A Quantitative Approach to Calculating the Energetic Heterogeneity of

1984

J. Phys. Chem. B 2000, 104, 1984-1997

A Quantitative Approach to Calculating the Energetic Heterogeneity of Solid Surfaces from an Analysis of TPD Peaks: Comparison of the Results Obtained Using the Absolute Rate Theory and the Statistical Rate Theory of Interfacial Transport Wladyslaw Rudzinski,*,† Tadeusz Borowiecki,‡ Tomasz Panczyk,† and Anna Dominko‡ Department of Theoretical Chemistry and Department of Chemical Technology, Faculty of Chemistry, Maria Curie-Sklodowska UniVersity, pl. Marii Curie-Sklodowskiej 3, Lublin, 20-031, Poland ReceiVed: July 29, 1999; In Final Form: December 16, 1999

We present a comparison of the quantitative information obtained about the energetic heterogeneity of silicasupported nickel catalysts by an analysis of an experimental TPD peak using the absolute rate theory, in the form of the Wigner-Polanyi equation, and the statistical rate theory of interfacial transport. A new method of evaluating the desorption energy distribution, based on an improved condensation approximation approach and applicable to thermodesorption kinetics in both the above approaches, has been developed for this purpose. To make the comparison, a literature report of a study of adsorption equilibria in the hydrogen/SiO2-Ni system is used together with a TPD study of our own sample of the catalyst. The use of the Wigner-Polanyi approach resulted in the recovery of energy distribution functions showing the solid surface to be more energetically heterogeneous than it is when one uses the SRTIT approach on the same data. The broader surface energetic heterogeneity calculated using the Wigner-Polanyi approach is especially dramatic in the systems where significant readsorption occurs. This is because readsorption affects TPD peaks in a similar way to that due to surface heterogeneity. Despite this, the Wigner-Polanyi approach is commonly applied, neglecting the readsorption term, because of fundamental problems connected with using the full ART formulation, with the readsorption term included. We find that in general, for the same set of physical parameters characterizing a given gas/solid adsorption system, the Wigner-Polanyi approach will generate theoretical TPD peaks which are narrower, and shifted toward lower temperatures, compared with matching theoretical TPD peaks generated by the SRTIT approach. Clearly, one of these approaches is misleading, and on the basis of various considerations presented below, arguments are put forward favoring the use of the SRTIT approach for the interpretation of TPD peaks.

Introduction Nine years after the principles of thermodesorption, now known as temperature-programmed desorption (TPD), were first described by Amenomiya and Cvetanovic in 1963,1 the first paper on the application of that technique to the study of the energetic heterogeneity of catalysts surfaces was published by the same two authors.2 Temperature-programmed desorption has since gained enormous popularity, leading to the publication of many papers, including several extensive reviews.3-9 However, to this day, characterization of the energetic properties of catalysts surfaces, based on an analysis of TPD peaks, is normally done in a semiqualitative way. Only a very few papers have attempted a quantitative analysis of catalysts surface heterogeneity using TPD data. Until very recently, all the attempts to elicit information about surface energetic heterogeneity from experimental TPD spectra, were based on the use of the absolute rate theory (ART) to describe the adsorption/desorption kinetics10

dθ/dt ) Kap(1 - θ)s exp(-a/kT) - Kdθs exp(-d/kT) (1) where θ is the fractional coverage of adsorption sites, s is the * Corresponding author. Phone: +48 81 5375633. Fax: +48 81 5375685. E-mail: [email protected]. † Department of Theoretical Chemistry. ‡ Department of Chemical Technology.

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

θ(e)(p,T) )

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

(2)

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

( )

d dθ ) -Kdθs exp dt kT

(3)

This truncated ART rate expression is commonly called the Wigner-Polanyi equation, though such a simplification hardly

10.1021/jp9926903 CCC: $19.00 © 2000 American Chemical Society Published on Web 02/12/2000

Energetic Heterogeneity of Solid Surfaces

J. Phys. Chem. B, Vol. 104, No. 9, 2000 1985

deserves a name. What it means is that any readsorption which might be occurring simultaneously with the desorption was neglected. It is a surprise to us to find a total lack of consideration aimed at showing how neglecting the readsorption term may affect the analysis of experimental TPD peaks from energetically heterogeneous solid surfaces. Application of the Wigner-Polanyi equation to generate information about surface energetic heterogeneity from experimental TPD peaks has usually been based on the assumption that desorption from an energetically heterogeneous surface proceeds in stepwise fashion, in the sequence of increasing activation energies for desorption d. This picture envisions a sharp “desorption front” which, at each surface coverage θ, is clearing adsorption sites characterized by a fixed activation energy for desorption d. Although such a picture is not far from the truth, it was improperly interpreted to allow the use of eq 3 at any surface coverage θ using a suitably chosen value of d. As a result, eq 3 was rewritten to the following form

-(dθt/dt) d ) ln Kd θt kT

(4)

ln

and treated as a kind of Arrhenius plot. At a fixed surface coverage θt the lhs of eq 4 was often a fairly linear function of T-1, the tangent of which was assumed to be the value d(θt), and the intercept was used to yield the parameter Kd. This method gained great popularity judging by the literature. Not surprisingly, it was observed that d varies with θt. However, it was also observed that Kd varies over several orders of magnitude. This can be seen in the review by Zhdanov.11 Seebauer et al.12 have reviewed different theoretical representations for the preexponential factor Kd. None of them was able to account for such strong variations of Kd with θt. In fact, it is not difficult to show the source of the conceptual error which leads to this surprise: it is related to the use of the Arrhenius plot of eq 4. The proper analysis of TPD peaks should be based on a more general “integral equation” for surface coverage, one which is commonly used in theories of adsorption equilibria on energetically heterogeneous solid surfaces:13-16

θ(e) t (p,T) )

∫0∞θ(e)(,p,T)χ() d

(5)

where θ(e) is the average coverage of surface sites on a t heterogeneous solid surface and χ() is the differential distribution of the number of adsorption sites among corresponding values of adsorption energy χ(), normalized to unity

∫0∞χ() d ) 1

(6)

Equation 5 can be treated as an integral equation for χ() provided that the function θt(p,T) is known from experiment and the kernel θ(,p,T) can be postulated a priori on some rational basis. Thus, when the Wigner-Polanyi equation is used to analyze the experimentally monitored rate of desorption from an energetically heterogeneous surface (dθt/dt), the following relationship should be used

dθt ) dt

( )

( )

∞ d χ(d) dd ) -∫0 Kdθ(d) exp χ(d) dd ∫0∞ dθ dt kT

as, it has already been suggested in some papers.17-25

(7)

