6000
J. Phys. Chem. C 2007, 111, 6000-6008
Influence of Re-adsorption and Surface Heterogeneity on the Microkinetic Analysis of Temperature-Programmed Desorption Experiments Xinyu Xia, Jennifer Strunk, Sergey Litvinov, and Martin Muhler* Laboratory of Industrial Chemistry, Ruhr-UniVersity Bochum, D-44780 Bochum, Germany ReceiVed: NoVember 22, 2006; In Final Form: February 28, 2007
A profound microkinetic analysis of the influence of re-adsorption and of surface energetic heterogeneity on temperature-programmed desorption (TPD) experiments is performed on the basis of the Wigner-Polanyi equation. Exact solutions for TPD experiments under both re-adsorption-free and re-adsorption-dominated conditions from energetically homogeneous surfaces are presented. TPD experiments from energetically heterogeneous surfaces with uniform energy distribution are analyzed considering the surface diffusion of adsorbates. Convenient mathematical approaches are derived to evaluate the TPD profiles for first-order and second-order desorption. It is shown that, when the heterogeneity is strong and the coverage is not close to 0 or 1, the slope in the plot of ln(Tp/β) versus 1/(RTp) yields the differential heat of adsorption for freely occurring re-adsorption at Θ ) Θp. In the absence of re-adsorption, the differential desorption energy at Θ ) Θp can be derived from the slope. Two case studies show that the thermodynamic parameters, especially the surface energy distribution, can be obtained successfully from TPD experiments based on the derived methods.
1. Introduction Temperature-programmed desorption (TPD) is a widely used method to study the interaction between adsorbates and adsorbents.1 Many of the analyses in the literature deal with the condition that re-adsorption is absent, which is suitable for single-crystal surfaces in an ultrahigh vacuum (UHV) setup. An extensive quantitative analysis of such UHV data was established by Weinberg and co-workers.2,3 For porous catalyst samples, the situation is more complicated concerning two issues: first, re-adsorption is usually nonnegligible, no matter whether in a flow setup or in a vacuum setup;4-7 second, the adsorption sites generally have a broad energy distribution. Here, the energetic heterogeneity on the surface does not only refer to the change of desorption barriers with increasing coverage due to the lateral interaction between the adsorbates,1,8 which exists on both crystallographic planes and polycrystalline powder, but also refers to the energetic heterogeneity prior to adsorption. Such heterogeneity has been analyzed by the line-shape method,9-11 and a coverage dependent desorption energy can be derived from TPD spectra.12 Corresponding to the method to study thermodynamics on an energetically heterogeneous surface,13 a thorough method to approach the kinetics on a heterogeneous surface is based on the integral of kinetics on local sites. Rudzinski et al.14 recently studied the TPD from heterogeneous surfaces in this way. However, they applied statistical rate theory, which is not common in the kinetic study of catalytic processes. Seebauer15 also studied the extraction of the energy distribution and the pre-exponential factor of desorption from TPD spectra; however, in this study, re-adsorption was not considered, and an approximation similar to the “condensation approximation’’ (CA) was applied. In this contribution, we first consider the effect of re-adsorption on an energetically homogeneous surface, based on the * Corresponding author. Phone: +49 234 3228754. Fax: +49 234 3214115. E-mail:
[email protected]. URL: http://www.techem.rub.de.
