6386
Langmuir 1999, 15, 6386-6394
A New Quantitative Interpretation of Temperature-Programmed Desorption Spectra from Heterogeneous Solid Surfaces, Based on Statistical Rate Theory of Interfacial Transport: The Effects of Simultaneous Readsorption Władysław Rudzin´ski,*,† Tadeusz Borowiecki,‡ Anna Dominko,‡ and Tomasz Pan´czyk† Departments of Theoretical Chemistry and Chemical Technology, Faculty of Chemistry, UMCS, Pl. Marii Curie-Sklodowskiej 3, 20-031 Lublin, Poland Received January 5, 1998. In Final Form: April 20, 1999 We have recently proposed a new method of estimating the surface energetic heterogeneity of solids from temperature-programmed desorption (TPD) spectra analyzed in terms of the statistical rate theory of interfacial transport (SRTIT) (Chem. Anal. 1996, 41, 1057; In Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces; Elsevier: New York, 1996.). SRTIT is a new theoretical approach linking the rate of desorption kinetics to the chemical potential of the adsorbed molecules and the molecules in the gas phase. While applying SRTIT to analyze TPD spectra, one arrives at the condensation approximation (CA) for the adsorption energy distribution. Having obtained the CA function, one can calculate the exact adsorption energy distribution using well-known methods (Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992). The analysis of TPD spectra in terms of SRTIT is improved here by taking into account the readsorption kinetics. As an example of its application, an analysis of a TPD spectrum of hydrogen desorption from an alumina-supported catalyst is presented.
1. Introduction TPD (temperture-programmed desorption) is probably the most frequently used method of investigating the energetic heterogeneity of catalyst surfaces. By “surface energetic heterogeneity” we understand the variation in the gas-solid potential function across a solid surface. In chemisorption, such systems as those involved in catalysis, the adsorbing molecules are “trapped” in deep local minima of the gas-solid potential functionsthe “adsorption sites”. In such cases the energetic surface heterogeneity is considered in terms of localized adsorption on a lattice of energetically different adsorption sites. A quantitative measure of this surface energetic heterogeneity is the differential distribution of the number of adsorption sites among corresponding values of adsorption energy , χ(), usually applied in its form normalized to unity,
∫Ω χ() d ) 1
(1)
where Ω is the physical domain of on a solid surface. The experimentally observed fractional coverage of the solid surface θt(p,T) is then given by the following average,
θt(p, T) )
∫Ω θ(, p, T) χ() d
(2)
where θ(, p, T) is a theoretical expression describing the fractional coverage of the sites whose adsorption energy lies within the interval (, + d ), commonly called the “local adsorption isotherm”. The function χ() is commonly called the “adsorption energy distribution”. * To whom correspondence should be addressed. E-mail:
[email protected]. † Department of Theoretical Chemistry. ‡ Department of Chemical Technology.
A variety of expressions have been used to represent the “local” adsorption (isotherm), depending on the physicochemical nature of a gas-solid interface under consideration. Equation 2 can be treated as an integral equation for the unknown function χ(), in which θ(, p, T) is known from theory and θt(p, T) is known from experiment. A variety of methods have been used to solve eq 2, and their description can be found in two recently published monographs.3,4 Determining the function χ() from the isotherm θt(p, T) can easily be done in the case of physisorption, where the solid-adsorbate bonds are relativly weak and measurements of the adsorption isotherms can be carried out at low or room temperatures. In the case of chemisorption, measurements of θt(p, T) involve many technical problems to be solved. To study the surface energetic heterogeneity in chemisorption systems, a different technique has been developedsthe temperature-programmed desorption. The principles of that method were published by Amenomiya and Cvetanovic in 1963,5 but it was not until 9 years later that the first theoretical paper on the application of this method to study the surface energetic heterogeneity was published by Cvetanovic and Amenomiya.6 (1) Rudzinski, W.; Borowiecki, T.; Dominko A.; Zientarska, M. Chem. Anal. 1996, 41, 1057. (2) Rudzinski, W. A New Theoretical Approach to AdsorptionDesorption Kinetics on Energetically 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. (3) Rudzinski, W.; Everett, D. M. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (4) Jaroniec, M.; Madey, E. Physical Adsorption on Heterogeneous Solids; Elsevier: New York, 1989. (5) Amenomiya, Y.; Cvetanovic, R. J. J. Phys. Chem. 1963, 67, 144. (6) Cvetanovic, R. J.; Amenomiya, Y. Catal. Rev.-Sci. Eng. 1972, 6, 21.