Carter17 assumed that χ(d) is a linear combination of a number of Dirac delta functions, corresponding to the number of distinct kinds of adsorption sites existing on a solid surface. That procedure requires solving a large system of linear equations to recover χ(d). In practice, this method frequently yields a singular matrix of coefficients,18 leading to incorrect solutions.19 To get around this, Carter et al.20 proposed a procedure which can be used when the heating rate in the TPD experiment is a linear function of temperature. But in practice, TPD experiments are carried out at a constant heating rate, that is, the heating rate is a linear function of time. Recent papers concerning this problem have been published by Dondur and Fiedler,21 Karge and Dondur,22 and by Hunger et al.23,24 The problem was made more complex when Ma et al.25 pointed to certain difficulties in recovering the correct function χd(d). The situation is very similar to that faced by those investigating adsorption equilibria. Equation 5 is the first kind of a Fredholm integral equation, and the difficulties in finding its solution have been well described in many publications. (for reviews see refs 13-16). In practice, one theoretical approach leads to stable solutions. This is the condensation approximation and its elaborations.26-31 The condensation approximation (CA) has been used in theories of adsorption equilibria for decades but it was only 5 years ago that Seebauer32 first applied it to solve the integral eq 7. Seebauer also pointed out the incorrect use of the CA in previous work.33 A year later, Southwell and Seebauer34 extended their work by applying the procedure to other chemisorption systems. The CA approach has recently been applied by Rudzinski et al.35 to examine the possibility of using the Polanyi-Wigner equation for a quantitative analysis of TPD peaks from energetically heterogeneous solid surfaces. However, neglecting readsorption can distort a quantitative analysis of TPD peaks from energetically heterogeneous solid surfaces. At the same time, application of the full ART expression (1) to adsorption on heterogeneous solid surfaces raises a problem. This involves a question: how does a change with changing d when going from one to another type of adsorption site. Is there any correlation between the two? The same fundamental question is inherited in all the improvements of ART, some of which invoke the existence of “precursor states”.36-38 At the beginning of the 1980s, a new family of approaches to adsorption/desorption kinetics appeared. A common fundamental feature of these approaches is that they relate the rate of adsorption/desorption to the chemical potential of the adsorbed molecules. These ideas are developed in works by Nagai,39-43 Kreuzer and Payne,5,6,44-46 and Ward and coworkers.47-51 In his works, Ward outlines the difficulties connected with the use of the ART approach. He quotes Rublof’s work,53 for instance, which showed that the ART is not able to predict the dependence of adsorption rate on coverage. Such observations resulted in the introduction of the concept of “sticking coefficient”.54,55 But then, it was also reported that such sticking coefficients may themselves depend on the instantaneous surface coverage.54,56 Recently, Ward and co-workers have launched a new approach, called the “statistical rate theory of interfacial transport” (SRTIT), which aims to eliminate these difficulties. Another objective of that approach is to relate the kinetic behavior of adsorption/desorption to the behavior of the system at equilibrium. One important consequence of this approach is that the

1986 J. Phys. Chem. B, Vol. 104, No. 9, 2000

Rudzinski et al.

activation energies for adsorption and desorption no longer appear in the Ward’s SRTIT approach. Instead, the terms related to adsorption and desorption kinetics are functions of the energy of adsorption . This eliminates the problem of establishing a relationship between a and d on a heterogeneous solid surface while allowing one to simultaneously describe the equilibrium isotherm and both the desorption and the readsorption processes before equilibrium is reached. This has encouraged us to apply the SRTIT approach in developing a theoretical description of the kinetics of adsorption/desorption on energetically heterogeneous solid surfaces. In a series of recent papers, Rudzinski et al.57-63 have shown how the SRTIT approach may be generalized to describe both the kinetics of isothermal adsorption/desorption on energetically heterogeneous solid surfaces and the kinetics of thermodesorption from such surfaces. In one of their recent papers,63 the present authors have shown that using the SRTIT approach, instead of the ART approach, can lead to significant differences in the theoretical description of the kinetics of isothermal adsorption on energetically heterogeneous solid surfaces. No one has yet shown what differences may appear in the theoretical interpretation of nonisothermal thermodesorption from energetically heterogeneous solid surfaces, if one uses the SRTIT rather than the Wigner-Polnayi approach. Such a comparison is the purpose of the work below.

(kT )

qs ) qs0 exp

Equation 8 now takes then the form

1 θ 1-θ dθ ) Kap exp exp - Kd × dt θ kT p1-θ kT [1 - θ(e)]p(e) (13)

[

( )]

( )

where

Ka ) Kgsqs0 exp

()

0

1

1 - θ(e) )

1 + Kp exp (e)

(

)

(8)

where µg is the chemical potential of the molecules in the gas phase and µs is the chemical potential of the adsorbed molecules, K′gs is a constant related to the exchange rate between the gas phase and the solid surface once the system has reached equilibrium. To a good approximation, K′gs can be written as the following product50,51

K′gs ) Kgsp(e)(1 - θ(e))

(9)

where the equilibrium state is defined as that to which a system isolated at a surface coverage θ and gas phase concentration with partial pressure p would evolve. The first term on the rhs of eq 8 describes the rate of adsorption, while the second term on the rhs describes the rate of desorption. One important assumption inherent in the SRTIT approach is that the adsorbed phase is at “quasi-equilibrium”; i.e., all the surface correlation functions are the same as they would be at equilibrium when the surface coverage is the same. As in our previous publications, we assume ideal gas behavior in the gas phase. So we write

µg ) µg0 + kT ln p

(10)

Then for the Langmuir model of adsorption we have

θ µs ) kT ln s q (1 - θ)

where the molecular partition function qs is the following product

( )

( ) ( )

(15)

(16)

At thermodynamic equilibrium c ) -kT ln Kp(e), but at the “quasi-equilibrium” conditions, assumed by Ward and Findlay, only the following relationship holds, (see eq 11)

c ) -µs - kT ln qs0

(17)

where µs is now a function of the average surface coverage θt. This comes from the postulate that at the assumed “quasiequilibrium” all surface correlation functions are the same, as their equilibrium counterparts, at the same surface coverage θt. The function c is known to play a fundamental role in CA approach. The quasi-equilibrium θ(,c,T) defined in eq 16 describes the “local” surface coverage of sites having adsorption energy , when the total surface coverage is θt. The relationship between θ() and θt is governed by the adsorption energy distribution χ(), through the following equation

θt )

∫0∞θ(,c,T)χ() d

(18)

Thus

c ) c(θt,T)

(19)

While applying CA approach, yet another function plays a fundamental role. This is the χc(c) function

χc(c) ) (11)

kT

- c kT θ(,c,T) ) - c 1 + exp kT

Principles of the SRTIT Approach. The SRTIT approach leads to the following expression for adsorption/desorption kinetics

)

(14)

As we will use the CA approach in our interpretation of equilibria and kinetics of adsorption on a heterogeneous solid surface, we rewrite Langmuir’s equation in the following form:

exp

(

( )

Kgs µg0 µg0 , Kd ) s exp kT kT q

At equilibrium, when dθ/dt ) 0, eq 12 yields the Langmuir isotherm (2), in which K ) xKa/Kd ) qs0 exp(µg0/kT). Thus

Theory

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

(12)

∂θt ∂c

(20)

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

χc(c) )

χ() d ∫0+∞(∂θ ∂ )

(21)



and called the “condensation function”, being a certain approximation of the real adsorption energy distribution χ(). In the limit T f 0, when the derivative (∂θ/∂) becomes the Dirac delta function δ( - c), χc(c) becomes exactly χ(). Application of SRTIT to the Study of Thermodesorption from Energetically Heterogeneous Solid Surfaces. In TPD experiments at every temperature θt is found from

N0 F θt(T) ) Nm βNm

∫TTc(T) dT

(22)

0

where β is the heating rate and is commonly the “ramping” function T ) T0 + βt, F is the volumetric flow rate of the carrier gas, N0 is the amount of the preadsorbed species, and Nm is the monolayer capacity. The instantaneous gas phase concentration, c(T), of the desorbing species is given by

c(T) ) -

β dθt FNm dT

(23)

Because the thermodesorption is assumed to run at quasiequilibrium conditions, defined as the conditions at which the surface correlation functions are practically the same as those at equilibrium, dθt/dT is evaluated from eq 18.

dθt ∂θt ∂θt dc ) + dT ∂T ∂c dT

(24)

which we developed in previous publications and used to analyze experimental TPD peaks.

