Fax: (48)-81-537 56 85. E-mail: RUDZINSK@ HERMES.UMCS.LUBLIN.PL. † Department of Theoretical Chemistry. ‡ Department of Chemical Technology. X Abstract published in Advance ACS Abstracts, May 15, 1997. (1) Falconer, J. L.; Schwarz, J. A. Catal. Rev. Sci. Eng. 1983, 25, 141. (2) Kreuzer, H. J. Langmuir 1992, 8, 774. (3) Bhatia, S.l Beltramini, N.; Do, D. D. Catal. Today 1990, 8, 309. (4) Cerofolini, G. The Intrinsically Heterogeneous Nature of surfaces of Catalytic Interest. In Adsorption of New and Modified Inorganic Sorbents; Dbrowski, A., Tertykh, V. A., Eds.; Elsevier: New York, 1996; p 435. (5) Tovbin, Yu. Theory of Adsorption-Desorption Kinetics on Flat Heterogeneous Surfaces. In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: New York, in press. (6) Amenomiya, Y.; Cvetanovic, R. J. J. Phys. Chem. 1963, 67, 144. (7) Cvetanovic, R. J.; Amenomiya, Y. Catal. Rev. Sci. Eng. 1972, 6, 21. (8) Czanderna, A. W.; Biegen, J. R.; Kollen, W. J. Colloid Interface Sci. 1970, 34, 406. (9) Carter, G. Vacuum 1962, 12, 245. (10) Grant, W. A.; Carter, G. Vacuum 1965, 15, 13. (11) Dawson, D. T.; Peng, Y. K. Surf. Sci. 1972, 33, 565.
S0743-7463(96)00921-3 CCC: $14.00
activation energy for desorption with the decreasing surface coverage. King13 was the first to assume that the activation energy for desorption may be a more complex function of the surface coverage. Also Tokaro et al.14 followed that assumption in their second work on that problem. An interesting paper in this field was published by Kno¨zinger and Ratnasamy.15 After 1980, the papers dealing with that problem were published by Davydov et al.,16 Unger et al.,17 Malet et al.,18 Leary et al.,19 Ma et al.20 and Salvador and Merchan.21 Ten years later two excellent reviews were published by Tovbin.22,23 Among the most theoretically advanced papers, special attention should be given to the papers published by Cordoba and Luque24 and by Sales and Zgrablich.25 The latter authors draw attention to the fact that the most theoretically advanced papers focused usually on the effects of adsorbate-adsorbate interactions on the TPD spectra.26-30 Although less numerous, there have also been published some advanced papers taking into account simultaneously surface energetic heterogeneity and the lateral interactions between the adsorbed molecules.23,31-33 (12) Tokoro, Y.; Misono, M.; Uchijima, T.; Yoneda Y. Bull. Chem. Soc. Jpn. 1978, 51, 85. (13) King, D. A. Surf. Sci. 1975, 47, 384. (14) Tokoro, Y.; Uchijima, T.; Yoneda, Y. J. Catal. 1979, 56, 110. (15) Kno¨zinger, H.; Ratnasamy, P. Catal. Rev. Sci. Eng. 1978, 17, 31. (16) Davydov, V. Ya; Kiselev, A. V.; Kiselev, S. A.; Polotryuk, V. O. V. J. Colloid Interface Sci. 1980, 74, 378. (17) Unger, K. K.; Kittelman, U. R.; Kreis, W. K. J. Chem. Technol. Biotechnol. 1981, 31, 435. (18) Malet, P.; Munuera, G. In Adsorption at the Gas-Solid and Liquid-Solid Interface; Rouquerol, J.; Sing, K. S. W., Eds.; Elsevier: Amsterdam, 1982; p 383. (19) Leary, K. J.; Michaels, J. N.; Stacy, A. M. AIChE J. 1988, 34, 263. (20) Ma,, M. C.; Brown, T. C.; Haynes, B. S. Surf. Sci. 1993, 297, 312. (21) Salvador, F.; Merchan, D. React. Kinet. Catal. Lett. 1994, 52, 211. (22) Tovbin, Yu. K. Prog. Surf. Sci. 1990, 34, 1. (23) Tovbin, Yu. K. Theory of Physical Chemistry Processes at a GasSolid Interface; CRC Press: Boca Raton, FL, 1991. (24) Cordoba, A.; Luque, I. I. Phys. Rev. 1982, B26, 4028. (25) Sales, J. I.; Zgrablich, G. Surf. Sci. 1987, 187, 1. (26) Goymour, C. G.; King, D. A. J. Chem. Soc., Faraday Trans 1 69, 749. (27) Adams, D. L. Surf. Sci. 1974, 42, 12. (28) Zhdanov, V. P. Surf. Sci. 1983, 133, 469. (29) Redondo, A.; Zeiri, Y.; Goddart, W. A., III. Surf. Sci. 1984, 136, 41. (30) Helsing, B.; Mallo, A. Surf. Sci. 1984, 144, 336. (31) Tovbin, Yu. K. Kinet. Katal. 1979, 20, 1226.