10.1021/la9800147 CCC: $18.00 © 1999 American Chemical Society Published on Web 07/10/1999
Interpretation of TPD Spectra from Solid Surfaces
Langmuir, Vol. 15, No. 19, 1999 6387
The TPD technique was first used 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. Since then, the interpretation of TPD spectra has remained largely on a qualitative level. Attempts to draw quantitative information about surface energetic heterogeneity from experimental TPD spectra were mostly based on the ART (absolute rate theory) approach,7 used to describe the kinetics of adsorption and desorption.8-11 As that approach is still commonly used in the interpretation of the TPD spectra, certain comments on this approach are necessary. We start by discussing the fundamental equation for the rate of adsorption/desorption offered by the ART approach. In the interest of clarity, we consider the simplest case when only one molecule is adsorbed on one site. Then, according to the ART,7-11
dθ ) Kap(1 - θ) exp(-a/kT) - Kdθ exp(-d/kT) (3) dt where t is the 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 related pre-exponential constants. T and k are the absolute temperature and the Boltzmann constant, respectively. The first term on the right-hand side (r.h.s.) of eq 3 represents the rate of adsorption, and the second term represents the desorption rate. At equilibrium, when (dθ/dt) ) 0, one arrives at the Langmuir adsorption isotherm, θ(e),
θ(e)(, p, T) )
Kp exp(/kT) 1+ Kp exp(/kT)
(4)
where K ) Ka/Kd, ) (d - a), and the superscript (e) stands for equilibrium. The inapplicability of the Langmuir equation to represent the gas/solid equilibria in real adsorption systems was reported as early as at the beginning of the 20s. As a result, instead of the Langmuir equation, various empirical isotherms were used.3,4 For example, it was found that in systems with strongly heterogeneous surfaces the Temkin equation can be used to describe θt(p, T). In the case of the systems with less heterogeneous surfaces the Langmuir-Freundlich isotherm is good for correlating experimental data. When the solutions of the integral equation (2) were first obtained in the 40s (eqs 3 and 4), it became clear that when the Temkin empirical isotherm represents θt(p, T) in eq 2, the recovered adsorption energy distribution χ() is a rectangular function, whereas when the Langmuir-Freundlich isotherm is to be used for θt(p, T), the recovered adsorption energy distribution is a Gaussian-like function. In all such cases the local isotherm was taken to be the Langmuir isotherm. (7) Clark, A. The Theory of Adsorption and Catalysis; Academic Press: New York, 1970. (8) Falconer, J. L.; Schwarz, J. A. Catal. Rev.-Sci. Eng. 1983, 25, 141. (9) Bhatia, S.; Beltramini, N.; Do, D. D. Catal. Today 1990, 8, 309. (10) Cerofolini, G. The Intrinsically Heterogeneous Nature of Surfaces of Catalytic Interest. In Adsorption of New and Modified Inorganic Sorbents; Dabrowski, A., Tertykh, V. A., Eds.; Elsevier: New York, 1996; p 435. (11) 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, 1996.
The inability of the two terms on the r.h.s of eq 3 to correlate the observed kinetics of adsorption and desorption, respectively, was reported as early as in the 20s.12 These two terms are commonly believed to represent the so-called “Langmuirian kinetics” because they lead to the Langmuir isotherm at equilibrium. In practice, instead of the “Langmuirian” term for adsorption kinetics in eq 3, various empirical equations were used. Then, it was observed that in the systems where adsorption equilibria could be described by the Temkin isotherm, the Elovich empirical equation appeared to be a good formula for describing the adsorption kinetics.13,14 In cases where the adsorption equilibria were described by the LangmuirFreundlich equation, the power-law equation appeared to be the best formula to describe the adsorption kinetics.15-18 So it seems that it is the energetic heterogeneity of the real solid surfaces that is responsible for the wide applicability of the Elovich and of the power-law equations to adsorption kinetics. Recently,2 the application of the statistical rate theory of interfacial transport (SRTIT) made it possible to prove that the Elovich equation represents the isothermal adsorption kinetics in the systems characterized by a rectangular adsorption energy distribution, whereas the power-law equation holds for the systems with a Gaussian-like energy distribution. At an early stage of thermodesorption experiments, it was commonly assumed,6,19-23 that surface energetic heterogeneity manifests itself as a linear variation (increase) of the activation energy for desorption with decreasing surface coverage. King24 and Tokoro et al.25 then proposed that the activation energy for desorption may be a more complex function of the surface coverage. An interesting paper on this topic was published by Kno¨zinger and Ratnasamy.26 After 1980, papers concerning this problem were published by Davydov et al.,27 Unger et al.,28 Malet and Munuera,29 Leary et al.,30 Ma et al.,31 and Salvador and Merchan.32 Very interesting reports concerning this issue were presented at the “Second International Symposium on Surface Heterogeneity Effects in Adsorption and Catalysis (12) Low, M. J. Chem. Rev. 1960, 60, 267. (13) Roginski, S. Z. Adsorption and Catalysis on Heterogeneous Surfaces; Izd-vo Akademii Nauk SSSR: Moscow-Leningrad, 1948. (14) Aharoni, C.; Tompkins, F. Adv. Catal. 1970, 21, 1. (15) Bangham, D. H.; Burt, F. F. Proc. Royal Soc. 1924, A105, 481. (16) Zukhovitskii, S. I. Adsorption of Gases and Vapours; ONTI: Moscow, 1935. (17) Elovich, S. Yu.; Kharakhorin, F. F. Problems Kinet. Catal. 1937, 3, 322. (18) Cerofolini, G. F.; Re, N. J. Colloid Interface Sci. 1995, 174, 428. (19) Czanderna, A. W.; Biegen, J. R.; Kollen, W. J. Colloid Interface Sci. 1970, 34, 406. (20) Carter, G. Vacuum 1962, 12, 245. (21) Grant, W. A.; Carter, G. Vacuum 1965, 15, 13. (22) Dawson, D. T.; Peng, Y. K. Surf. Sci. 1972, 33, 565. (23) Tokoro, Y.; Misono, M.; Uchijima T.; Yoneda, Y. Bull. Chem. Soc. Jpn. 1978, 51, 85. (24) King, D.A. Surf. Sci. 1975, 47, 384. (25) Tokoro, Y.; Uchijima, T.; Yoneda, Y. J. Catal. 1979, 56, 110. (26) Kno¨zinger, H.; Ratnasamy, P. Catal. Rev.-Sci. Eng. 1978, 17, 31. (27) Davydov, V. Ya.; Kiselev, A. V.; Kiselev, S. A.; Polotryuk V. O. V. J. Colloid Interface Sci. 1980, 74, 378. (28) Unger, K. K.; Kittelman, U. R.; Kreis, W. K. J. Chem. Technol. Biotechnol. 1981, 31, 435. (29) 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. (30) Leary, K. J.; Michaels, J. N.; Stacy, A. M. AIChE J. 1988, 34, 263. (31) Ma, M. C.; Brown, T. C.; Haynes, B. S. Surf. Sci. 1993, 297, 312. (32) Salvador, F.; Merchan, D. React. Kinet. Catal. Lett. 1994, 52, 211.