(27)

For the purpose of our further consideration, we remark yet that in the CA approach and its improvements, the function c is found from the condition

( ) ∂2θ ∂2

)0

)c,θ)1/2

1 dc 4kT dt

(30)

The rhs of eq 30 is then compared with the value of (dθ/ dt)θ)1/2,)c, evaluated from eq 13. We can consider two extreme cases: (1) the features of the gas/solid system are “volume dominated”, and (2) the features of the gas/solid system are “solid dominated”. In the first instance, the amount of the desorbed species in the gas phase above the surface dominates over the adsorbed portion. In that case, when the system is isolated and equilibrated, the equilibrium pressure p does not change, so that p(e) ) p. Changes in equilibrium surface coverage θ(e) are then given by eq 15 in which p(e) ) p. Thus, in cases of the volumedominated systems, from eq 13, we obtain

( ) dθ dt

)c,θ)1/2

[

) Kap2 exp

()

( )][ ( )]

c c - Kd exp kT kT

Kp exp

(28)

)c

Here we will demonstrate the applicability of the approximation represented in eq 26. Note that at a given surface coverage θt, the local coverage θ defined in eq 16 is a function of time t, because both T and θt in eq 19 change with time. Thus, from eq 16 we have

( - c)β dθ ∂θ dT ∂θ dc ) + ) θ(1 - θ) dt ∂T dt ∂c dt kT2 dc 1 θ(1 - θ) (29) kT dt Now, let us consider the value of (dθ/dt) on adsorption sites

1+

c kT

-1

(31)

By comparing eqs 30 and 31 and considering that dc/dT ) 1/β dc/dt, and p ) Pc(T) we arrive at the following expression

(25)

(26)

∂θt π2 ) - k2Tχ′(c) ∂T 6

)-

[

The second term on the rhs of eq 25 pops up as a correction to the standard CA expression

dc Fc(T) ) χc(c) βNm dT

(dθdt )

dc 4kT )[K P2c2(T)ec/kT - Kde-c/kT] 1 + dT β a

So, from eqs 20 and 24 we have

dc ∂θt Fc(T) ) χc(c) + βNm dT ∂T

whose adsorption energy ) c. On these sites θ ) 1/2, and from eq 29 we have

KPc(T) exp

( )] c kT

-1

(32)

the solution of which yields the function c(T) and hence T(c) since for physical reasons, c(T) must have a one-to-one relationship with T. Now let us consider the other extreme, the case of soliddominated systems. There the adsorbed amount is so large, compared to the amount of the desorbed species in the gas phase, that after isolating the system and equilibrating, θ remains practically unchanged, so that θ(e) ) θ. Because we are considering the situation when θ ) 1/2 and ) c, from eq 15 it follows that p(e) ) 1/K exp(-c/kT). Thus, for the case of the solid-dominated systems, the expression dc/ dT takes the following form

[

( )]

Kd 2c dc 2kT KaPc(T) exp )dT βK kT Pc(T)

(33)

As was the with eq 32, the above equation is the differential equation whose solution yields the function c(T), or T(c) for the other extreme: the solid-dominated systems. We see that the theoretical interpretation of the experimental TPD peaks will be affected to some extent by the technical conditions of the experimental setup. In the forthcoming sections we will investigate to what extent experimental conditions (i.e., the technical features of the experimental setup) may affect the properties of the experimental TPD peaks. For both the volume- and solid-dominated systems, the function (dc/dT) as a function of c is next inserted into eq 25 or 26. Finally, c(T) in eq 25 or 26 is expressed as a function of c, i.e., c(T(c)). Accepting the CA level of accuracy, eq 26 makes it possible to calculate the CA function χc(c), from an experimental TPD spectrum c(T).


χc(c) )

( )

Fc(T(c)) dc βNm dT

Rudzinski et al.

-1

θt(t,T) )

(34)

∫0∞θ(d,t,T)χ(d) dd

(36)

The function dc is found from the condition The exact energy distribution function χ() can be calculated by solving eq 21. Rudzinski and Jagiello have developed a simple method of solving this integral equation. We end this section by noting that in our recent papers64,65 yet another approximation has been applied. We assumed there that when thermodesorption is carried out slowly, θ ≈ θ(e), and p ≈ p(e). In that case the desorption term in eq 13 becomes identical with the Wigner-Polanyi expression. However, TPDtype thermodesorption is essentially a nonequilibrium process, whether it is a volume- or solid-dominated systems, so the application of the SRTIT approach presented in this paper should offer a more accurate approach. Application of the Condensation Approximation to the Analysis of TPD Peaks Using the ART Approach. Elliot and Ward51 have recently published a paper analyzing the differences which can appear in the interpretation of experimental TPD peaks depending on whether one uses the ART or the SRTIT approach to quantify the results. Their paper dealt, however, with a case of energetically homogeneous solid surfaces, represented by the adsorption system which they investigated. The present paper is aimed at reporting on such differences in the theoretical interpretation of TPD spectra, in the case of energetically heterogeneous solid surfaces. Thermodesorption studies employing the Wigner-Polanyi eq 1 neglect the readsorption term. One of the reasons for that is the problem of making sense of the relationship between “activation energy for adsorption” and the “adsorption energy”, used in the description of adsorption equilibria. In the case of energetically heterogeneous solid surfaces this concerns the kind of relationship which exists between a and d on adsorption sites characterized by various values of the adsorption energy . Does any kind of correlation exist at all? If not, one has to introduce into the theoretical description of TPD two functions: one describing the differential distribution of site activation energy for adsorption χ(a), and the second for the activation energies for desorption, χ(d). This means that if one would like to consider desorption and readsorption occurring simultaneously, a two-dimensional energy distribution χ(a,d), would have to be introduced into the theoretical formulation. Thus, in order to make the comparison of SRTIT and ART we searched the literature for an experimental adsorption system and experimental conditions where readsorption could be neglected. We found that such conditions are fulfilled in the case of hydrogen thermodesorption from a silica-supported nickel catalysts examined below. We begin by following the generally accepted strategy of neglecting the readsorption term in eq 1. Then, we introduce a function describing the “activation energy distribution for desorption”, χ(d), and assume that d should be close to . Thus, we assume that at a fixed temperature T, the average rate of desorption from the whole heterogeneous surface, dθt/ dt, is given by eq 7. In the ART the function θ(d) is found by integrating the Wigner-Polanyi equation 1. Assuming that at t ) 0, θ ) 1, we have

[

( )]

θ(d,t,T) ) exp -Kdt exp and

d kT

(35)

( ) ∂2θ ∂d2

)0

(37)

dc ) kT ln Kdt

(38)

d)dc

So

And, further

θt(dc,T) )

[ (

∫0∞exp -exp

)]

dc - d χ(d) dd kT

(39)

Therefore

dθt ∂θt ∂θt ddc ) + dT ∂T ∂dc dT

(40)

From eq 39 we deduce that

∂θt )∂dc

( )

∫0∞ ∂∂θd χ(d) dd ) -χc(dc)

(41)

where χc(d) is the condensation approximation for χ(d). Thus

ddc ∂θt dθt ) -χc(dc) + dT dT ∂T

(42)

where (∂θt/∂T) is like before a negligible correction

∂θt ) ∂T

( )

∫0∞ ∂∂θd

dc - d χ(d) dd T

(43)

and the derivative (ddc/dT) is calculated from eq 38, in which t ) (T - T0)/β. The accuracy of the expressions, obtained from the WignerPolanyi equation, by applying the condensation approximation is discussed in the Appendix. Studies of Adsorption Equilibria as a Source of Important Auxiliary Information in Fundamental Studies of Thermodesorption Kinetics. For decades, studies of adsorption equilibria were the main source of quantitatiVe information about surface energetic heterogeneity in gas/solid systems. Hundreds of papers have been published on that subject, including exhaustive reviews and monographs.13-15 It seems to us that among the reasons for employing equilibrium adsorption isotherms as the source of the desired information was the following. Theories of adsorption equilibria led to more rigorous expressions for the experimentally monitored quantities than did the existing formulations of adsorption kinetics. Consider the problem of the coverage dependence of adsorption and desorption terms. This issue is responsible for many disputes recorded in the literature. While dozens of papers have addressed this and other issues dependent on the quantitative determination of surface energetic heterogeneity from equilibrium adsorption isotherms, only a few papers have been published on employing the kinetics of thermodesorption for that purpose. However, both for experimental and theoretical reasons, TPD studies of desorption have several advantages over studies of adsorption equilibria.