© 1997 American Chemical Society
3446 Langmuir, Vol. 13, No. 13, 1997
All these papers were based on the application of the absolute rate theory (ART) to describe the adsorptiondesorption processes on solid surfaces. Meanwhile, some difficulties related to the application of that theory were reported more and more frequently, which led to reconsidering the fundamentals of the adsorption-desorption kinetics. Thus, in 1957 Kisliuk34 launched his idea of the “precursor states”, which 20 years later was developed further by King,35 assuming that two kinds of precursor states may exist, and a year later (in 1978) another edition of that idea was proposed by Gorte and Schmidt.36 A comprehensive review of the works of that kind can be found in the excellent review by Lombardo and Bell.37 The improvements offered by the theories of the precursor states did not gain, however, much interest of the scientists studying TPD spectra from the hetrogeneous surfaces. At the beginning of the eighties a new family of approaches to the adsorption-desorption kinetics appeared. These were the papers by Nagai,38,39 Kreuzer and co-workers,40-44 and Ward and co-workers.45-48 Starting with various physicochemical principles, all these three approaches led to the expressions relating the rate of desorption to the chemical potential of the adsorbed molecules. One common assumption accepted there was that the process of desorption runs under the quasiequilibrium conditions. Starting with the theory developed by Ward and coworkers,36 Rudzin˜ski and Aharoni49 have been able to provide a satisfactory theoretical explanation for some well-known features of adsorption kinetics. Recently Rudzinski et al.50,51 have proposed the first quantitative analysis of the TPD spectra from the heterogeneous surfaces, based on the approach developed by Ward and Findlay. The present paper brings some further development of this approach to obtain more accurate characteristics of surface energetic heterogeneity and to study predictive features of that approach. (32) Tovbin, Yu. K.; Votyakov, E. V. Zh. Fiz. Khim. 1992, 66, 716. (33) Tovbin, Yu. K.; Votyakov, E. V. Surf. Phys. Chem. Mech.1993, 10, 17; 1993, 11, 31. (34) Kisliuk, P. J. Phys. Chem. 1957, 3, 95. (35) King, D. A. Surf. Sci. 1977, 64, 43. (36) Gorte, R.; Schmidt, L. D. Surf. Sci. 1978, 76, 559. (37) Lombardo, S. I.; Bell, A. T. Surf. Sci. Rep. 1991, 13, 1. (38) Nagai, K. Phys. Rev. Lett. 1985, 54, 2159. (39) Nagai, K. Surf. Sci. 1986, 176, 193. (40) Kreuzer, H. J.; Payne, S. H. Surf. Sci. 1988,198, 235; 1988, 200, L433. (41) Payne, S. H.; Kreuzer, H. J. Surf. Sci. 1988, 205, 153. (42) Kreuzer, H. J.; Payne, S. H. Thermal Desorption Kinetics. In Dynamics of Gas-Surface Interactions; Retter, C. T., Ashfold, M. N. R., Eds.; Royal Society of Chemistry Cambridge, 1991; Chapter 6. (43) Kreuzer, H. J. Langmuir 1992, 8, 774. (44) 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: New York, 1966. (45) Ward, C. A.; Findlay, R. D. J. Chem. Phys. 1982, 76, 5615. (46) Elliott, J. A.; Ward, C. A. Statistical Rate Theory and the Material Properties Controlling Adsorption Kinetics on Well Defined Surfaces. In Equilibria and Dynamics of Gas adsorption on Heterogeneous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: New York, 1996. (47) Ward, C. A.; Elmoselshi, M. Surf. Sci. 1986, 176, 457. (48) Elliott, J. A.; Ward, C. A. J. Chem. Phys., in press. (49) Rudzinski, W.; Aharoni, C. Langmuir, in press. (50) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Zientarska, M. Chem. Anal. 1996, 41, 1057. (51) Rudzinski, W. A New Theoretical Approach to AdsorptionDesorption Kinetics on Energetically Heterogeneous Flat Solid Surfaces Based on the 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.
Rudzinski et al.
Theory We start with discussing the fundamental equation for the rate of the interfacial transport, offered by the ART approach. For the purpose of clarity, we consider first the simplest case when one molecule is adsorbed on one site. Then, according to the ART,
dθ ) Kap(1 - θ) exp(-a/kT) - Kdθ exp(-d/kT) dt
(1)
where t is time, θ is the fractional surface coverage, p is the pressure in the gas phase, a and d are the activation energies for adsorption and desorption respectively, and Ka and Kd are the appropriate constants. T and k are the absolute temperature and the Boltzmann constant, respectively. The first term on the right hand side of eq 1 represents the rate of adsorption, and the second term represents the desorption rate. At equilibrium, (∂θ/∂t) ) 0, and then one arrives at the Langmuir adsorption isotherm, θ(e),
θ(e)(,p,T) )
Kp exp(/kT) 1+ Kp exp(/kT)
(2)
where K ) Ka/Kd, ) (d - a), and the superscript (e) stands for equilibrium. The average (“total”) coverage of the surface at equilibrium, θ(e) t is given by
θ(e) t )
∫θ Ω
(e)
(,p,T) χ() d
(3)
where χ() is the differential distribution of the number of adsorption sites among the corresponding values, usually called the “adsorption energy distribution”. This function is used in its form normalized to unity
∫ χ() d ) 1 Ω
(4)
where Ω is the physical domain of . It is commonly taken to be the interval (0, +∞), for the purpose of mathematical convenience. At low (partial) pressures p and high coverages θt (the conditions which are met usually in the TPD experiments), the first term of eq 1 is expected to be small compared to the second one. Thus, to a good approximation, the experimentally observed desorption rate should be given by
-
dNt ∂θ(n) t ) -Nm ) NmKdθ(n) t exp(-d/kT) dt ∂t
(5)
where the superscript (n) stands for the nonequilibrium conditions. From now on, we will omit that superscript whenever nonequilibrium is considered. In eq 5 Nt is the adsorbed amount, and Nm is the maximum adsorbed amount when θt ) 1. Experimentally, the desorption rate is monitored as the concentration of the desorbed species in the carrier gas, c, so
-
dNt ) Fc dt
(6)
where F is the volumetric flow rate of the carrier gas. Of course, eq 5 is valid for a hypothetical homogeneous solid surface with the same constant desorption energy d across the surface. Meanwhile, it was generally observed that, while applying eq 3, the estimated d values changed with the surface coverage, θt, so d in eq 3 was considered to be generally a function of θt, d(θt). The shape of the experimential TPD spectra suggested that d may, in general, be a complicated function of θt; this function could not be easily modeled by an analytical function with a small number of parameters. So, one could see departure from that strategy later on, and the attempts to determine the exact shape of d(θt) from the measured TPD spectra. Various methods were applied for that purpose,1,3 but it seems that the most popular one was that based on applying the following Arrhenius plot
Solid Surface Energetic Heterogeneity
Langmuir, Vol. 13, No. 13, 1997 3447
-(dθt/dt) d ) ln Kd θt kT
(7)
ln