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on Solids” held in Zakopane-Levocˇa (Poland-Slovakia) in the autumn of 1995.33 The shapes of the experimential TPD spectra suggested that d may, in general, be so complicated of a function of θt that it could not be modeled by an analytical function with a small number of parameters. As a result, some attempts have been made to determine the exact “experimental” shape of d(θt) from measured TPD spectra. Various methods have been applied for that purpose,20,34-43 but it seems that the most popular is that one was based on applying the following Arrhenius plot,
ln
-(dθt/dt) d ) ln Kd θt kT
(5)
along with a variable heating rate. (We will limit here our considerations to one-site-occupancy adsorption). When that method was applied to the experimental TPD spectra, it was found that not only d varies with surface coverage. Surprisingly, Kd can also vary over several orders of magnitude. This can best be seen in the excellent review by Zhdanov,44 where he collected d(θt) and Kd(θt) functions determined for a variety of adsorption systems. Seebauer et al.45 have reviewed different theoretical representations for the pre-exponential factor Kd. None of them is able to account for such strong variations of Kd with θt. Such difficulties related to the use of the ART approach while describing the kinetics of adsorption/ desorption on/from the heterogeneous surfaces accumulated were driving scientists to undertake fundamental studies on adsorption/desorption kinetics. As early as in 1957 Kisliuk46 proposed the idea of a “precursor state”. He assumed that, before being adsorbed, molecules form a weakly bound “precursor phase”. Molecules which do not strike empty sites but do already adsorbed molecules may still have a chance to be adsorbed by entering such a precursor state and then jumping to a neighboring empty site. Twenty years later in 1977 King47 made further effort to improve that idea by assuming that two precursor states may exist: one over a filled adsorption site and the other over an empty site. Still another version of that theory was proposed in 1978 by Gorte and Schmidt.48 At the time “precursor state” theories did not appear to solve the problem of large variations in Kd with coverage, nor did they attract much attention among scientists analyzing TPD spectra from heterogeneous solid surfaces.49 (33) Proceedings of the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids; Zakopane-Levocˇa, Poland-Slovakia, Autumn 1995; Brunovska, A., Rudzinski, W., Wojciechowski, B. W., Eds. (34) Bown, T. C.; Lear, A. E.; Ma, M.C.; Haynes, B. S. In Fundamental Issues in the Control of Carbon Gasification Reactivity; Lahaye, J., Ehrburger, P., Eds.; Kluwer: Boston, 1990; p 307. (35) Witkopf, H. Vacuum 1984, 37, 819. (36) Britten, J. A.; Travis, B. J.; Brown, L. F. Adsorption and Ion Exchange AIChE Symposium; AIChE: New York, 1983; p 7. (37) Carter, G.; Bailey, P.; Armour, D. G. Vacuum 1982, 32, 233. (38) Du, Z.; Sarofin, A. F.; Longwell, J. P. Energy Fuels 1990, 4, 296. (39) Ma, M. C.; Brown, T. C.; Haynes, B. S. Surf. Sci. 1993, 297, 312. (40) Seebauer, E. G. Surface Sci. 1994, 316, 391. (41) Southwell, R. P.; Seebauer, E. G. Surf. Sci. 1995, 340, 281. (42) Southwell, R. P.; Seebauer, E. G. Surf. Sci. 1995, 329, 107. (43) Cerofolini, G. F.; Re N. J. Colloid Interface Sci. 1995, 174, 428. (44) Zhdanov, V. Surf. Sci. Rep. 1991, 12, 183. (45) Seebauer, E. G.; Kong, A. C. F.; Schmidt, L. D. Surf. Sci. 1988, 193, 417. (46) Kisliuk, P. J. Phys. Chem. 1957, 3, 95. (47) King, D. A. Surf. Sci. 1977, 64, 43. (48) Gorte, R.; Schmidt, L. D. Surf. Sci. 1978, 76, 559. (49) Lombardo, S. I.; Bell, A. T. Surf. Sci. 1991, Rep. 13, 1.