J. Phys. Chem. B, Vol. 104, No. 9, 2000 1989 isotherms and the differential heat of adsorption leads to the determination of the adsorption energy distribution χ(). Such an analysis, however, requires that the isotherms and heats of adsorption be measured with high accuracy. The data reported by Prinsloo and Gravelle do not meet the required standard and as a result the determination of the exact shape of χ() is not possible using their data. However, the data are good enough to determine an approximate form of the function χ(), and some parameters essential for our theoretical analysis. We approximate that distribution by the following Gaussian-like function

( ) [ ( )]

- 0 1 R χ() ) R - 0 1 + exp R

Figure 1. Spectrum of the hydrogen desorbed from our silica-supported nickel catalyst. The temperature ramping rate β was 40 K/min (]).

The TPD is a quick and elegant technique compared to the time-consuming and cumbersome experimental studies of adsorption equilibria. Moreover, there is a fundamental theoretical reason why the amount that desorbs is much more sensitive to temperature changes, used in TPD studies, than to the changes of pressure commonly used in the isothermal studies of adsorption equilibria. Temperature appears in the exponential terms of the appropriate thermodynamic expressions, whereas pressure is essentially a multiplying factor in these expressions. Thus, for both experimental and theoretical reasons, studies of thermodesorption kinetics seem to be a much more promising way of deducing a quantitative information about surface heterogeneity of solid surfaces. However, contrary to the impressive progress that has been made on the experimental side, theoretical interpretation of TPD peaks from heterogeneous solid surfaces cries out for further fundamental study. It appears that simultaneous studies of adsorption equilibria and thermodesorption kinetics may be very helpful for establishing certain principles which could be later employed in the routine analyses of TPD data. In choosing a specific material to study we have focused our attention on silica-supported nickel catalysts. Such catalysts are frequently investigated because of their numerous applications in industrial catalytic reactions. One type of such catalysts is obtained by impregnation of SiO2 with nickel nitrate solution and was the subject of an equilibrium adsorption study by Prinsloo and Gravelle.66 The authors carried out measurements of both the equilibrium isotherms, and the related differential heats of hydrogen adsorption. Having at our disposal such a nice set of equilibrium adsorption data, we decided to prepare a similar silica-supported nickel catalyst and to carry out hydrogen TPD from that nickel catalyst. In addition, our theoretical analysis showed that in this system and at the experimental conditions applied by us, readsorption plays a negligible role. That makes the comparison between ART and SRTIT approaches possible. Combining the analysis of equilibrium adsorption data with data for thermodesorption kinetics should help to clarify some fundamental issues. At the end of this paper we present a model investigation showing how readsorption can affect the quantitative information about surface energetic heterogeneity calculated from experimental TPD peaks. The preparation of our catalyst was similar to that by Prinsloo and Gravelle and has been described in detail in one of our recent publications.62 Differences in the reagents used as well as in preparation details must have surely resulted in certain differences between the two catalysts but we believe that they are substantially similar. Figure 1 shows the results of TPD we carried out using an Altamira TPD setup. In one of their recent publications, Rudzinski et al.29 have shown how an appropriate analysis of the equilibrium adsorption

exp

(44)

2

centered at ) 0, the variance of which is related to the heterogeneity parameter R (the variance is equal to Rπ/x3). Accepting the CA level of accuracy, we have

[

θt(c) ) 1 + exp

)]

(

c - 0 R

-1

(45)

At equilibrium c is given by eq 15, so

[

Kp(e) exp

θt(p(e),T) )

[

( )] ( )]

kT/R

0 kT

0 1 + Kp exp kT (e)

kT/R

(46)

Equation 46 is the well-known Langmuir-Freundlich isotherm equation, which is commonly used to correlate experimental adsorption isotherms. The isosteric heat of adsorption Qst(θt) is given by

[

Qst(θt) ) -k

]

∂ ln p(e) ∂(1/T)

(47)

θt

So, the isosteric heat of adsorption, corresponding to χ() in eq 44 is given by

Qst ) Q0st + R ln

1 - θt θt

(48)

where

∂ ln K Q0st ) k + 0 ∂(1/T)

(49)

Figure 2 shows the coverage dependence of Qst predicted by eq 48. The features of that coverage dependence are to be compared with experimental data, reported by Prinsloo and Gravelle. It is well recognized that 0 constitutes the main contribution to Q0st. This can easily be shown when the adsorbing species is a monatomic ideal gas. Then

K ) qs0

[(

) ]

2πmkT h2

3/2

kT

-1

(50)

We consider qs0 to be the vibrational partition function of a three-dimensional isotropic Einstein oscillator. Then, except for high temperatures, we may write

qs0 ≈ e-3hν/2kT

(51)


Rudzinski et al.

Figure 2. Coverage dependence of Qst in eq 48, calculated for three values of the heterogeneity parameter R: 5 kJ/mol (---), 10 kJ/mol (- - -), 20 kJ/mol (s), and Q0st ) 81.8 kJ/mol. The black circles are the experimental points measured by Prinsloo and Gravelle, plotted versus the relative surface coverage θt, by assumning that the monolayer capacity Nm is equal to 11.5 µmol/m2. The values of the parameter Q0st and Nm used in the presentation are those found in our theoretical analysis which will be described soon in detail.

Figure 3. Results of fitting Prinsloo and Gravelle’s experimental heats of adsorption by eq 48: Nm ) 11.5 µmol/m2, Q0st ) 81.8 kJ/mol, and R ) 8.8 kJ/mol.

where ν is the oscillator frequency. Thus

∂ ln K 5 k ) /2kT - hν ∂(1/T)

(52)

The parameter hν/k is called the “characteristic temperature”. For monatomic crystals, for instance, hν/k is of about 300 K. Accepting such an estimate for hν/k, we have

k

∂ ln K 5 ) /2kT - (kT)T≈300K ∂(1/T)

(53)

Thus, for monatomic crystals and measurements carried out at about 300 K, the contribution from the term k(d ln K/dT-1) on the rhs of eq 49 would be about 3/2(300R) ≈ 3.7 kJ/mol. The studies of equilibria of hydrogen adsorption on the SiO2 supported Ni catalyst, reported by Prinsloo and Gravelle, were carried out at relatively low temperatures. Thus, in analyzing Prinsloo and Gravelle’s experimental data we have accepted the low-temperature expression given by eq 51 to represent the vibrational partition functions of the adsorbed molecules. When we come to analyze TPD data we should use the hightemperature expression for qs0

qs0 )

(hνkT)