rigorous mathematical treatment of the Wigner-Polanyi equation. Then, the TPD from a heterogeneous surface is analyzed. The effects of re-adsorption and surface diffusion are taken into account. The uniform energy distribution, from which the Temkin isotherm is derived, is applied here, because it is usually quite efficient to describe polycrystalline catalyst surfaces.13 The TPD profile may depend on both adsorption/desorption and mass transfer. The effects of the latter have widely been discussed. Gorte16 pointed out that it is difficult or sometimes impossible to derive adsorption enthalpies or desorption activation energies from TPD experiments because of the demanding mathematical analysis of the desorption process including the transport processes and re-adsorption phenomena occurring simultaneously. Some assumptions typically used to analyze the data from porous catalysts were found to be often wrong.17 Demmin and Gorte18 provided reliable criteria for negligible convective and diffusive lags, negligible gradients, and the CSTR assumption. Hinrichsen et al.19,20 studied TPD processes with both the PFR and the CSTR models, and they found that there are just minor differences in peak positions between these two models under typical experimental conditions. Kanervo et al.21 recently derived a criterion for negligible intraparticle diffusion limitations. These studies showed that the mass transfer effects can be avoided under a broad range of experimental conditions and emphasized the relative significance of the adsorption/desorption kinetics over mass transfer in the explanation of TPD spectra. This is why the kinetic results derived in the theoretical part of this contribution for a CSTR can be directly applied under most experimental conditions. In the case studies of this contribution, we additionally examined the effect of external and internal mass transfer by including the corresponding terms in the continuity equation for the sake of completeness. 2. Effect of Re-adsorption on TPD from an Energetically Homogeneous Surface For the TPD from an energetically homogeneous surface with a heating ramp β
10.1021/jp0677748 CCC: $37.00 © 2007 American Chemical Society Published on Web 04/04/2007
Re-adsorption and Surface Heterogeneity
J. Phys. Chem. C, Vol. 111, No. 16, 2007 6001
dT )β dt
the complete form of the Wigner-Polanyi equation with reaction order ν becomes
β
( )
( )
∆H ) a - d ∆SQ ) R ln
AapQ Ad
Since the adsorption entropy is a function of coverage and pressure, which is not the case for the adsorption enthalpy, the former should be referred to standard conditions.22 Re-adsorption in a TPD setup depends on the adsorptive partial pressure p. Using a CSTR model has been shown to be a good approximation due to the very low partial pressure of the desorbing species.19,20 Therefore, p is proportional to the total amount of active sites on the sample Nm and the reaction rate, as derived from the ideal gas law, provided that the inlet partial pressure of the adsorptive gas is 0:
p)-
NmRTaβ dθ νV˙ dT
(5)
where Ta is the ambient temperature, V˙ is the volume flow rate of carrier gas in a flow setup or the flow conductance of adsorptive gas in a vacuum setup; the latter is given by23
d3 φ) 6l
x
2πRTa M
β
( )
d dθ ) -Ad exp θ dT RT
(7)
Considering Ad as a function of T according to the transition state theory24
Ad )
kBT h
(8)
then the solution of eq 7 is
{
θ0 exp -
kBT2 2hβ
[(
1-
) ( ) ( ) ( )]}
d d d 2 d exp + E1 RT RT RT RT
(9) where E1 is the exponential integral function:
(10)
( ) kBTp2 βh
(11)
where W is the Lambert W function: y ) W(x) if x ) y exp(y). This solution means that Tp is determined by d and β, independent of initial coverage. If the change of Ad with T is neglected, the analytical solution of eq 7 is
θ ) θ0 exp
{ [ ( )
( )]}
AdT d d d E exp β RT RT 1 RT
(12)
The error is quite small, when Ad is taken as kBTp/h. At d2θ/dT2 ) 0,
d ) RTpW
( ) AdTp β
(13)
its approximation is the Redhead formula.25 When kaNmRTa/V˙ . 1, re-adsorption occurs freely, which was considered in several studies.4,6,7 Under this condition, eq 2 becomes
β
(
)
pQV˙ ∆H ∆SQ dθ θ exp )dT 1 - θ NmRTa RT R
(14)
Equation 14 is equivalent to eq 17 in ref 4. If the change of ∆SQ with T is neglected, then its solution is
[
{
θ ) -W -θ0 exp -
(
)
pQT ∆H ∆SQ + exp NmV˙ β RT R
(
) ]}
(15)
)]
(16)
pQ∆H ∆SQ ∆H - θ0 E1 NmV˙ βR R RT
(6)
d and l refer to the tubing from the reactor to the pump. 2.1. First-Order Adsorption/Desorption. For first-order adsorption/desorption on an energetically homogeneous surface, the solution of eq 2 coupled to eq 5 depends on kaNmRTa/V˙ . Here, ka ) Aa exp(- a/RT) is the adsorption rate constant. When kaNmRTa/V˙ , 1, eq 2 becomes
exp(-xy) dx x
d ) RTpW
(3) (4)
∫1∞
At T ) Tp, d2θ/dT2 ) 0,