Then, at the beginning of the 80s, a new family of approaches to adsorption/desorption kinetics appeared. A common fundamental feature of all these approaches is that they relate the rate of desorption kinetics to the chemical potential of the adsorbed molecules. These were the approaches proposed by Nagai,50,51 Kreuzer and coworkers,52-56 and Ward and co-workers.57-60 The appearance of that idea resulted in a heated debate.61-64 Using the hard hexagon adsorption model, Nagai65 showed, on one hand, that the simple ART approach neglects the role of entropy changes as an important factor affecting the kinetics of adsorption-desorption processes. On the other hand, the approaches using the chemical potential of adsorbed molecules do account for the effect of entropy changes. Recently, Rudzinski and co-workers have published the first attempts to generalize the SRTIT approach, developed by Ward and co-workers,66-68 to heterogeneous surfaces. By applying that approach, Rudzinski et al. have been able to show that the Elovich and the power law, originally proposed as empirical equations, correspond to the Temkin and Langmuir-Freundlich equations for adsorption equilibria. The authors then developed a method of quantitative analysis of surface energetic heterogeneity using experimental TPD spectra. What is perhaps more important is that the new method of analysis of TPD spectra does not involve the assumption that Kd will vary with surface coverage. In this paper this method is further generalized by taking readsorption into account. 2. Theory With the assumption that transport between two phases at thermal equilibrium results primarily from single molecular events, the equation for the rate of transport between gas and a solid phase was developed by Ward and Findlay57 using a first-order perturbation analysis of the Schro¨dinger equation and the Boltzmann definition of entropy, (50) Nagai, K. Phys. Rev. Lett. 1985, 54, 2159. (51) Nagai, K. Surf. Sci. 1986, 176, 193. (52) Kreuzer, H. J.; Payne, S. H. Surf. Sci. 1988, 198, 235; 1988, 200, L433. (53) Payne, S. H.; Kreuzer, H. J. Surf. Sci. 1988, 205, 153. (54) 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. (55) Kreuzer, H. J. Langmuir 1992, 8, 774. (56) Kreuzer, H. J.; Payne, S. H. Theories of Adsorption-Desorption Kinetics on Homogeneous Surfaces. In Equilibria and Dynamics of Gas Adsorption on Heterogenous Solid Surfaces; Rudzinski, W., Steele, W. A., Zgrablich, G., Eds.; Elsevier: New York, 1966. (57) Ward, C. A.; Findlay, R. D. J. Chem. Phys. 1982, 76, 5615. (58) 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. (59) Ward, C. A.; Elmoselshi, M. Surf. Sci. 1986, 176, 457. (60) Elliott, J. A.; Ward, C. A. J. Chem. Phys. 1997, 106, 5667. (61) Zhdanov, V. P. Surf. Sci. 1986, 171, L461. (62) Nagai, K.; Hirashima, A. Surf. Sci. 1986, 171, L464. (63) Cassuto, A. Surf. Sci. 1988, 203, L656. (64) Nagai, K. Surf. Sci. 1988, 203, L659. (65) Nagai, K. Surf. Sci. Lett. 1991, 244, L147. (66) Rudzinski, W.; Aharoni, C. Langmuir 1997, 13, 1089. (67) Rudzinski, W.; Borowiecki, T.; Dominko, A.; Panczyk, T.; Gryglicki, J. Polish J. Chem. 1998, 72, 2103. (68) Rudzinski, W.; Panczyk, T. Surface Heterogeneity Effects on Adsorption Equilibria and Kinetics: Rationalization of Elovich Equation. In Surfaces of Nanoparticles and Porous Materials; Schwarz, J., Contescu, A., Eds.; Marcel Dekker: New York, 1999; Chapter 15, pp 355-390.
Interpretation of TPD Spectra from Solid Surfaces
[ (
)
)]
(
(µg - µs) (µs - µg) dθ ) K′gs exp - exp dt kT kT
Langmuir, Vol. 15, No. 19, 1999 6389
(6)
where µg and µs are the chemical potentials of the adsorbate in the gas phase and in the adsorbed (surface) phase, respectively. Then, K′gs is a constant describing the rate of the elementary adsorption/desorption processes at equilibrium. To a good approximation, K′gs can be written as the following product,
K′gs ) Kgsp(e)(1 - θ(e))
(7)
where the equilibrium state is defined now to be that one, to which the system would evolve, after making it isolated, at a certain adsorbed amount θ, and the amount of the species in the gas phase. Until thermodesorption is not very fast, p(e) and θ(e) should not be far from their nonequilibrium values p and θ. So, we will assume p(e) = p and θ(e) = θ in our further considerations. An important assumption made by Ward and Findlay was that the transient surface configurations of the adsorbed molecules during adsorption are close to those at equilibrium at the same surface coverage. Let µg be the chemical potential of an ideal gas, and then
µg ) µg0 + kT ln p
(8)
Let µs be the expression for the chemical potential of the adsorbed molecules corresponding to the Langmuir model of adsorption (one-site-occupancy monolayer adsorption with no interactions between the adsorbed molecules). Then,
µs ) kT ln
θ , qs ) qs0 exp kT qs(1 - θ)
( )
(9)
where qs is the molecular partition function of the adsorbed molecules. The energetic heterogeneity of a solid surface is manifested mainly as a variation of the adsorption energy in the molecular partition function qs,3. For molecules adsorbed on different sites, eq 6 takes then the following form,
(1 - θ)2 dθ ) K′ap2 - K′dθ dt θ
K′a ) Kgsqs exp
( ) kT
and K′d )
Kgs qs
exp
K)
( ) -µg0 kT
x
( )
Ka µg0 s ) q0 exp Kd kT
(14)
As thermodesorption is assumed to run at quasiequilibrium conditions, we can study the temperature dependence of the coverage of adsorption θ, on sites having adsorption energy equal to by considering eq 4. For that purpose we rewrite this equation to the following form,
θ() )
(10)
where
µg0
of view, it seems strange that two different kinetic processes can lead to the same behavior of a physical system at equilibrium. Then again, if the Langmuir equation can result from varius kinetic equations, the question arises, which kinetic expression is correct? In a recent review,68 the authors have presented an exhaustive analysis of the difficulties and paradoxes which one faces while trying to generalize the classical ART equations,3 to describe the kinetics of adsorption/desorption processes, carried out at isothermal conditions. Then, it was shown there that the application of the SRTIT makes it possible to develop in an easy and consistent way all the empirical equations (Elovich, power law), which have been used so far to correlate the data for the isothermal kinetics of adsorption/desorption processes on/from heterogeneous solid surfaces. Our theoretical considerations were focused on the Elovich equation, which is still used most commonly to correlate experimental data for isothermal adsorption kinetics. We have shown there that the Elovich equation can easily be developed by using SRTIT, neglecting the second desorption term, and assuming a rectangular adsorption energy distribution. Next, we generalized the Elovich equation by taking into account also the second desorption term. At equilibrium, the generalized equation led us to the adsorption isotherm expression which has been known in the literature as the Temkin equation, obtained by generalization of the Langmuir isotherm equation for the case of heterogeneous surfaces characterized by a rectangular adsorption energy distribution. At equilibrium, when dθ/dt ) 0, the SRTIT equation (12) leads to a Langmuir isotherm (4), and the meaning of the Langmuir constant K is the following:
exp{( - c)/kT} 1 + exp{( - c)/kT}
(15)
where
(11)
With this notation eq 10 takes the form
(1 - θ)2 dθ ) Kap2 exp(/kT) - Kdθ exp(-/kT) (12) dt θ where
Ka ) Kgs Kgsqs0 exp(µg0/kT) and Kd ) s exp(-µg0/kT) (13) q0 Note that, at equilibrium, when (dθ/dt) ) 0, eq 12 takes the form of the Langmuir isotherm (4). The fact that two different kinetic equations lead to the same form of the equilibrium isotherm is intriguing. From the physical point
c ) -kT ln Kp(e)
(16)
and where the quasiequilibrium pressure p is equal to c(T)P, i.e., to the atmospheric pressure P multiplied by the recorded concentration c of the desorbed species in the carrier gas. Figure 1 shows the temperature dependence of the function θ(),
θ() )
exp(r/τ) 1 + exp(r/τ)
(17)
where r ) ( - c)/kT0 and τ ) T/T0 are appropriate dimensionless values. Figure 1 shows that the kernel θ() becomes a step function in the hypothetical low-temperature limit T f 0:
6390 Langmuir, Vol. 15, No. 19, 1999
Rudzin´ ski et al.