3

(54)

which leads to a small negative value for this term, -1.2 kJ/ mol. The true value lies somewhere between these two estimates. The small size of both correction terms indicates that 0 can safely be identified with Q0st. For diatomic molecules the proof is more complicated but the final conclusion would be similar. Even when the free rotations in the gas phase degenerate into vibrations in the adsorbed state, the temperature effect should be smallsin fact, none at higher temperatures. This stems from the fact that rotational partition functions are proportional to the first power of temperature T. Therefore, the value of Q0st can be closely identified with 0 for diatomic molecules as well. From eq 48, it follows, that Q0st ≈ 0 is the value of Qst at the surface coverage at which a plot of Qst vs surface coverage θt has an inflection point. Thus, looking at Figure 2 we can see, that half coverage occurs at about 6 µmol/m2 and the Q0st value at this point is about 80 kJ/mol. Assuming that the monolayer capacity is about 11.5 µmol/m2, the value of R is found by fitting the calorimetric

Figure 4. Left-hand side of eq 55, versus the logarithm of pressure p(e) with Nm ) 11.5 µmol/m2, 0 ) 81.8 kJ/mol, and R ) 8.8 kJ/mol, giving ln K ) -29.4.

data in Figure 2 using eq 48. The result of these best-fit exercises is shown in Figure 3. Accepting the estimated values of Nm, 0, and R, we estimate the value of K using the following linear representation for the experimental adsorption isotherm.

θ(e) 0 R t ln - ) ln K + ln p(e) (e) kT 1 - θ kT

(55)

t

That linear regression is shown in Figure 4. Looking at Figures 3 and 4 we can see systematic deviations of experimental data from the predicted linear behavior. This indicates that the true form of the distribution of the energies of hydrogen adsorption on the SiO2/Ni catalyst is not well described by eq 44. However, we will let things stand as they are for the following reason. Theoretical studies of the equilibria of adsorption on energetically heterogeneous solid surfaces show14 that it is the parameter K and the variance of χ() which mainly determine the behavior of equilibrium isotherms and the heats of adsorption. The shapes of χ(), i.e., higher moments of χ(), have a smaller effect on their behavior. Therefore, the approximate shape of χ() used here should not significantly affect the estimate of the parameter K. That knowledge of K will be important in the forthcoming analysis of our TPD peak since it will decrease the number of the best-fit parameters in the SRTIT analysis, so that their number will be the same as the number of best-fit parameters in the theoretical analysis of the same TPD peaks done in terms of the Wigner-Polanyi (ART) approach. Determination of the Adsorption (Activation) Energy Distribution from Experimental TPD Peaks. In theoretical studies of TPD peaks based on the Wigner-Polanyi approach, the activation energy for desorption is frequently identified with



Figure 5. Comparison of the condensation function χc(c) (- - -), and the function χ() (s), calculated from the TPD spectrum shown in Figure 1 with the condition that the lowest values of should be close to 60 kJ/mol. The value of Kd is 1010 min-1. Because the value of K estimated from Prinsloo and Gravelle’s data is equal to 1.7 × 10-13 Pa-1, the corresponding value of Ka, calculated from the relation K ) xKa/Kd, was 2.9 × 10-16 Pa-2‚min-1.

the negative of the adsorption enthalpy. Thus, the activation energies for desorption d should be close to the values of the adsorption energies . This is equivalent to assuming that a is nearly zero. In both the Wigner-Polanyi and SRTIT approach, the most essential step is the determination of the c(T) or dc(T) function. Having determined c(T), the condensation function χc(c) is calculated from eq 34, and a fairly accurate form of χ() is found from the RJ eq 14.

χ() ) χc() -

π2 (kT)2χ′′c() 6

Figure 6. Comparison of χc(d) (- - -) and χ(d) (s), calculated from eq 57. The value of Kd is 1010 min-1.

Figure 7. A comparison of χ() (s) and χ(d) (---) functions calculated using the SRTIT and Wigner-Polanyi approaches, respectively.

(56)

For the SiO2/Ni catalyst investigated here, the lowest values of of the calculated function χ() should be close to the lowest values of Qst measured experimentally, i.e., about 60 kJ/mol. Thus, when calculating c(T) by solving eq 32 or 33, the parameters Kd and Ka, should be such that the above condition is fulfilled. Since we have already estimated the value of the parameter K ) xKa/Kd from the analysis of the Prinsloo and Gravelle’s data, we can only try various values of Kd to find the one that satisfies the above conditions. The solution of either differential eq 32 or 33 requires one more best-fit parameter: the boundary condition. One can show that this boundary condition is dictated by the requirement that the calculated condensation function χc(c) be normalized to unity, a point that has been discussed in detail in one of our recent papers.65 In the case of the Wigner-Polanyi approach the situation is even simpler. Here the function dc(T) is given by eq 38 in which t ) (T - T0)/β and χc(dc) is calculated from eq 41. Also the parameter Kd must be chosen in such a way that the lowest activation energy in the distribution χ(d) is close to 60 kJ/mol. The true distribution χ(d) can, to a good approximation, be calculated from χc(dc) using following equation (see Appendix)

χ(d) ) χc(d) - 0.577(kT)χ′c(dc) - 0.988(kT)2χ′′c(dc) (57) Our calculations show that the SRTIT approach, using either eq 32 or 33, yields practically the same function c(T) for the system hydrogen/silica-Ni catalyst investigated here. We will investigate this matter in more detail at the end of this section. Figure 5 shows the function χc() calculated from the TPD spectrum shown in Figure 1, together with the function χ() calculated from eq 56.

Figure 8. A comparison of the functions c(T) (s) and dc(T) (- - -), corresponding to the SRTIT and Wigner-Polanyi approaches, respectively.

Figure 6 shows the functions χc(d) and χ(d) calculated from eq 57, with the condition that the lowest value of d of the function χ(d) should be 60 kJ/mol. Finally, Figure 7 shows a comparison of the functions χ() and χ(d) calculated by using SRTIT and Wigner-Polanyi approach, respectively, while Figure 8 shows a comparison of the related c(T), and dc(T) functions. The function c(T) was found by solving eq 32, with the boundary condition taken in such a way that the function χc() is normalized to unity. Solving eq 33 yields almost the same function c(T). In calculating the functions χ(), and χ(d) shown in Figure 7, only one parameter, Kd, was adjusted. So, the difference between these two functions has to be ascribed to different equations used to calculate these two functions. The function χ(d) suggests that the surface of the Ni/SiO2 catalyst is more energetically heterogeneous than is suggested by the function χ(). This would suggest that the application of the WignerPolanyi equation might generally lead to higher estimates of surface energetic heterogeneity. Now we are going to check to what extent the uncertainty accompanying our estimation of the value of the parameter K


Rudzinski et al.