a d ν dθ ) Aa exp p(1 - θ)ν - Ad exp θ (2) dT RT RT
where θ is the fractional coverage on energetically homogeneous surfaces; Aa and Ad are the pre-exponential factors of adsorption and desorption, respectively; and a and d are the energy barriers of adsorption and desorption, respectively. They are related to the adsorption enthalpy (∆H) and standard adsorption entropy (∆SQ) by
θ)
E1(y) )
(1)
At d2θ/dT2 ) 0
[
(
NmTa∆H Tp2 ∆SQ ∆H + ln )+ ln β RTp R pQV˙ or
∆H ) -RTpW
[
(
)]
V˙ pQTp ∆SQ exp NmβRTa R
(17)
Its form is quite similar to eq 13. For measurements with porous catalysts in a flow setup, the following ranges of experimental parameters are typical: the value of ∆SQ is limited by the standard entropy of adsorptive gas and two-dimensional “adsorbate gas’’,26 and therefore -200 J mol-1 K-1 < ∆SQ < -80 J mol-1 K-1; Tp is usually between 200 and 1000 K, as reported in most published spectra; and 0.1 K/min < β < 20 K/min, 10 NmL/min < V˙ < 100 NmL/min are common conditions for a flow TPD setup, and 1 µmol < Nm < 2000 µmol is reasonable concerning typical amounts of sample and adsorption site densities. Within this range, the value of ln(-∆H/RTp) is between 2.5 and 3.7; thus, it can be written in a Redhead-like approximation:
6002 J. Phys. Chem. C, Vol. 111, No. 16, 2007
-
Xia et al.
V˙ pQTp ∆H ∆SQ ≈ ln -3 RTp NmβRTa R
(18)
Estimates for the value of ∆SQ can be derived from statistical thermodynamics. When kaNmRTa/V˙ is neither , 1 nor . 1, none of the items in eq 2 can be omitted; this equation can only be solved numerically unless a ) 0. Neither d nor ∆H can be obtained analytically. Such conditions should be avoided in measurements by adjusting the sample mass and flow rate. Figure 1 presents the effect of re-adsorption on the first-order TPD from an energetically homogeneous surface. This figure indicates that, in case kaNmRTa/V˙ < 0.1, the variations of Nm and V˙ do not influence the TPD curve, and re-adsorption can be neglected; when kaNmRTa/V˙ > 10, the free re-adsorption model is quite accurate. 2.2. Second-Order Adsorption/Desorption. When ν ) 2, with constant Ad and omitting re-adsorption, eq 2 becomes
β
( )
d 2 dθ θ ) - Ad exp dT RT
(19)
Its solution is
θ)
β/T d d d β Ad exp - Ad E 1 + RT RT RT Tθ0
( )
( )
(20)
At d2θ/dT2 ) 0
(
d ) RTpW
)
2AdTpθp β
(21)
Because of the symmetric peak shape of second-order TPD, the coverage at the desorption peak (θp) is equal to θ0/2, so
d ) RTpW
(
)
AdTpθ0 β
(22)
Equation 22 implies that the desorption peak position depends on the initial coverage under these conditions. With freely occurring re-adsorption, eq 2 becomes
β
( )
(
)
θ 2 2V˙ pQ ∆H ∆SQ dθ exp )dT 1 - θ NmRTa RT R
[
(
)
]
θ0 2pQV˙ Tp ∆SQ exp NmRTa β R (1 - θ /2)3 0
(24)
which can also be written in Redhead-like form. For a group of measurements with various β, if the change of ∆SQ with T is omitted, the linear form of eq 24 is applicable:
ln
[
]
(1 - θ0/2) NmTa(-∆H) Tp ∆SQ ∆H + ln )+ ln + β RTp R θ0 2pQV˙ (25) 2
In the plot of ln Tp2/β versus 1/RTp, both ∆H and ∆SQ can be derived; for the latter, Nm and V˙ should be known; and ∆SQ obtained in this way is close to ∆SQ(Tp). 3. Effect of Surface Heterogeneity 3.1. Energy Distribution Model. For an energetically heterogeneous surface, eq 2, which contains two activation energies, should be integrated over energy. If both of their distributions are considered, the two-dimensional energy distribution will cause serious difficulties in the mathematical treatment. However, when adsorption is strongly activated, readsorption would be negligible, and it is not necessary to consider the distribution of a. On the other hand, when adsorption is weakly activated, the transition state of adsorption on different sites is presumably rather identical. Thus, it can be assumed that a is constant. The small change of a with coverage can be compensated by the change of d. With constant a, the distributions of d and -∆H are parallel. [An intermediate a value changing with adsorption sites renders the mathematical models too flexible to evaluate TPD spectra uniquely and is not considered in this contribution.] The overall coverage on a heterogeneous surface with uniform energy distribution is given by
Θ(T) )
(23)
There is no analytical solution for this equation. At d2θ/dT2 ) 0 with 2θp ≈ θ0
∆H ) -RTpW
Figure 1. Accuracy of free re-adsorption and nonre-adsorption approximations in evaluating simulated TPD spectra from energetically homogeneous surfaces, with different values of kaNmRTa/V˙ (numbers in the plot). Solid lines: exact solution (from eq 2); dashed-dotted lines: approximation of free re-adsorption (from eq 14); dashed line: approximation of nonre-adsorption (from eq 7). Parameters: d ) 70 kJ/mol; a ) 20 kJ/mol; ∆SQ ) -150 J-1 mol-1 K; Ad ) 1014 s-1; β ) 0.1 K/s.