Figure 1. Temperature dependence of the function θ(, c). The dimensionless temperatures are τ1 ) 0.5 (s), τ2 ) 4 (- - -), and τ3 ) 8 (- - -).
limθ() ) θc() ) Tf0
{
0, 1,
< c g c
(18)
This in turn means that desorption from a heterogeneous solid surface proceeds in a stepwise-like fashion. In the limit T f 0 the kernel θ() in eq 15 becomes the step function (18), so that θt takes the form
θt )
∫
∞ c
χ() d ) χ(c)
∫
χ() d
( )
-
()
∫0∞ ∂θ ∂
(20)
( )[
∫0∞ ∂θ ∂
χ() d ) -
χ(c) +
( )
(∂χ∂) ( - ) + c
c
]
1 ∂2χ ( - c)2 + ... d (23) 2 ∂2 c
Thus,
θt ) -χ(c) -
and the integration constant is chosen in such a way that χ(∞) ) 0. The assumption that Ω is the interval (0, +∞) is commonly made in adsorption literature for purposes of mathematical convenience. It does not affect the calculated results much except at very low, (θt f 0), or very high, (θt f 1) surface coverages.3 θt calculated in this way has been known in the past as the application of the “condensation approximation”.3 At finite temperatures θ() is not an exact step function, but it is a frequent practice to replace the true kernel θ under the integral (2) by a step function θc located at ) c defined by the condition,
∂2θ ∂2
maximum,
(19)
where χ() is the integral of χ(),
χ() )
Figure 2. Derivatives (∂θ/∂) of isotherms θ(, c) drawn in Figure 1.
( )∫( ) ( )∫( ) ∂χ ∂
∞
c
0
∂θ ( - ) d ∂ c
1 ∂2χ 2 ∂2
∂θ ( - )2 d + ... (24) ∂ c
∞
c
0
The first term on the r.h.s. of eq 24 is exactly the result previously obtained by applying the condensation approximation (CA), i.e., assuming the hypothetical limit to be T f 0. Because of the symmetry of the function (∂θ/∂), the second term on the r.h.s. of eq 24 disappears, and the first nonvanishing correction to -χ(c) is the third term. For the Langmuir model of adsorption, to a good approximation,3 2
π ( - )2 d ) (kT)2 ∫0∞(∂θ ∂ ) c 3
(25)
The function χc(c) calculated from the relation
)0
(21)
)c
χc(c)) -
Figure 2 shows the derivatives (∂θ/∂) of the isotherms θ(, c) drawn in Figure 1. It has been shown in the literature on adsorption equilibria that, surprisingly, the above-described practice leads to fairly exact values of θt for many adsorption regimes reported in the literature. The reason is that -χ(c) is the leading term of an expansion of θt at finite temperatures.3 Here, we will briefly review the principles of that expansion. The integral (2) is evaluated by parts:
θt(p, T) ) θ(, p, T) χ()|∞0 -
χ() d ∫0∞(∂θ ∂ )
(22)
It can be shown that the first term on the r.h.s. of eq 22 is negligible compared to the second one (except for very small and very high values of θt3). In that case the second term is evaluated by expanding χ() into its Taylor series around the point ) c, at which (∂θ/∂) reaches a
dθt 1 dNt(c) )dc Nm dc
(26)
is usually called the condensation approximation of the exact adsorption energy distribution χ() and is defined as follows:3
χc(c) )
χ() d ∫0∞(∂θ ∂ )
(27)
In the limit T f 0 the derivative (∂θ/∂) becomes the Dirac δ function δ( - c) and χc(c) becomes χ(). In studies using temperature-programmed desorption, at each temperature T, the remaining adsorbed amount Nt(T) is found from the relation
Nt(T) ) N0 -
∫TTc(T) dT
F β
0
(28)
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
Interpretation of TPD Spectra from Solid Surfaces
Langmuir, Vol. 15, No. 19, 1999 6391
constant. We will now show how the condensation approximation function χc(c) can be evaluated from TPD spectra. We assume for simplicity that at T0 the surface coverage θt is complete so that N0 ) Nm, and that at the final temperature, Tk, θt is zero. Then,
Nm )
∫TT c(T) dT
F β
k
(29)
0
The derivative (dNt/dt) calculated from eq 28 is given by
dNt Fc(t(T)) )dT β
(30)
Now, since
θt )
exp(( - )/kT)
∫Ω1 + exp(( -c )/kT)χ() d
(31)
c
the derivative (dNt/dt) can also be expressed as follows:
[
]
∂θt ∂θt dc dNt ) Nm + dT ∂T ∂c dT
(32)
From eqs 24 and 25, it follows that