(

)

β2 c/kT 2 2 dc ∂θt 2 K e Nm P χc + - Kde-c/kT 2 a dc dT ∂T 4kT F (58) )dT β dc ∂θt β c/kT 1 + Ke NmP χc + F dT ∂T

(

)

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

∂θt F2 + βe(c+in)/kTNmP(FK + 8χcKakTNmP) dc ∂T + )(c+in)/kT dT 2βχce NmP(FK + 4χcKakTNmP) Figure 9. (s) Function χ() calculated by assuming K ) 1.7 × 10-13 Pa-1, estimated from the linear plot in Figure 4, i.e., it is the same χ() function as the one shown in Figure 7. (---) Function χ() calculated by assuming K ) 10-12 Pa-1. (- - -) Function calculated for K ) 10-14 Pa-1.

may affect results of our analysis. For that purpose, we have calculated two other yet functions χ(), for two K values different from the value K ) 1.7 × 10-13 Pa1- estimated from the linear plot in Figure 4. First we have taken K ) 10-12 Pa1-, thus assuming strong readsorption to exist, and next K ) 10-14 assuming, on the contrary, that very small readsorption exists. The corresponding χ() functions are shown in Figure 9. The assumption that strong readsorption exists yields χ() function which is much narrower, indicating, thus, much smaller degree of surface energetic heterogeneity of the investigated catalyst surface. Next, the assumption that very small readsorption exists yields χ() which is even closer to the function χ(d), provided that ≈ d. However, taking still lower values of K does not practically change χ() any longer, so χ() and χ(d) will never be the same. However, it seems rather impossible that the uncertainty in the estimation of the parameter K could yield K values of one or 2 orders of magnitude different than the true value of the parameter K. Predictive Features of the Wigner-Polanyi and SRTIT Approaches. The TPD techniques have been developed as a tool to extract information about physicochemical features of an adsorption (chemisorption) system from experimental TPD peaks. At present, an evaluation of the qualitatiVe characteristics of surface energetic heterogeneity only is commonly carried out. Seeking to improve such evaluations so as to obtain quantitatiVe information, model investigations are frequently carried out to see how various physicochemical features, such as surface heterogeneity, interactions between adsorbed molecules, etc., affect the shape of TPD peaks. So far, the Wigner-Polanyi approach has been employed for that purpose almost exclusively. As the main purpose of this publication is to compare the Wigner-Polanyi and SRTIT approaches, we will now present model calculations showing the differences in the shape of TPD peaks predicted by the Wigner-Polanyi and SRTIT approach. We consider a model chemisorption system in which the dispersion of the adsorption energies is described by the Gaussian-like function given in eq 44 with the same values of the 0 and R as in the case of the hydrogen/SiO2/Ni system studied in the previous section. First, we will show that the terms (∂θt/∂T) appearing on the rhs of eqs 25 and 42 can safely be neglected. Although we have already neglected this term in previous publications here, for the first time we will justify that procedure. Replacing c(T) in eq 32 with its value calculated from eq 25, we arrive at the following quadratic equation for (dc/dT)

{[

( (

F F2 + NmP 16χcFKKdkT + 64χc2KaKd(kT)2NmP + ∂θt 2βFK + 16βχcKakTNmP + ∂T ∂θt (c+in)/kT 1/2 e / β2e(c+in)/kTK2NmP ∂T (c+in)/kT {2βχce NmP(FK + 4χcKakTNmP)} (59)

)] }

)

The above equation serves for the volume-dominated systems. For the solid-dominated system, the expression for dc/dT is obtained by replacing c(T) in eq 33 by its value calculated from eq 25. After solving the resulting quadratic equation for dc/ dT, we arrive at the following expression for its physically meaningful root:

∂θt (FK + 4χcKakTNmP) dc ∂T + F NmP 8χcFKKdkT + )dT 2χc(FK + 2χcKakTNmP) ∂θt 2 1/2 16χc2KaKd(kT)2NmP + β2e2(c+in)/kTK2NmP / ∂T {2βχce(c+in)/kTNmP(FK + 2χcKakTNmP)} (60)

{[ ( ( ) )] }

The condensation function χc(c) is then calculated from the following equation

( ) [ ( )]

- c ∞1 kT χc(c,T) ) 0 kT - c 1 + exp kT

∫

exp

2

χ() d

(61)

and inserted into either eq 59 or eq 60, respectively. As we have promised, our model investigation will be carried out for the Gaussian-like function χ() given in eq 44. Thus, we will use the parameters found for the Ni/SiO2 catalyst, but we will assume that the function χ() has a symmetrical Gaussian-like shape. When solving eqs 59 and 60, the boundary condition is dictated by the requirement that the calculated function χc(c) be normalized to unity. Figure 10 shows the two functions c(T) obtained by solving the eqs 59 and 60, respectively. Figure 11 shows the Gaussian-like function (44) and the two condensation functions χc(c), obtained by inserting dc/dT calculated from eqs 59 and 60 into eq 25. Finally, Figure 12 shows the differences in the TPD peaks calculated from eq 25, by using dc/dT calculated from eq 59 and 60, respectively. The model calculations have been done twice. Once by assuming that ∂θt/∂T ) 0, and then by using the expression for ∂θt/∂T shown in eq 27. Thus, we can see that retaining the term (∂θt/∂T) introduces only small corrections to the analysis of TPD peaks. At the same time, a comparison of the solid lines in part A and B suggests,


Figure 10. Functions c(T) obtained by solving the differential eq 59, for the volume-dominated systems (- - -), and eq 60 for the soliddominated systems (---), respectively. The Gaussian-like function (44) was characterized by the values 0 ) 81.8 kJ/mol and R ) 8.8 kJ/mol, representing a rough estimate from the data of Prinsloo and Gravelle for their Ni/SiO2 catalyst. The parameters Ka, Kd, and K are the parameters found in our analysis of the TPD spectrum of hydrogen desorption from our Ni/SiO2 catalyst, and are listed in the description of Figure 5.

Figure 11. Gaussian-like function (44) χ() (s), and its two condensation functions χc(c), calculated from eq 61, by using the functions c(T) calculated from eq 59, for the volume-dominated systems (- - -), and eq 60 for the solid-dominated systems (---), respectively. Other parameters are the same as in Figure 10.


Figure 13. A comparison of TPD peaks obtained by taking into account readsorption (s) and neglecting readsorption (- - -) in the system hydrogen/Si/Ni. All other parameters are the same as in Figure 12: (A) comparison of the TPD peaks calculated by using c(T), evaluated from eq 59; (B) comparison of the TPD peaks corresponding to eq 60.

investigated here by us, we have repeated all the calculations by putting the readsorption constant Ka ) 0. Figure 13 shows the comparison of the TPD peaks calculated previously with the physically incorrect TPD peaks calculated by neglecting readsorption. We may therefore conclude that in this particular case of hydrogen thermodesorption, from SiO2/Ni catalyst, readsorption plays a negligible role. Thus, the difference between the χ() and χ(d) functions shown in Figure 7 has to be ascribed to different expressions offered by Wigner-Polanyi and SRTIT approaches. We now compare TPD peaks corresponding to the Gaussianlike adsorption energy distribution (44) calculated by using the SRTIT and Wigner-Polanyi approaches. We assume that the χ(d) function is function (44) in which ) d. As in the case of SRTIT approach, we assume that the other parameters are those found in our analysis of the TPD peaks of hydrogen desorption from the Ni/SiO2 catalyst. In the case of the WignerPolanyi approach, the function χc(dc,T) is calculated from the equation

χc(dc,T) )

Figure 12. The two TPD peaks calculated by using c(T), evaluated from eq 59 by assuming the volume-dominated behavior (A), and from eq 60, i.e., assuming the solid-dominated behavior (B). The solid lines (s) represent the TPD peaks calculated by retaining the term (∂θt/∂T), whereas the broken lines (- - -) are the TPD peaks calculated by neglecting the term (∂θt/∂T).

that for the fully symmetrical function χ(), accepted in the present model calculations, the assumption that the system is solid- or volume-dominated has a minor effect on the shape of TPD peaks. An inspection of eqs 32 and 33 shows that the difference in the behavior of the solid- and volume-dominated systems decreases as K increases. When KPc(T) exp(c/kT) . 1, eq 32 reduces to eq 33. To investigate the role of the readsorption term in the SRTIT approach, for the particular system hydrogen/Ni-SiO2 catalysts