3
1 δ
∫
d,max
d,max-δ
θ(T,d) dd
(26)
θ(T,) d
(27)
or
Θ(T) )
1 δ
∫
max
max-δ
Assuming that the composition of the adsorption sites does not change, then d is independent of T, so
dΘ 1 ) dT δ
∫
d,max
d,maxδ
∂θ(T,d) dd ∂T
(28)
This integral-differential equation is fundamental for kinetics on heterogeneous surfaces with uniform energy distribution. 3.2. Distribution of Adsorbates on Energetically Different Sites. During a TPD experiment, the distribution of adsorbates on energetically different sites depends not only on the
Re-adsorption and Surface Heterogeneity
J. Phys. Chem. C, Vol. 111, No. 16, 2007 6003
adsorption/desorption but also on the surface diffusion between different kinds of sites. The latter is further dependent on the geometric distribution of the sites, which can be highly irregular for polycrystalline samples. The conditions without surface diffusion between different kinds of sites is discussed in the following subsections. For the condition with surface diffusion between these sites, quasiequilibrium of adsorbate distribution on different surface sites is assumed, since the diffusion barrier is considerably smaller than the desorption barrier. The kinetic equation under this condition has been derived recently,31 for first-order TPD
β
dΘ ) kap(1 - Θ) - AdKdiff(1 - Θ) dT
(29)
For second-order TPD
β
dΘ ) kap(1 - Θ)2 - AdKdiffΘ(1 - Θ) dT
(30)
θ0(d, Θ0) ≈
Kdiff )
(
)
(
)
d,max - Θδ d,max - exp RT RT δ 1 - exp -(1 - Θ) RT
[
]
(31)
When δ . RT, that is, when the surface is strongly heterogeneous, and Θ is not close to 1, eq 31 becomes
(
)
(
d,max - Θδ d,max Kdiff ≈ exp - exp RT RT
)
(32)
For TPD measurements with heterogeneous surfaces, the precondition of eq 32 is commonly satisfied. Furthermore, if Θ is not close to 0, eq 32 becomes
(
Kdiff ≈ exp -
)
d,max - Θδ RT
( )
d θ exp d d,max-δ RT δ d
∫
d,max
)
β
dΘ ) -AdKdiff(1 - Θ) dT
(36)
where Kdiff(Θ) is expressed exactly by eq 31. If δ . RT and Θ is not close to 0 or 1, at d2Θ/dT2 ) 0, the following result is derived:
ln
diff Tp diff d d + ln ≈ β RTp Adδ(1 - Θp)
(37)
Therefore, in the plot of 1/RTp versus ln Tp/β, the slope is diff d corresponding to Θp. A simulation with the same parameters as used in Figure 2 is shown in Figure 3. It indicates that, when surface diffusion is taken into account, the TPD peaks shift to lower T. For second-order TPD at surface diffusion equilibrium, when Θ is not close to 0 and 1 and δ . RT, in a similar way, it can be derived that at d2Θ/dT2 ) 0
(33)
Equation 33 is equivalent to the CA, but it is invalid when the coverage is small for a uniform energy distribution31 and is not suitable to evaluate TPD spectra. 3.3. TPD from Heterogeneous Surfaces without Readsorption. In the absence of re-adsorption, if surface diffusion is also absent, the desorption rate of the total surface can be obtained by integrating eq 7 over d, considering first-order desorption, and further omitting the change of Ad with T and :
dΘ ) -Ad β dT
(
(35)