along with a variable heating rate. (We limit our consideration to the one-site-occupancy adsorption). While applying this method (or some other ones) to the experimental TPD spectra, it appeared that not only d varied with the surface coverage. Surprisingly, Kd also varied over several orders of magnitude as a rule. This can best be seen in the excellent review by Zhdanov,52 where he collected d(θt) and Kd(θt) functions determined for a variety of adsorption systems. Seebauer et al.53 have reviewed different theoretical representations for the pre-exponential factor Kd. None of them were able to account for such strong variations of Kd with θt. It is very exciting in Zhdanov’s review that for the same well-defined adsorption system sometimes drastically different functions d(θt) and Kd(θt) were reported by various authors. It could hardly be explained by certain differences in the adsorbent sample preparations. Different physiochemical regimes, in which the experiments were carried out by different researchers, were more evident. However, different experimental regimes should not affect so much the determined d(θt) and Kd(θt) functions. Some of the authors trying to generalize eq 3 for the case of the actual, energetically heterogeneous solid surfaces wrote it in the following form20,54,55
-
dθt ) Kd dt
∫
Ωd
θ(d) exp(-d/kT) χd(d) dd
(8)
where χd(d) is the distribution of the number of adsorption sites among various values of d, normalized to unity,
∫
χ ( ) Ωd d d
dd ) 1
(8a)
where Ωd is the physical domain of d, and the subscript t in ∂θt/∂t means the average of ∂θ/∂t over the “total” heterogeneous solid surface. Equation 8 could be treated as an integral equation for χd(d), provided that the kernel θ(d) is known and that there exists a functional relationship between d and a on a heterogeneous solid surface. One can even raise the question whether a and d on various adsorption sites are correlated at all. The only thing that is sure is the fact that Langmuirian kinetics of adsorption (represented by the first term on the right hand side of eq 1 was not found in the real adsorption systems. Instead, Elovich, power law, or other empirical equations for adsorption kinetics had to be applied, and they were reviewed in the exhaustive article by Low.56 In terms of ART, it means that there must exist a dispersion of a values on different adsorption sites. The concept of the dispersion of the activation energies for adsorption, a, was accepted as the most fundamental one by the authors trying to explain the theoretical origin of these empirical equations.54,57-60 The TPD experiments, on the other hand, reveal a dispersion of d values on the actual solid surfaces. So, generally, the correct generalization of eq 3 should read
-
dθt ) Kd dt
∫ ∫ θ( , ) exp(- /kT) χ Ωd
a
Ωa
∫∫ Ωa
d
χ ( , ) Ωd ad a d
d
ad(a,d)
da dd ) 1
da dd (9) (9a)
where χ(a,d) is a two-dimensional differential distribution of (52) Zhdanov, V. P. Surf. Sci. Rep. 1991, 12, 183. (53) Seebauer, E. G.; Kong, A. C. F.; Schmidt, L. D. Surf. Sci. 1988, 193, 417. (54) Temkin, M. I. Zh. Fiz. Khim. 1938, 11, 169; 1941, 15, 296. (55) Roginsky, S. Z. Adsorption and Catalysis on Heterogeneous Surfaces. Izd. Acad. Nauk SSSR 1948. (56) Low, M. J. D. Chem. Rev. 1960, 60, 267. (57) Aharoni, C.; Tompkins, F. C. Adv. Catal. 1970, 21, 1. (58) Aharoni, C.; Ungarish, M. J. Chem. Soc., Faraday Trans. 1 1977, 73, 1943; 1981, 77, 957; 1983, 79, 119. (59) Tovbin, Yu. K.; Fedyanin, V. K. Kinet. Katal. 1978, 19, 1202. (60) Tovbin, Yu. K. Surf. Phys. Chem. Mech. 1982, 10, 45.
the number of adsorption sites among corresponding pairs of values of a and d, and Ωd and Ωa are the physical domains of d and a, respectively. However, the theoretically correct eq 8 is, in fact, useless, until the analytical expression for θ(a,d) is known. Statistical theories of adsorption (at equilibrium) provide one with θ as a function of (d - a). The interpretation of TPD spectra from a heterogeneous solid surface was based on an intuitive assumption, which was correct to some extent and under certain conditions, as we will see soon. That common interpretation was based on the assumption that, at every moment (at each adsorbed amount), the observed desorption rate can still be described by eq 3. The value of d corresponding to that adsorbed amount was still called the “activation energy for desorption” and was given the same physical meaning as for a homogeneous solid surface characterized by this d value. Looking into eq 8, one can deduce easily that such an interpretation could be accepted only if the following two conditions are fulfilled: (1) There exists a functional relationship between d and a. (2) The desorption proceeds in an ideally stepwise fashion, in the sequence of increasing activation energies for desorption. Some authors argue5 that d and a should be linear functions of the adsorption energy ) d - a, whereas some others suggest21 that a may not be a simple function of d. Sometimes d is identified with the enthalpy of adsorption,44 and sometimes it is assumed to be a linear function of adsorption enthalpy.61 The generalization of ART on the basis of eq 8 does not involve the assumption that the desorption process proceeds in an ideally stepwise fashion but involves making an assumption about the functional relationship between d and a. Making that assumption is not involved if one applies the approaches relating the kinetics of desorption to the chemical potential of the adsorbed molecules, µs. At the beginning of the eighties a new family of approaches to the adsorption/desorption kinetics appeared. Although they represented different theoretical considerations, they led to the following expression for the rate of desorption Rd:
( )
Rd ) Φ exp
µs kT
(10)
where µs is the chemical potential of the adsorbed molecules and Φ is a certain function of temperature. In one case, it was assumed that Φ may also be a function of the surface coverage θ. Although all the approaches appeared at approximately the same time, the approach proposed by Nagai38,39 should be considered as the first. This is because the idea of that approach may be traced back to de Boer.62,63 Nagai’s work was aimed at improving ART to account for the effects of the entropy changes in the adsorbed phase on the rate of desorption Rd. The appearance of Nagai’s approach aroused a strong dispute in the literature.64-67 Recently, however, Nagai68 has shown an impressive example where his expression is superior over the traditional ART expression to reproduce the experimental TPD spectra. Also, Kreuzer and co-workers40-44 arrived at expression 10, though their derivation was based on different physical arguments. In Kreuzer’s approach, Φ is assumed to be a product of the temperature dependent factor and a sticking coefficient, which cannot be obtained from the thermodynamic arguments but must be calculated in some way from a microscopic theory or be postulated in a phenomenological approach based on the experimental evidence for a particular system or some simple arguments. Still another approach leading to eq 10 was proposed by Ward and co-workers.45-48 Under the assumption that the transport between two phases at the thermal equilibrium results primarily from single mo(61) Aharoni, C. J. Chem. Soc., Faraday Trans. 1 1978, 74, 1507. (62) Scholten, I. I. F.; Zwieterring, P.; Konvalinka, J. A.; de Boer, J. H. Trans. Faraday Soc. 1959, 55, 2166. (63) de Boer, J. H. Adv. Catal. 1956, 8, 89. (64) Zhdanov, V. P. Surf. Sci. 1986, 171, L461, L469. (65) Nagai, K.; Hirashima, A. Surf. Sci. 1986, 171, L464. (66) Cassuto, A. Surf. Sci. 1988, 203, L656. (67) Nagai, K. Surf. Sci. 1988, 203, L659. (68) Nagai, K. Surf. Sci. Lett. 1991, 244, L147.