∂θt π2 ) - (k2T)χ′(c) ∂T 3
(33)
From eqs 16, 30, 31, and 33 we have
-
( )
∂θt dc π2 2 Fc(t(T)) ) - k Tχ′(c) Nmβ ∂c dT 3
(34)
Note that the second term on the r.h.s. of eq 34 is a secondorder correction term. Like in the case of adsorption equilibria, in most cases it can be ignored, compared to the first term. In view of the definition of χc(c) in eq 26, we now arrive at
χc(c) )
[ ]
Fc(T(c)) dc βNm dT
-1
(35)
Now, it will be very essential for our consideration to remark that c is the value of on the adsorption sites, whose local coverage θ is equal to 1/2. So, we define the function F(c, t, T):
1 F(c, t, T) ) θ(c, t, T) - ) 0 2
(36)
Then
[ ]
(∂F/∂t)c (∂θ/∂t) dc ))dt (∂F/∂c)t (∂θ/∂)t
(37)
θ)1/2,)c
( )
1 1 ) Kap2ec/kT - Kde-c/kT c,θ)1/2 2 2
(38)
Further, when ) c and θ ) 1/2
[(∂θ∂) ]
t θ)1/2,)c
and finally
)
1 4kT
(39)
( )
Equation 40 is a first-order differential equation for the function c(T) because, for physical reasons, c must be a one-to-one function of T. Its solution yields also the function T(c). At the same time c(T) in eqs 35 and 40 can also be expressed as a function of c. By inserting (∂c/∂T) from eq 40 into eq 35 and expressing c(T) as c(c), we can calculate χc(c). The “condensation” function χc(c) is an approximation of the true adsorption energy distribution χ(). The function χc() becomes χ() in the limit T f 0, and also when (δθ/δχ) f 0, where δθ is the variance of the derivative (∂θ/∂), whereas δχ is the variance of χ(). In other words, the more strongly heterogeneous the solid surface, the closer is the function χc(c) to χ(). Having calculated χc(c), one can calculate a better approximation for χ() using a variety of methods proposed in the literature.69-76 All of them involve calculating functions which contain derivatives of χc(c). The solution of the equation (40) yields the function T(c) which is to be inserted into eq 35 to calculate -χc(c), from the experimental function c(T). The form of the solution c(T) will vary due to 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 Kd. To incorporate this temperature dependence, one has to know the temperature dependence of the molecular partition function of the adsorbed molecules qs0. It is well-known3 that an a priori theoretical determination of the temperature dependence of qs0 is a difficult theoretical problem. However, it is commonly assumed that qs0 is a slowlyvarying function of low powers of T. For that reason, we will neglect the temperature dependence of K ˜ d ) 2kTKd in our forthcoming analysis of TPD spectra. Let us consider the predictive features of our new method of quantitative analysis of TPD spectra. Comparing eqs 35 and 40 we have
( )
2kTKd c Fc(T) exp ) β kT χc(c)βNm
( )
c 2kTKaP2c2(T) exp (41) β kT Equation 41 is a quadratic equation with respect to c(T). Of its two solutions, only the following one has physical meaning:
c(T) )
To represent (∂θ/∂t)c we use the SRTIT equation 12. Thus,
∂θ ∂t
( )
dc 2kTKd c 2kTKaP2c2(T) c ) exp exp dt β kT β kT (40)
xF2 + 4K˜ aK˜ dχc2(c)Nm2P2 - F 2K ˜ aχc(c)ec/kTNmP
(42)
To calculate a theoretical c(T) spectrum when the function χ() is known, we take the following steps: (1) The function (69) Cerofolini, G. F. Z. Phys. Chem. (Leipzig) 1978, 259, 1020. (70) Jagiello, J.; Schwarz, J. A. J. Colloid Interface Sci. 1991, 146, 415. (71) Re, N. J. Colloid Interface Sci. 1994, 166, 191. (72) Nederlof, M. N.; Van Riemsdijk, W. H.; Koopal, L. K. J. Colloid Interface Sci. 1990, 135, 410. (73) Hsu, C. C.; Wojciechowski, B. W.; Rudzinski W.; Narkiewicz, J. J. Colloid Interface Sci. 1978, 66, 292. (74) Rudzinski, W.; Narkiewicz J.; Patrykiejew, A. Z. Phys. Chem. (Leipzig) 1979, 260, 1097. (75) Rudzinski, W.; Jagiello, J. J. Low Temp. Phys. 1981, 45 1. (76) Rudzinski, W.; Jagiello, J.; Grillet, Y. J. Colloid Interface Sci. 1982, 87, 478.
6392 Langmuir, Vol. 15, No. 19, 1999
Rudzin´ ski et al.