Figure 2 shows a simulation result of TPD spectra: when the initial coverage is high enough, the desorption peak broadens and its apex disappears. This indicates the superposition of desorption from sites in the heterogeneous energy distribution model. Unlike the first-order TPD from an energetically homogeneous surface, Tp increases with decreasing Θ0 (initial fractional coverage of TPD from heterogeneous surface) here, which can be explained by the change of d with Θ. At small Θ0, the increase of Tp with β is still clearly visible (Figure 2, right plot). When the surface diffusion is in quasi-equilibrium, for firstorder TPD, from eqs 29 and 1 and omitting re-adsorption, we have
where Kdiff is a coefficient reflecting the surface diffusion equilibrium:
exp -
1 d,max - d - Θ0δ 1 + exp RT
diff Tp diff d d + ln ln ) β RTp AdδΘp(1 - Θp)
Thus, diff d can also be obtained from the 1/RTp versus ln Tp/β plot. 3.4. TPD from Heterogeneous Surfaces with Re-adsorption. In this case, p(Θ) is expressed as
p)(34)
θ can be obtained from eq 9 or eq 12; thus, eq 34 changes to a pure integral problem, which can be solved numerically. Here, the initial distribution θ0 is a needed parameter. Grant and Carter28 assumed a “uniform distribution’’ of occupied sites after pre-adsorption. However, this distribution cannot represent the real case, because the coverage on sites with high d should be substantially higher than on that with lower d. According to our previous work,13 on a strongly heterogeneous surface with δ . RT with a medium Θ0 (not close to 0 and 1), in equilibrium with the adsorptive gas phase, the distribution of adsorbates on the energetically heterogeneous sites takes the Fermi-Dirac form as a function of both and qdiff; consequently the θ0 here can be expressed as
(38)
NmRTa dΘ νV˙ dt
(39)
If the surface diffusion is absent, in no way has it been found to simplify the integral-differential problem of eq 27. For real processes, this quasi-equilibrium might easily be approached, according to our observation on isothermal adsorption with changing pressure.31 Under this condition, from eqs 29 and 39,
Ad NmRTa dΘ dΘ ) - ka (1 - Θ) - Kdiff (1 - Θ) (40) dT V˙ dT β When kaNmRTa/V˙ . 1 (free re-adsorption conditions)
(
)
pQV˙ ∆SQ dΘ )exp Kdiff dT NmRTa β R
(41)
6004 J. Phys. Chem. C, Vol. 111, No. 16, 2007
Xia et al.
Figure 2. Simulation of first-order TPD from a heterogeneous surface in nonre-adsorption, nonsurface-diffusion mode (eq 34). Upper plot: with different Θ0 (numbers in the plot) and β ) 0.1 K/s; lower plot: with different β (numbers in the plot), solid lines: Θ0 ) 0.30; dashed lines: Θ0 ) 0.15. Parameters: Ad ) 1013 s-1, d,max ) 70 kJ/mol, δ ) 70 kJ/mol, pre-adsorption at 100 K.
Figure 4. Simulation of first-order TPD from a heterogeneous surface in the re-adsorption mode (eq 40). Upper plot: with different Θ0 (numbers in the plot). Lower plot: with different heating rates (numbers in the plot). Solid lines, Θ0 ) 0.25; dashed lines, Θ0 ) 0.15. Parameters: δ ) 70 kJ/mol; max ) 70 kJ/mol; β ) 0.1 K/s; ∆SQ ) -120 J-1 mol-1 K; Nm ) 50 µmol; and V˙ ) 100 NmL/min.