3448 Langmuir, Vol. 13, No. 13, 1997
Rudzinski et al.
lecular events, the equation for the rate of transport between a gas and a solid phase was developed by Ward and Findlay,45 using the first-order perturbation analysis of the Schro¨dinger equation and the Boltzmann definition of entropy,
[ (
g
)
(
s
dθ µ -µ ) K′gs exp dt kT
- exp
s
)]
µ )
µg0
µ -µ kT
+ kT ln p
(11)
θ q (1 - θ)
(13)
s
where qs is the molecular partition function of the adsorbed molecules. Equation 11 takes then the following form
dθ 1-θ 1 θ ) K′ap - K′d dt θ p1-θ
(14)
where
K′a ) K′gsqs exp(µs0/kT)
and
K′d )
K′gs qs
exp(µs0/kT)
(14a)
The energetic heterogeneity of the actual solid surfaces is demonstrated mainly as the variation of the adsorption energy in the molecular partition function qs,71 for molecules adsorbed on different sites,
qs ) qs0 exp(/kT)
Ka ) K′gsqs0 exp(µs0/kT)
(15)
With this notation eq 14 takes the form (69) Talbot, J.; Jin, X.; Wang, N.-H. L. Langmuir 1994, 10, 663. (70) Harris, L. B. Surf. Sci. 1968, 10, 129; 1968, 13, 377; 1969, 15, 182. (71) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992.
and
Kd )
K′gs qs0
exp(-µs0/kT) (16a)
As has already been mentioned, in the typical TPD experiments the (partial) pressure of the desorbed species (concentration c in the carrier gas) is usually small, so the desorption rate is usually represented only by the second term on the right hand side of eq 16. Thus, we assume that in the case of a homogeneous solid surface and adsorption energy
dθ 1 θ ) -Kd exp(-/kT) dt p1-θ
(17)
Equation 17 can be rewritten in the following Langmuir-like form
(
xp exp(/kT) dθ ) dt 1 + xp exp(/kT)
(18)
1 dθ Kd dt
(19)
θ p,T,,
)
where
x)-
The average (experimentally monitored), nonequilibrium coverage of the whole heterogeneous surface, θt(p,T,dθt/dt), will be given by
(
)
dθt ) dt
θt p,T,
χ() d ∫ θ(p,T,,dθ dt ) Ω
(20)
The function θt(p,T,,dθt/dt) in eq 18 can be rewritten in the following form
θ() )
(12)
and µs be the expression corresponding to the Langmuir model of adsorption (one-site-occupancy monolayer adsorption with no interactions between the adsorbed molecules). Then,
µs ) kT ln
where
g
where µg and µs are the chemical potentials of the adsorbate in the gas phase and in the adsorbed (surface) phase, respectively. Then, it was emphasized that eq 9 does not apply at very low, and very high (close to unity) values of θ. The next essential assumption made by Ward and Findlay was that the transient surface configurations of the adsorbed molecules are close to the equilibrium ones, corresponding to the same surface coverage. That assumption is, in fact, inhered in all the theoretical approaches leading to expression 10. Otherwise, it could not be possible to relate Rd to the surface coverage θ in a simple way. This assumption has recently received impressive support from the computer simulations of adsorption kinetics reported by Talbot et al.69 More support came from the recent work published by Rudzin˜ski and Aharoni.49 The latter authors accepted the first term on the right hand side of eq 11 to describe the kinetics of adsorption. They generalized it next for the case of the heterogeneous solid surfaces using a method very similar to that used in the present paper, to describe the kinetics of desorption from a heterogeneous solid surface. In this way, they developed the empirical Elovich and power law kinetic equations. They showed that, in the adsorption systems characterized by Temkin’s isotherm, the kinetics of adsorption will be described by the Elovich equation, whereas, in the systems characterized by the Langmuir-Freundlich isotherm, the kinetics will obey the power law equation. The fact that the well-known empirical equations for the adsorption kinetics have been developed rigorously as corresponding to the well-known isotherm equation for adsorption equilibria supports strongly the assumption that in the real adsorption systems the transient states must be close to the equilibrium ones. Let µg be the chemical potential of an ideal gas g
1-θ 1 θ dθ ) Kap exp(/kT) - Kd exp(-/kT) (16) dt θ p1-θ
exp(( - c)/kT) 1 + exp(( - c)/kT)
(21)
where
(
c ) kT ln
)
-p dθ Kd dt
(22)
Figure 1 shows the temperature dependence of the function θ(c)
θ() )
exp(r/τ) 1+ exp(r/τ)
(23)
where r ) ( - c)/kT0 and τ ) T/T0 are the appropriate dimensionless values. Figure 1 shows that the kernel θ() becomes the step function θc in the hypothetical low-temperature limit T f 0,
{
0, < c lim θ(,c) ) θc(,c) ) 1, g c Tf0
(24)
It means that, at not too high temperatures, the desorption from a heterogeneous solid surface proceeds in a stepwise-like fashion. At finite temperatures, c is the value of at which the function θ(,c) is the sharpest. This is when
( ) ∂2θ ∂2
)0
(25)
)c