∫0∞kT1

{ [ ( exp -exp

)]}

dc - d × kT dc - d exp χ(d) dd (62) kT

(

)

Figure 14 shows a comparison of the TPD peaks calculated by using SRTIT and Wigner-Polanyi approach, respectively. Thus, using the same values of the physical parameters, the SRTIT approach predicts TPD peaks which are significantly different from those predicted by the Wigner-Polanyi approach. How Neglecting Readsorption May Affect the Quantitative Determination of Adsorption Energy Distribution. The fundamental reasons for advocating for the use of the SRTIT approach have been discussed in the works of Ward and coworkers.47-52 Here we would like to emphasize one important advantage of SRTIT approach in the formulation of adsorption kinetics. In the case of energetically heterogeneous solid surfaces, the use of SRTIT solves the problem of taking into account readsorption phenomena, which may play a role in some TPD experiments. This can be done by adjusting the value of parameter Ka in the equations developed above. For the purpose


Figure 14. Comparison of the TPD peaks calculated for the Gaussianlike adsorption energy distribution (44), and accepting the parameters R, 0, and K, found in the analysis of the equilibrium data for the SiO2/ Ni catalyst, reported by Prinsloo and Gravelle. The solid line is the TPD spectrum calculated by using the Wigner-Polanyi approach. The same value of Kd, 1010 min-1, was used as that used to calculate χ(d) in Figure 6. The two broken lines have been calculated using the SRTIT approach, using the same Kd value 1010 min-1 as in Figure 4, and assuming the volume-dominated (- - -) and the solid-dominated behavior (---), respectively.

Figure 15. Effects of readsorption, predicted by the SRTIT approach. The solid and broken lines are theoretical TPD peaks, calculated for the Gaussian-like adsorption energy distribution (44), by accepting R ) 8.8 kJ/mol and 0 ) 81.8 kJ/mol, as found in our analysis of the equilibrium Prinsloo and Gravelle data, and using the parameter value Kd ) 1010 min-1 found in the analysis of our TPD peak. The solid line (s) is for Ka ) 0, i.e., when no readsorption occurs, the dashed line (- - -) is for Ka ) 10-13 Pa-2‚min-1, i.e., for moderate readsorption, whereas the dotted line (---) is for Ka ) 10-10 Pa-2‚min-1, indicating strong readsorption phenomena. For every Ka value, the parameter K was taken as xKa/Kd. (A) Volume-dominated systems (eq 59), and (B) solid-dominated systems (eq 60).

of illustration, in Figure 15 we show some theoretical TPD peaks calculated for hypothetical systems in which significant readsorption is taking place. Looking at Figure 15, one can see that readsorption affects TPD peaks in a similar way to surface energetic heterogeneity. To show this more clearly, we have performed the model calculation shown in Figures 16 and 17. There we show that growing surface energetic heterogeneity has a very similar effect on the TPD peaks as growing readsorption. The general conclusion which can be drawn from Figures 15-17 is that in systems where readsorption plays a significant role, ignoring readsorption will lead to an erroneous estimate of the adsorption energy distributions χ() or χ(d). The calculated distribution will show a larger dispersion of adsorption energies than really exists. And as we have emphasized, the form of the ART commonly used, i.e., the Wigner-Polanyi eq 3, neglects the readsorption term and may be subject to similar distortions.

Rudzinski et al.

Figure 16. Three adsorption energy distributions χ() which were used to generate the three TPD peaks shown in Figure 7. The solid line (s) is for 0 ) 81.8 kJ/mol and R ) 8.8 kJ/mol, i.e., for the parameters characterizing the hydrogen adsorption on our SiO2/Ni catalyst, when the surface energetic heterogeneity is approximated by the Gaussianlike function (44). The dashed line (- - -) is for a more heterogeneous surface characterized by 0 ) 100 kJ/mol and R ) 12 kJ/mol, whereas the dotted line (---) is for a still more heterogeneous surface characterized by 0 ) 120 kJ/mol and R ) 15 kJ/mol.

Figure 17. Three TPD peaks generated by SRTIT approach, corresponding to the three adsorption energy distributions χ(), shown in Figure 16. The solid line (s) is for 0 ) 81.8 kJ/mol and R ) 8.8 kJ/mol, the dashed line (- - -) is for 0 ) 100 kJ/mol and R ) 12 kJ/mol, whereas the dotted line (---) is for 0 ) 120 kJ/mol and R ) 15 kJ/mol. The other parameters used in this calculation are those characterizing hydrogen desorption from our SiO2/Ni catalyst, i.e., Ka ) 2.89 × 10-16 Pa-2‚min-1 and Kd ) 1010 min-1. (A) Volumedominated systems, and (B) solid-dominated systems.

An example of a real system in which significant readsorption exists has been presented in a recent publication.64 There we have investigated hydrogen desorption from an R-Al2O3/Ni catalyst. Here, however, we have chosen to analyze a physical system in which readsorption can be neglected, otherwise, we could not compare the fundamental features of the WignerPolanyi and SRTIT approaches. In systems where readsorption plays a negligible role, the use of Wigner-Polanyi does not yield a very different picture of surface heterogeneity as can be seen in Figure 7. Of course, one may still raise the question: which of the two functions χ() or χ(d) in Figure 7 represents a more correct picture of the actual surface energetic heterogeneity? To examine that question, we have done the calculations presented in Figure 18. Calculating Equilibrium Thermodynamic Quantities from Experimental TPD Peaks. In Figure 18 we compare the two theoretical heat of adsorption curves Qst(θt), calculated by using the two distribution functions χ() and χ(d), obtained by using the Wigner-Polanyi and SRTIT approach, respectively, and



Figure 18. Comparison of the two heat of adsorption curves Qst(θt), calculated by using SRTIT (- - -), i.e., the adsorption energy distribution χ() in Figure 7, and by using the Wigner-Polanyi approach (---), i.e., the function χ(d) also shown in Figure 7. The value of the parameter K used in this calculation was the same as that used to calculate χ() and χ(d) from our experimental TPD peak. The parameter Q0st was set equal to zero. The black circles are the experimental data reported by Prinsloo and Gravelle.

shown in Figure 7. To calculate the theoretical curves Qst(θt), we used the following equations.

( )

Qst(θt) ) -k

(∂θt/∂T)p ) -kT2 (∂θt/∂ ln p)T θt

∂ ln p ∂(1/T)

Qst(θt) ) -kT2

(63)

∂θ ∫0∞(∂T )χ() d

χ() d ∫0∞(∂ ∂θ ln p)

Kp exp(/kT)

)

[

]

d ln K + χ() d ∫0∞[1 + Kp exp(/kT)]2 kd(1/T)

Summary

Kp exp(/kT)

∫0∞[1 + Kp exp(/kT)]2χ() d )

[Q0 + ]χ() d ∫0∞(∂θ ∂ ) st

)

χ() d ∫0∞(∂θ ∂ )

Q0st

+

χ() d ∫(∂θ ∂ ) χ() d ∫(∂θ ∂ )

) f(c,T)

about surface energetic heterogeneity. It may be that lowering the initial desorption temperature T0, and improving the TPD experiment in other ways will help to decrease uncertainty in the shape of initial part of the peak, and hence of χ() in the region of lowest energies of adsorption. Finally, an important question remains about the practical use of TPD peaks for the quantitative determination of χ() when the requisite equilibrium data are not at hand. The problem stems from the required knowledge of K (or Ka), so that one can take into account the possible effects of readsorption. In one of our very recent publications,62 we have tried to use TPD peaks recorded at different heating rates to find a solution. There Kd and Ka were chosen in such a way that the difference between χ() functions calculated from TPD peaks recorded at different heating rates is minimized. However, this raises the problem of how best to define the similarity of two χ() functions. Thus, in seeking a reliable way to differentiate between the Wigner-Polanyi and SRTIT approaches, we decided to rely on auxiliary equilibrium experiments to determine K (Ka). So far we have not discussed the role of another physical factor which may affect the shape of TPD peaks. This is the possible interaction between adsorbed molecules. We did this on the assumption that in typical chemisorption systems gassolid interactions are dominant. We have therefore neglected the possible effects of admolecule-admolecule interaction in our analysis of hydrogen desorption from SiO2/Ni catalysts. Extending the SRTIT approach to account for interactions between molecules adsorbed on a heterogeneous solid surface will be considered in future publications.