χc(c) is calculated from eq 27. (2) The calculated function χc(c) is inserted into eq 42, and the function c(T) is inserted into eq 35 to give
( )
FxF2 + 4K dc ˜ aK ˜ dχc2(c)Nm2P2 - F2 ) dT 2K ˜ χ 2( )βec/kTN 2P2 a c
c
(43)
m
(3) After solving eq 43, the calculated c(T) function is inserted into eq 42 to calculate c(T). 3. Experimental Results and Discussion The wide range of applications of supported nickel catalysts in hydrogenation and various stabilization and hydrotreating processes is the reason why a great number of studies aimed at their description and characterization has appeared in the literature. In view of that we have taken a hydrogen TPD spectrum from an alumina-supported nickel catalyst (15% Ni) obtained by impregnating R-Al2O3 INS Puławy (surface area, 3.3 m2 g-1) with 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 passed through an 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 with molecular sieves 4A and 5A and with MnO/γ-Al2O3. Temperature-programmed desorption measurements were conducted with an AMI1 TPD apparatus (Altamira Instruments). A thermal conductivity detector was used as the detector. The experimental procedure was as follows: A sample of 0.5 g of catalyst was heated to 700 °C at a 25 K/min heating rate in a stream of inert gas flowing at a rate of 30 cm3/min. At 700 °C argon was replaced with hydrogen. The reduction process lasted 3 h. The sample was then cooled in the hydrogen stream to 25 °C. Next, hydrogen was replaced with argon for 20 min. After that, the temperature-programmed desorption was carried out at a heating rate of 20 K/min. The preparatory step before calculating χc(c) involves eliminating the noise in the experimental TPD curve, which will be enhanced by the subsequent differentiations. For that purpose, the TPD spectrum is approximated by a sum of Gaussian-like functions:
c(T) ) where
∑Rici(T)
(44)
{ ( )}
(T - T0i )ri-1 T - T0i exp ci(T) ) ri Ei Eiri
ri
(45)
In eq 45, Ei is the variance of ci(T) and ri governs the shape of ci(T). This equation produces a fairly symmetrical Gaussian-like function for r ) 3, a right-hand widened one for r < 3 and a left-hand widened one for r > 3. Figure 3 shows the decomposition of the TPD diagram for hydrogen desorption from the alumina-supported nickel catalyst, done by using the linear combination (44) of five functions ci(T) defined in eq 45. The parameters found by computer fitting in the course of that decomposition are collected in Table 1. The TPD spectrum considered here is one that has already been subjected to analysis in our previous publications where we neglected readsorption, i.e., the
Figure 3. Best fit of the experimental TPD diagram (O) for the hydrogen desorption from a nickel catalyst by the linear combination (46) of the function (47). The best-fit parameters found by computer are collected in Table 1.
Figure 4. Functions χc(c) calculated by assuming that K ˜a ) 0 and K ˜ d ) 105 kJ mol-1 min-1 for the three values of cb: 33 kJ/mol (- - -), 30 kJ/mol (- - -), and 20 kJ/mol (s). Table 1. Parameters Found by Computer Fitting in the Course of the Decomposition Defined in Equations 44 and 45 of the TPD Diagram of Hydrogen Desorbed from the Silica-Supported Nickel Catalyst no.
Ri
ri
T0i (°C)
Ei (deg)
i)1 i)2 i)3 i)4 i)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
second term on the r.h.s. of eq 10. Now, we will see how introducing readsorption affects the results of our analysis using the statistical rate theory of interfacial transport. Equation 40 was solved by using the Runge-Kutta method.77 In this method, the boundary condition is the value of the solution at a chosen value of the argument. In the following calculations, the value of c ) cb at the temperature 300 K was taken as a boundary conditionparameter. Figures 4 and 5 show how the choice of Kd and cb affects the calculated function χc(c) when readsorption is neglected. The parameter values used in our calculations have been chosen in such a way that the c values would lie in a physically reasonable range. Figure 6 shows the effect of the readsorption parameter K ˜ a on the calculated CA function χc(c). Having calculated χc(c), one may arrive at a still better approximation for χ(), using the variety of methods mentioned before.69-76 (77) Stoer, J.; Bulirsch, R. In Einfu¨ hrung in die Numerische Mathematik II; PWN: Warszawa, 1980.
Interpretation of TPD Spectra from Solid Surfaces
Langmuir, Vol. 15, No. 19, 1999 6393
Figure 5. Functions χc(c) calculated by assuming that K ˜a ) 0 and cb ) 20 kJ/mol, for three values of the parameter K ˜ d: 104 kJ mol-1 min-1 (- - -), 105 kJ mol-1 min-1 (- - -), and 106 kJ mol-1 min-1 (s).
Figure 7. Fitting by the linear combination (47) of the functions (49), the “experimental” χ(c) function calculated by assuming K ˜ d ) 107 kJ mol-1 min-1, cb ) 20 kJ/mol, and K ˜ a ) 5 kJ mol-1 atm-2 min-1. The corresponding best-fit parameters are collected in Table 2. Table 2. Parameters Found by Computer Fitting of the Function χc(Ec) Calculated by Assuming Ecb ) 20 kJ mol-1, K ˜ d ) 107 kJ mol-1 min-1, and K ˜ a ) 5 kJ mol-1 atm-2 min-1 no.
γi
li (kJ mol-1)
-1 m i (kJ mol )
i)1 i)2 i)3 i)4 i)5
0.4737 0.2797 0.1937 0.0412 0.0438
46.24 54.89 67.75 81.77 103.78
48.94 57.57 71.80 88.21 112.88
The condensation function χc(c) calculated from a TPD diagram will then have the form, n
Figure 6. Functions χc(c) calculated by assuming that K ˜d ) 106 kJ mol-1 min-1 and cb ) 20 kJ/mol, for three values of the -1 -2 -1 parameter K ˜ a: 100 kJ mol atm min (- - -), 10 kJ mol-1 atm-2 min-1 (- - -), and 0 kJ mol-1 atm-1 min-1 (s).