This happens when θ ) 1/2. Thus, in the course of desorption, the sharp “desorption front” is on the sites the energy of which is equal to c, and their relative coverage is then equal to 1/2. Now, let us remark that this statement will also be true when
Solid Surface Energetic Heterogeneity
Langmuir, Vol. 13, No. 13, 1997 3449 the limit T f 0, or in the limit of a very heterogeneous solid surface, they will become identical.71 At every temperature T, the still existing adsorbed amount Nt(T) is found from the relation
Nt(T) ) N0 -
Figure 1. Temperature dependence of the function θ(,c). The dimensionless temperatures are τ1 ) 0.5 (s), τ2 ) 4 (- -), and τ3 ) 8 (- - -). readsorption occurs, if the process runs at the quasi-equilibrium. At the equilibrium, eq 21 takes the same form as eq 2 except that c is then defined as follows
c ) -kT ln(Kp(e))
dθ dt
( )
1 ) Kap exp(c/kT) - Kd exp(-c/kT) c p
(27)
Replacing θ(,c) by the step function θc(,c) is a very useful approximation in the theories of adsorption on the energetically heterogeneous surfaces. It was first proposed by Roginski55 at the beginning of the forties and next elaborated further by Harris70 at the end of the sixties. That approximation is commonly called the “Condensation Approximation”. It allows us to elucidate in an easy way the approximate quantitative information about the surface energetic heterogeneity. In the limit T f 0 the kernel θ() becomes the step function eq 24, so θt takes the following simple form
θt )
∫ χ() d ) -χ( ) ∞
c
c
(28)
∫χ() d
(32)
∑R c (T)
c(T) )
(33)
i i
where
( ( ))
(T - T0i )ri-1 T - T0i ci(T) ) ri exp Ei Eri
ri
(33a)
In eq 33a, Ei is the variance of ci(T), and ri governs the shape of ci(T). This is a fairly symmetrical Gaussian-like function for r ) 3, a right-hand widened Gaussian-like function for r < 3, or a left-hand widened Gaussian-like function for r > 3. Let us assume that at T0 the surface coverage θt is equal to unity, so N0 ) Nm, and that at the final temperature, Tk, θt is equal to zero. Then
∫
F β
Nm )
Tk
T0
c(T) dT
(34)
The partial pressure p of the adsorbate is given by the relation p ) cP (where P is the atmospheric pressure). Further, the derivative (dNt/dt) calculated from eq 32 is given by
dNt Fc(t(T)) )dT β
(35)
Because
θt )
exp(( - c)/kT)
∫ 1+ exp(( - )/kT)χ() d Ω
(36)
c
the derivative (dNt/dt) is expressed as follows:
[
where χ() is the integral of χ()
χ() )
T
T0
where β is the heating rate, β ) dT/dt, and N0 is the amount adsorbed at T ) T0. In typical TPD experiments β is a constant. Below we are going to show how the condensation approximation for χ(), χc(c) can be evaluated from TPD spectra. The preparatory step to calculate χc(c) is to eliminate the noise in the experimental TPD curve, which would be multiplied in the subsequent differentiations. For that purpose, the TPD spectrum is approximated by a sum of Gaussian-like functions:
(26)
where p(e) is the value of the equilibrium pressure corresponding to the same surface coverage. The term “quasi-equilibrium” means that, at certain surface coverages, all the correlation functions in the adsorbed phase are practically the same as those at equilibrium and the same surface coverage. Thus in the course of desorption running under the quasi-equilibrium conditions, the occupancy of sites, i.e. the function θ(,c), will be the same as that at equilibrium. At equilibrium, adsorption and desorption play an equal role (Rd ) Ra), and eq 21 remains still valid. Thus when readsorption (the first term on the right hand side of eq 15 is taken into account, c should be defined by the condition θ()c) ) 1/2. Then, instead of eq 22, from eq 15 we obtain
∫ c(T) dT
F β
]
∂θt ∂θt dc dNt ) Nm + dT ∂T ∂c dT
(37)
exp(( - c)/kT) c - ∂θ ) 2 ∂T [1 + exp(( - c)/kT)] kT2
(38)
(29) Because,
The function χc(c) calculated from the relation
∂θt 1 ∂Nt(c) χc(c) ) )∂c Nm ∂c
(30)
is usually called the “Condensation Approximation” for the actual adsorption energy distribution χ() and is defined as follows:71
χc(c) )
∫( ) ∞
0
∂θ χ() d ∂
(31)
In the limit T f 0 the derivative ∂θ/∂ becomes the Dirac delta function δ(-c), and χc(c) becomes the exact function χ(). The essential condition for this approximation to be applicable is not solely the condition T f 0 but the condition (δθ/δχ) < 1, where δθ is the variance of the derivative (∂θ/∂) and δχ is the variance of χ(). Thus, to a certain degree of accuracy, the function χc(c) calculated in eq 30 can be compared to the function χ(). In
then
∂θt ) ∂T
∫( ∞
0
)
c - χ() d T
(39)
We now expand the function φ()
φ() )
c - χ() T
(40)
into its Taylor series around the point ) c, where the isotherm derivative dθ/d reaches its sharp maximum,
3450 Langmuir, Vol. 13, No. 13, 1997 ∂θt ) ∂T
∫
[
∞
0
φ()c) +
Rudzinski et al.