(64)

When the Wigner-Polanyi approach was used, the function χ(d) was inserted insted of χ(), and d was set equal to . Figure 18 suggests that in the case of no readsorption, Wigner-Polanyi and SRTIT both lead to good agreement with the experimental heats of adsorption, though the curve Qst(θt) calculated using the SRTIT approach seems to be closer to the experimental points. However, both the theoretical Qst(θt) curves calculated from the TPD peak fail to do a good job of representing the behavior of the experimental heats of adsorption at surface coverages θt > 80%. This presents an important caution for the practical use of TPD peaks to quantify the adsorption energy disttribution. For the reasons which are related to both technical problems and phenomena such as longitudinal diffusion, one should treat with caution the part of the TPD peak corresponding to the start of desorption. It is probably not a reliable source of information

The statistical rate theory of interfacial transport (SRTIT) offers the possibility of describing the kinetics of thermodesorption by using the same thermodynamic quantities that appear in theories of adsorption equilibria. In particular, it solves the problem of taking into account the kinetics of simultaneously occurring readsorption. Depending on the technical details of a particular TPD experimental setup, various equations resulting from the SRTIT approach can be applicable. The two extreme situations considered in our paper are the volume- and the solid-dominated physical conditions. It appears, however, that this factor has only a minor effect on the properties of experimental peaks. The major effect appears when, instead of the SRTIT approach, the traditional ART approach due to Wigner-Polanyi is used to analyze and describe the behavior of experimental TPD peaks. For both SRTIT and the traditional Wigner-Polanyi approach an easy and stable method of calculating the distribution of adsorption (and therefore desorption activation) energies has been proposed, based on the condensation approximation. Application of the Wigner-Polanyi equation yields an adsorption energy distribution which suggests that the surface of the adsorption system is more energetically heterogeneous than is suggested by the energy distribution calculated by using the SRTIT approach. The overestimation of surface energetic heterogeneity using the Wigner-Polanyi equation will increase in systems where readsorption plays a significant role. This is because readsorption affects TPD peaks in a similar way to surface energetic heterogeneity, while the Wigner-Polanyi formulation neglects readsorption effects. This is made clear when we note that for the same values of adsorption parameters, the Wigner-Polanyi approach generates TPD peaks which are


Rudzinski et al.

narrower and shifted to lower temperatures when compared to TPD peaks generated by the SRTIT approach. In the systems where readsorption plays a negligible role, the Wigner-Polanyi and SRTIT methods yield a similar picture of surface energetic heterogeneity. It seems, however, that the use of SRTIT is slightly better at recovering the behavior of various thermodynamic quantities from experimental TPD peaks. Acknowledgment. The authors express their thanks to Prof. B. W. Wojciechowski from the Department of Chemical Engineering, Queen’s University, Kingston, Canada, for his assistance during the preparation of this manuscript.

The integral (39) is calculated by integrating by parts.

∫

dm dl

( )

∂θ χ(d) dd ∂d

So, the integral In takes the following form

In ) (-kT)n

In ) -(-kT)n

( )

∫

∫

( )

( )

We can see that the first term in the RJ expansion for θt is the CA expression θt. We replace the integration interval (dl, dm) by the interval (-∞, +∞), as is done in theories of equilibrium adsorption. However, the integrals in eq I.3 behave in a slightly different way than those considered in the theories of adsorption equilibria. The derivative (∂θ/∂d) can be rewritten to the following form

∂θ -1 θ ln θ ) ∂d kT

(I.4)

While evaluating the integrals, In

In )

( )

∫-∞+∞ ∂∂θd (d - dc)n dd

(I.5)

it will be convenient to change the integration variable from d to θ. So, we express d by θ

1 d ) kT ln Kdt - kT ln ln θ

[ ]

I1 ) (kT)Eu

[ ]

and

(I.12)

where Eu is Euler’s constant; Eu ≈ 0.5772. The second integral, I2, reads

(

I2 ) (kT)2 Eu2 +

)

π2 ) 1.977(kT)2 6

(I.13)

θt ) -χ(dc) - 0.577(kT)χ(dc) - 0.988(kT)2χ′(dc) (I.14) From eq I.14 we have

∂θt ) -χc(dc) ) -χ(dc) - 0.577(kT)χ′(dc) ∂dc 0.988(kT)2χ′′(dc) (I.15) Applying the Rudzinski-Jagiello idea of approximating the correction forms on the rhs of eq I.15 by χ′c(dc), χc′′(dc), we arrive at the counterpart of eq 27

χ(dc) ) χc(dc) - 0.577(kT)χ′c(dc) - 0.988(kT)2χ′′c(dc) (I.16) The appearance of the second term on the rhs of eq I.16 is due to the lack of symmetry in the function (∂θ/∂d). As to the term (∂θt/∂T) on the rhs of eq 46, it is given by

( )

∂θt ) -0.577kχ(dc) - 0.988k2Tχ(dc) ∂T

(I.17)

References and Notes (1) Amenomiya, Y.; Cvetanovic, R. J. J. Phys. Chem. 1963, 67, 144. (2) Cvetanovic, R. J.; Amenomiya, Y. Catal. ReV. Sci. Eng. 1972, 6,

(I.6) 21.

We can see that when θ f 0, then d f -∞, and when θ f 1, then d +∞. Further

1 d - dc ) -kT ln ln θ

(I.11)

Thus, retaining the two first two correction terms to the CA result χ(dc), the expression for θt takes the form

dm ∂θ ∂θ χ(d) dd ) -χ(dc)-χ(dc) ( dl ∂d ∂d d dm ∂θ dc) dd - (1/2)χ′(dc) ( - dc)2 dd + ... (I.3) dl ∂d d dm

∫0∞(lnn x)e-x dx

For n ) 1

Then

dl

(I.9)

1 ln ) x or θ ) e-x, and therefore dθ ) -e-x dx θ (I.10)

χ(d) ) χ(dc) + χ(dc)(d - dc) + (1/2)χ′(dc)(d - dc)2 + ... (I.2)

∫

∫01lnn[lnθ1] dθ

Now we switch from the integration variable θ to the new integration variable x defined as follows

(I.1)

It has been shown that the first term on the rhs of eq I.1 can be neglected compared to the second term, except for the extreme situations when θt is either very close to zero or to unity. Thus, χ(d) is expanded into its Taylor series around d ) dc, where the derivative (∂θ/∂d) reaches its maximum.

-

(I.8)

This is because the expressions for In take then a form which can be found in the published tables of integrals

Appendix

θt ) θ(d)χ|dm dl

dθ dd ) -kT θ ln θ

(I.7)

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A Quantitative Approach to Calculating the Energetic Heterogeneity of

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