Now we propose yet another method of calculating the true adsorption energy distribution. A first step toward such a calculation will be a “decomposition” of the calculated condensation function χc(c) into overlapping one-modal distributions, corresponding to certain kinds of adsorption sites. A decomposition of the condensation function χc(c) was reported by Cases and co-workers78,79 for a number of adsorption systems. These authors came to the conclusion that the general physical situation is as follows. If even one can distinguish, on a given surface, a number of distinct kinds of adsorption sites (or surface domains), there will still exist a distribution of surface properties (adsorption energies) within each domain. Therefore, one should consider the following representation for an actual adsorption energy distribution χ(), n
χ() )
γiχi(), ∑ i)1
n
γi ) 1 ∑ i)1
(46)
where χi() is the adsorption energy distribution for the ith kind of adsorption sites, but is not necessarily a Dirac δ function. (78) Villie´ras, F.; Cases, J. M.; Franc¸ ois, M.; Michot, L. J.; Thomas, F. Langmuir 1992, 8, 1789. (79) Villie´ras, F.; Michot, L. J.; Cases, J. M.; Franc¸ ois, M.; Rudzinski, W. Langmuir, in press.
χc(c) )
()
∫0∞ ∑ i)1 γi
∂θi ∂
n
χi() d )
γiχci() ∑ i)1
(47)
where θi() is the isotherm equation describing adsorption on the ith kind of adsorption sites. We will use “histograms” to represent the true adsorption energy distributions χi(), i.e., the following rectangular functions:
χi() )
1 for li < < m i l m i i
(48)
Then χci(c) takes the form,
χci(c) )
{[
(
)]
c - m i 1 1 + exp m l kT i - i
[
-1
-
( )] }
c - li 1 + exp kT
-1
(49)
in which T has, for a given value of c, to be the value T(c) found by solving the differential equation (40). Figure 7 shows the example of the decomposition of the condensation function χc(c), calculated by allowing readsorption. The corresponding best-fit parameters are collected in Table 2. Figure 8 shows the “true” adsorption energy distribution χ() in the form of histograms. They show the existence of five different adsorption sites on the catalyst surface. We postpone speculations concerning their physical nature because this is not the main task of the present publication. Although diffusional effects are commonly neglected, they can also be taken into account, to calculate a more
6394 Langmuir, Vol. 15, No. 19, 1999
Figure 8. True adsorption energy distribution depicted by histograms, corresponding to the condensation function χc(c) shown in Figure 6. The parameters used to draw histograms are those collected in Table 2.
Figure 9. Two histograms drawn by the solid line (s) represent the true adsorption energy distribution χ(), whereas the broken line (- - -) is the corresponding “condensation” approximation χc(c), calculated from eqs 47 and 49.
accurate function c(T).80 It is known that TPD spectra can be affected by intraparticle diffusion and lateral interactions between the adsorbed molecules. It is even suspected that the two effects mentioned above can even be the source of “extra” peaks on a TPD diagram.30,81 In the present treatment we have neglected diffusional effects, which may be responsible for some broadening and overlapping of peaks as compared to their true shape determined solely by the function χ(). Diffusional effects, however, are usually neglected in the published analyses of TPD spectra from heterogeneous surfaces. This reflects the general belief that it is the energetic surface heterogeneity that governs the shape of a TPD diagram. As we have mentioned, the choice of the parameters K ˜ d, ˜ a ) 2kTKa was dictated by the requirement that cb, and K the c values found by solving the differential equation (40) could lie in a physically reasonable range. That range is determined mainly by the choice of K ˜ d. The other two parameters have more effect on the shape of the χc(c) function. That function is obviously a result of overlapping of a number of χci functions, the variance of any one of which cannot be smaller than the variance of the derivative (∂θi/∂). If the variance of any one χci is smaller than that of (∂θi/∂) (calculated for the related temperature range), we will have difficulties in fitting the “experimental” χc(c) function by the linear combination in eq 46 of the (80) Huang, Y. J.; Schwarz, J. A. J. Catal. 1986, 99, 149. (81) Yates, I. T., Jr. In Methods of Experimental Physics; Park, R. L., Ed.; Academic Press: New York, 1985; Vol. 22.
Rudzin´ ski et al.
Figure 10. TPD c(T) spectra calculated for the two-modal (two histograms) adsorption energy distribution χ() shown in Figure 9, by assuming that K ˜ d ) 106 kJ mol-1 min-1, cb ) 20 kJ/mol, and β ) 20 K/min. The solid line (s) corresponds to the situation when no readsorption occurs, i.e., K ˜ a ) 0, whereas the broken line (- - -) corresponds to the situation when readsorption exists -1 -2 and K ˜ a ) 100 kJ mol atm min-1.
Figure 11. Effect of the heating rate β on the TPD spectrum calculated for the two-modal adsorption energy distribution χ() shown in Figure 8. The parameters used in the calculation are the following: K ˜ d ) 106 kJ mol-1 min-1, K ˜ a ) 100 kJ mol-1 atm-2 min-1, cb ) 20 kJ/mol, β ) 20 K/min (s), β ) 40 K/min (- - -), and β ) 60 K/min (- - -).
functions χci(i) defined in eq 47. As a matter of fact, we faced such a difficulty for certain sets of the parameters ˜ a. K ˜ d, cb, and K Now, we are going to demonstrate the predictive features of our new method of analysis of TPD spectra from heterogeneous solid surfaces. Figure 9 shows two histograms used to represent a solid surface with two kinds of adsorption sites. There, the “condensation” function χc(c) is shown, calculated by using the equations (47) and (49). Figure 10 shows how readsorption affects the shape of experimental TPD spectra. Readsorption causes the TPD spectrum to become more “diffuse” and shifted toward higher desorption temperatures, as to be expected. Finally, Figure 11 shows how the heating rate affects experimental TPD spectra. One can see that changing β will affect not only the values of c(T) but also the shape of TPD spectra. This raises the hope that fitting a family of TPD spectra obtained by applying varying heating rates ˜ a, and may offer a way of determining the unknown K ˜ d, K cb parameters. We postpone, meanwhile, such numerical exercises to our future publications. LA9800147