() ]
()
c
c
(41)
Of course φ(c) ) 0, and the contribution from the second term within the square brackets under the integral in eq 41 is negligible. This is because,
c - ∂φ 1 ) - χ() + χ′() ∂ T T
(42)
and the corresponding contribution to the integral in eq 41 reads
χ(c) T
( - ) d ∫ (∂φ ∂ )
-
∞
(43)
c
0
Because of the symmetry of the function ∂θ/∂ with respect to c, the integral in eq 43 is practically negligible. The first nonvanishing contribution comes from the second (correction) term,
d ∫ 21(∂∂φ) ( - ) ∂θ ∂ ∞
2
2
2
0
(44)
c
c
Because
( )
c - 2 ∂2φ χ′′() ) - χ′() + 2 T T ∂
(45)
then
1 ∂θ d ) - χ′( )∫ ( )( - ) ∫ 21(∂∂φ) ( - ) (∂θ ∂ ) T ∂ 2
Ω
2
2
2
c
c
c
c
Ω
d (46)
and finally we have
∂θt π2 ) - (k2T)χ′(c) ∂T 3
( )
(48)
Now, let us remark that the second term on the right hand side of eq 48 is a kind of a second-order correction term to the nonexisting, in this case, condensation approximation (CA) leading term. It can be ignored when compared to the first term. Then, in view of the definitions of χc(c) in eq 30 and Nt(T) in eq 32,
[ ]
Fc(T(c)) dc βNm dT
-1
(49)
(Please note that c must be a one-to-one function of T.) Now let us consider the “local” rate of desorption from the sites the adsorption energy, which is equal to
( ) ( ) ( )( ) dθ dt
dθ β dT c
∂θ )β ∂ c c
c
)0
c
dc dT
(50)
∂θ ∂c
and
c
)
-1 4kT
( )
(53)
( ) dθ dt
{ }
-c 1 ) -Kd exp c p kT
(54)
Substituting (dθ/dt)c from eq 54 into eq 53, we obtain
( )
{ }
K ˜d dc -c ) exp dT kT βPc(T)
(55)
where K ˜ d ) 4kTKd. Equation 55 is a first-order differential equation for the function c(T). The form of the solution c(T) will be influenced primarily by the form of the experimentally determined function c(T). To some (secondary) extent, the solution c(T) will also be influenced by the temperature dependence of the constant K ˜ d. To incorporate this temperature dependence, one has to know the temperature dependence of the molecular partition function of the adsorbed molecules qs0. As is known from adsorption literature,71 an a priori theoretical determination of the temperature dependence of qs0 is one of the most difficult theoretical problems. It is commonly assumed that qs0 should be a slowly-varying function of low powers of T. For that reason, we will neglect the temperature dependence of K ˜ d in our forthcoming analysis of the TPD spectra. The solution of eq 55 makes it possible to express c(T) as the function c(c). Having determined the function c(c), we insert it into eq 49 and evaluate the “condensation” function χc(c). The “condensation” function χc(c) is of fundamental importance in the theories of gas adsorption (chemisorption) on the energetically heterogeneous solid surfaces. Having calculated χc(c), one can also calculate more accurate approximations for χ(), using a variety of methods proposed in the literature.72-78 For instance, one can use the method proposed by Rudzinski and Jagiello.71,76,77
[ ]
∂2χc π2 (kT)2 6 ∂2
(56)
Now, let us finally consider the reverse problem of calculating the TPD spectrum c(T) when the form of the adsorption energy distribution is known. From eq 49 we have
c(T) )
[ ]
βNm dc χ ( ) F c c dT
(57)
When the readsorption phenomena are neglected from eqs 53 and 54, we get
( )
( )
4kTKd dc -c exp ) dT βp kT
(58)
Remembering yet that p ) Pc(T), from eqs 57 and 58, we obtain
x
( )
NmK ˜d c χc(c(T)) exp FP kT
(59)
where the function c(T) is found by solving the differential eq 58, in which p is replaced by Pc(T) given by eq 59. Equation 58
(51)
From eq 18, we have
( )
c
-β dc 4kT dT
Now, let us remark that according to eq 18 θ()c) ) 1/2 so, when the readsorption rate is neglected, from eq 14 we have
c(T) )
because
(∂T∂θ)
)
χRJ() ) χc() +
∂θt Fc(t(T)) π2 ) - k2Tχ′(c) Nmβ ∂c 3
χc(c) )
dθ dt
(47)
From eqs 26, 35, 37, and 47 we have
-
( )
1 ∂2φ ∂φ ∂θ d ( - c) + ( -c)2 ∂ 2 ∂2 ∂
(52)
(72) Hobson, J. P. Can. J. Phys. 1965, 43, 1934, 1941. (73) Nederlof, M. M.; Van Riemsdijk, W. H.; Koopal, L. K. J. Colloid Interface Sci. 1990, 135, 410. (74) Hsu, C. C.; Wojciechowski, W. B.; Rudzinski, W.; Narkiewicz, J. J. Colloid. Interface Sci. 1978, 67, 292. (75) Rudzinski, W.; Narkiewicz, J.; Patrykiejew, A. Z. Phys. Chem. Leipzig 1979, 260, 1097. (76) Rudzinski, W.; Jagiello, J. J. Low Temp. Phys. 1981, 45, 1. (77) Rudzinski, W.; Jagiello, J.; Grillet, Y. J. Colloid Interface Sci. 1982, 87, 478. (78) Jagiello, J. Langmuir 1994, 10, 2778. (79) Re, N. J. Colloid Interface Sci. 1994, 166, 191.
Solid Surface Energetic Heterogeneity
Langmuir, Vol. 13, No. 13, 1997 3451
takes then the following form:
( ) x dc
1
K ˜ dF
β
PNm χc(c)
1
)
dT
( )
exp -
c
kT
(60)
So, the procedure of calculating the TPD spectrum, when the true energy distribution χ() is known, is the following: (1) First the “condensation” approximation (CA) function χc(c) is calculated from eq 30. (2) The calculated CA function is inserted into eq 60, and the solution of that differential equation yields the function c(T). (3) The calculated function is inserted into eq 59 to obtain the function c(T). Although diffusional effects are commonly neglected, they can also be taken into account to calculate the more accurate function c(t).80 Then, it is known that the TPD spectra may also be affected by the intraparticle diffusion and by the lateral interactions between the adsorbed molecules. It is suspected that the two effects mentioned above may even be a source of some extra peaks on the TPD diagrams.19 In the present treatment we neglect diffusional effects, which may be responsible for some broadening and stronger overlapping of peaks, compared to their true shape determined solely by the function χ(). Diffusional effects, however, are usually neglected in the published analyses of the TPD spectra from the heterogeneous surfaces. This reflects the general belief that this is the energetic surface heterogeneity which predominantly governs the shape of a TPD diagram in most cases.
Experimental Results and Discussion The purpose of the present publication is to present the applicability of our new theoretical approach for a quantitative determination of surface energetic heterogeneity from the experimental TPD spectra. For the limited volume of the publication, we will limit here ourselves to demonstrate the applicability of our new approach by analyzing one TPD diagram of hydrogen desorbed from an alumina-supported nickel catalyst. Temperature-programmed desorption measurements were conducted with AMI1 equipment (Altamira Instruments). A thermal conductivity detector was used as a detector. Experiments were conducted with an alumina-supported nickel catalyst (15 Ni) obtained by impregnation of R-Al2O3 INS Puławy (surface area ) 3.3 m2 g-1) by nickel nitrate solution. The catalyst made in this way was calcinated at 400 °C and reduced right before the experiment. Hydrogen was purified on OXICLEAR deoxidant made by PIERCE CHEMICAL Co. and was passed through the activated carbon absorber used as a reducing agent. The carrier gas was argon (99.99% pure) additionally purified over the same OXICLEAR deoxidant and then over 4A and 5A molecular sieves and with MnO/γ-Al2O3. The experimental procedure was as follows: First, a sample of 0.5 g of catalyst was heated to 700 °C with a 25 K/min heating rate in the stream of inert gas flowing at a rate of 30 cm3/min. After the desired temperature was obtained, argon was replaced by hydrogen. The reduction process lasted for 3 h. Then, the sample was cooled in the hydrogen stream to 25 °C. Next, hydrogen was replaced by argon for 20 min. After that, the temperatureprogrammed desorption was carried out at a heating rate of 20 K/min. Figure 2 shows the approximation of the experimental TPD diagram of hydrogen desorption from the aluminasupported nickel catalyst, done by using the linear combination (eq 33) of five functions ci(T) defined in eq (80) Huang, Y. J.; Schwarz, J. A. J. Catal. 1986, 99, 149.
Figure 2. Best fit of the experimental TPD diagram (O) of hydrogen desorption from the nickel catalyst, by the linear combination (eq 33) of the functions (eq 33a). The best-fit parameters found by computer are collected in Table 1. Table 1. Parameters Found by Computer Fitting in the Course of the Decomposition Defined in Eqs 33 and 33a of the TPD Diagram of Hydrogen Desorbed from the Silica-Supported Nickel Catalyst i
Ri
ri
T°i, °C
Ei, deg
1 2 3 4 5
10604 1405 2139 2004 1154
2.22 2.32 2.71 2.71 1.90
35.56 278.67 141.00 218.50 384.51
127.15 107.49 88.04 84.47 213.93
33a. The parameters found by computer fitting in the course of that decomposition are collected in Table 1. The analytical approximation (eq 33) of the experimental function c(T) was then used to solve the differential equation (eq 46). That differential equation was solved by using the wellknown Runge-Kutta method.81 In this method, the boundary condition is the value of the solution function at a chosen value of the argument. In the present calculations, the value of c ) max1 at the temperature corresponding to the maximum of the first peak (from the left) was taken as the boundary condition parameter. The typical shape of the obtained function c(T) has been shown in our previous publications.50,51 The choice of the parameters K ˜ d and max1 in our model calculations was dictated by the obvious requirement that the c values could lie in a physically reasonable range. It is, however, to be emphasized that these are still illustrative model calculations. Provided that the experimental function c(T) corresponds to the physical situation when N0 ) Nm (desorption starts from θt ) 1), the calculated function χc(c) should be normalized to unity. In general, however, it is expected that N0 < Nm, so the integral
∫0∞χc(c) dc
(61)
may be equal to unity or smaller than unity. Our numerical exercises showed that the value of the integral in eq 61 depends on the choice of the parameters K ˜ d and max1. Figure 3 shows how, for certain choices of the parameter K ˜ d, the value of the integral in eq 61 changes with the choice of max1. Figure 3 deserves certain comments. The reason why the value of the integral in eq 52 was not plotted for still higher or still lower vales of max1 is the following. For higher values of max1, χc(c) tends to the Dirac delta distribution. Meanwhile the variance of χc(c) cannot, for (81) Stoer, J.; Bulirsch, R. Einfu¨ hrung in die Numerische Mathematik II; PWN: Warszawa, 1980.
3452 Langmuir, Vol. 13, No. 13, 1997
Figure 3. The value of the integral in eq 61, plotted as a ˜ d ) 2000 atm kJ mol-1 min-1. function of max1, when K
Rudzinski et al.
Figure 5. Decomposition of the condensation function χc(c) ˜ d ) 2000 atm kJ mol-1 calculated for max1 ) 49.5 kJ/mol and K min-1, by the linear combination (eq 57) of the functions χci (- - -). The best-fit parameters found by computer are collected in Table 2. The peaks are numbered from the left to the right. Table 2. Parameters Found by Computer, while Decomposing the Function χc(Ec) Calculated by Assuming Emax1 ) 49.5 kJ mol-1 and K ˜ d ) 2000 atm kJ mol-1 min-1 i
γi
ri
°i, kJ mol-1
Ei, kJ mol-1
δ i ,a kJ mol-1
δ*i, kJ mol-1
1 2 3 4 5
0.32 0.33 0.164 0.125 0.061
3.6 2.1 4 3.0 2.0
41.5 48.5 47.4 51.9 90.0
7.05 8.0 21.2 32.0 38.0
1.9x10-3 5.7x10-3 7.9x10-2 5.07 11.12
7.3 9.7 8.9 8.98 17.44
a
Figure 4. Function χc(c) calculated by assuming that max1 ) 45 kJ/mol (s) and by assuming that max1 ) 49.5 kJ/mol (- -). In both cases K ˜ d ) 2000 atm kJ mol-1 min-1. For the function (- -) the value of the integral in eq 61 is equal to 0.999 84, whereas for the other function (s) the integral in eq 61 is equal to 0.7996.
obvious physical reasons, be smaller than the variance of the derivative ∂θ/∂ in this temperature (energy) range. Then, when max1 falls below a certain critical value, the calculated spectrum c(T) does not tend to zero from the left hand side but to infinity. This nonphysical behavior of c(T) is a check of using nonphysical values of max1. Figure 4 shows two functions χc(c) corresponding to two pairs of the values K ˜ d and max1. One of the functions ˜ d ) 2000 atm kJ mol-1 χc(c) calculated by assuming K -1 -1 min and max1 ) 49.5 kJ mol appears to be normalized to unity, whereas for the other function χc(c) calculated by assuming K ˜ d ) 2000 atm kJ mol-1 min-1 and max1 ) 45 kJ mol-1 the integral is smaller than unity. A closer inspection into Figure 4 suggests that the function for which the integral in eq 61 is close to unity represents a less realistic solution from a physical point of view. In order to show it, we make a “decomposition” of that condensation function χc(c) into overlapping one-modal distributions, each of them corresponding to a certain kind of adsorption site. Such a decomposition of the condensation function χc(c) was reported by Cases and co-workers82 for a number of adsorption systems. These authors came to the conclusion that the general physical situation will be the following. If even one may distinguish, on a given surface, a number of distinct kinds of adsorption sites (or surface domains), there will still exist a certain distribution of surface properties (adsorption energies) within each group (do(82) Villie´ras, F.; Cases, J. M.; Franc¸ ois, M.; Michot, L. J.; Thomas, F. Langmuir 1992, 8, 1789.
The variance of χc. b The variance of the ∂θi/∂.
main) of sites. Therefore, one should consider the following representation for an actual adsorption energy distribution χ() n
χ() )
n
γiχi(), ∑γi ) 1 ∑ i)1 i)1
(62)
where χi() is the adsorption energy distribution for the ith kind of adsorption site but is not necessarily a Dirac delta function. The “condensation” function χc() calculated from a TPD diagram will then have the form n
χc(c) )
γi∫0 ∑ i)1
∞
() ∂θi ∂
n
χi() d )
γiχci(c) ∑ i)1
(63)
where θi() is the isotherm equation describing adsorption on the ith kind of adsorption site. In their improved decomposition method (DIS), Cases et al.83 have used the following DA functions to represent χci()
χci(c) )
rci(c - °ci)rci-1 Erci
((
exp -
))
c - °ci Eci
rci
(64)
the features of which are the same as those of the functions in eqs 33a. (This is simply a kind of an analytical approximation.) Figure 5 shows an example of the decomposition (eq 63) of the condensation function χc(c). The corresponding best-fit parameters are given in Table 2. Table 2 reports also on the variances of the functions χci(c), along with the variance of their corresponding derivatives ∂θi/∂, the maximum of which is located at the same point of c at which the function χci(c) reaches its maximum. For obvious reasons, the variance of χci(c) must (83) Villie´ras, F.; Michot, L. J.; Cases, J. M.; Franc¸ ois, M.; Rudzin˜ski, W. Langmuir, in press.
Solid Surface Energetic Heterogeneity
Figure 6. Better approximation χRJ() for χ(), calculated using eq 56, from the “condensation” function χc(c) shown by the solid line in Figure 4.
Figure 7. Calculated theoretical TPD spectra of hydrogen desorption calculated by accepting two pairs of values: 1 ) 47kJ/mol, 2 ) 70 kJ/mol (- -) and 1 ) 35 kJ/mol, 2 ) 50 kJ/mol (s). The intensity is expressed in a molar fraction of hydrogen in the carrier gassargon.
be larger than that of its corresponding derivative ∂θi/∂, or equal when the function χci(c) is a Dirac delta function. Table 2 shows that this condition is not fulfilled for three functions χci(c) (i ) 1, 2, 3). On the contrary, the decomposition (eq 63) of the other function χc(c) in Figure 4, the integral (eq 61) of which is smaller than unity, yields the composite functions χci(c) having variances larger than the variances of their corresponding derivatives ∂θi/∂. But this would mean that the desorption in our experiment started from surface coverages smaller than unity. As the latter χc(c) function seems to be more physically realistic, we used it to calculate the χRJ() function according to eq 56. Figure 6 shows the calculated χRJ() function. That function shows clearly the existence of the five different kinds of adsorption sites, the existence (and mutual proportions) of which could not so clearly be deduced from the initial experimental TPD spectrum shown in Figure 2. While thinking about a possibility of determining the parameters K ˜ d and max1 in an independent way, we focus our hopes on possible accompanying independent calorimetric measurements. Calorimetric studies of the adsorption equilibria show that the main contribution to the isosteric heat of adsorption qst comes from the energies
Langmuir, Vol. 13, No. 13, 1997 3453
Figure 8. Theoretical TPD spectra of hydrogen desorption corresponding to various heating rates, obtained by accepting 1 ) 47 kJ/mol and 2 ) 70kJ/mol. The three solid lines from the top to the bottom were calculated by accepting the heating rate β ) 80, 40, and 20 K/min, respectively.
of adsorption and from their distribution. In particular, the shape of the experimental curve qst(θt) is related mainly to the shape of the adsorption energy distribution χ().71 And, as the shape of χ() calculated from the TPD spectrum is also affected by the choice of the parameters K ˜ d and max1, an independent calorimetric measurement of qst(θt) may carry the necessary information about the right choice of the parameters K ˜ d and max1. That possibility deserves obviously further theoretical studies and will be a subject of our forthcoming publication. Finally, we are going to demonstrate the predictive features of our new approach to the TPD spectra from the energetically heterogeneous surfaces. Let us consider a solid surface containing two kinds of adsorption sites, one characterized by the adsorption energy 1 and another one having an adsorption energy equal to 2. So, the adsorption energy distribution χ() is the following linear combination of two Dirac delta functions:
χ() ) X1δ( - 1) + X2δ( - 2)
(65)
where X1 and X2 ) (1 - X1) are the fractions of the surface sites having adsorption energies 1 and 2, respectively. The “condensation” distribution function χc(c) has then the following form:
( (
) )]
1 - c 1 exp kT kT χc(c) ) X1 1 - c 1 + exp kT
[
( (
) )]
2 - c 1 exp kT kT + X2 2 2 - c 2 1 + exp kT (66)
[
For the purpose of illustrative calculations we take X1 ) X2 ) 1/2. Figure 7 shows examples of two TPD spectra calculated by accepting two pairs of values, 1 and 2. Figure 8 shows how the heating rate β affects our theoretical TPD spectra. Generally, the two peaks are shifted toward higher temperatures and their overlapping increases, as usually occurs in experiment. LA960921G