coverage at the desorption peak maximum, which is quite close to the value estimated from max - δΘ0/2 (64.8 kJ/mol); ∆SQ is derived as -135 J mol-1 K-1, if ln qdiff p /δ is omitted. Figure 5 presents the effect of heterogeneity on TPD spectra: when heterogeneity increases, the peaks broaden and shift to lower T. Second-order TPD under these conditions can be treated in a similar way; when δ . RT and Θ is not close to 0 or 1, at d2Θ/dT2 ) 0, the following linear form is derived:
ln Figure 3. Simulation of first-order TPD from a heterogeneous surface in the nonre-adsorption and the free surface diffusion mode (eq 36), with different Θ0 (numbers in the plot). Parameters: Ad ) 1013 s-1, d,max ) 70 kJ/mol, δ ) 70 kJ/mol, β ) 0.1 K/s.
At
d2Θ/dT2
ln
)0
(
)
qdiff NmRTa Tp qdiff p ∆SQ p ) + ln Q + ln + β RTp R p V˙ δ
(42)
where qdiff p ) max - δΘp is the differential heat of adsorption at the desorption peak. In the plot of ln Tp/β versus 1/RTp, the diff slope is q for Θ ) Θp; the intercept is determined by ∆SQ, δ, and the experimental conditions. In Figure 4, the TPD spectra are simulated according to these conditions. The peaks are more symmetric compared with first-order TPD from a homogeneous surface or in nonre-adsorption mode; the peak width is larger, but the apex is more clearly visible compared with the corresponding one in the nonre-adsorption mode. The efficiency of eq 42 is tested with this simulation: by fitting to the Tp and β values for Θ0 ) 0.15, we obtain qdiff ) 64.3 kJ/mol for the
[
]
NmRTa qdiff Tp qdiff p ∆SQ p (1 - Θp) ) + ln + Q β RTp R 2p V˙ Θ δ
(43)
4. Case Studies The results of two TPD measurements on different Cu catalysts are discussed in this section. The essential features of the experimental setup were described elsewhere.29 4.1. TPD of CO from a Cu/ZnO/Al2O3 Catalyst. In our previous work, the coverage-dependent adsorption energies of CO on Cu catalysts obtained from microcalorimetry and from TPD were well-correlated, where the re-adsorption in TPD was also taken into account.30 The adsorption energy distribution for a Cu/ZnO/Al2O3 sample (labeled as CZA2 described in our recent publications13,31) was obtained on the basis of a calorimetric measurement.13 In the following, it is evaluated on the basis of the TPD spectra (Figure 6). The experimental conditions are listed in Table 1. The Tp values in Figure 6 are higher than that obtained in TPD from Cu single crystals in vacuum setups (190-230 K)32 because of the re-adsorption of the CO molecules within the porous catalyst particles of the fixed bed. Since β is constant in these experiments, max, δ, and ∆SQ cannot be obtained
Re-adsorption and Surface Heterogeneity
J. Phys. Chem. C, Vol. 111, No. 16, 2007 6005 TABLE 1: Experimental Parameters of TPD of CO from a Cu/ZnO/Al2O3 Sample parameter
value
sample mass specific surface Cu area (by N2O RFC) density of Cu adsorption sites density of catalyst size of the catalyst particles mean pore radius (BJH method) internal-particle porosity external-particle porosity inner diameter of reactor flow rate
0.1 g 5.5 m2/g 24.4 µmol/m2 5200 kg/m3 250-355 µm 20 nm 0.5 0.4 3.8 mm 10 NmL/min
TABLE 2: Thermodynamic Parameters of CO Adsorption on a Cu/ZnO/Al2O3 Sample Obtained from Flow TPD and Microcalorimetry max (kJ/mol) δ (kJ/mol) ∆SQ (J mol-1 K-1) a
Figure 5. Effect of heterogeneity on the shape of TPD peaks. Numbers in plots, δ in kJ/mol. Upper plot: first-order, calculated using eq 29, Θ0 ) 0.3, max ) 70 kJ/mol; β ) 0.1 K/s; ∆SQ ) -120 J-1mol-1, Nm ) 200 µmol, and V˙ ) 100 NmL/min. Lower plot: second-order, calculated using eq 30, Θ0 ) 0.4, max ) 60 kJ/mol; β ) 0.1 K/s; ∆SQ ) -180 J-1 mol-1 K, Nm ) 50 µmol, and V˙ ) 100 NmL/min.
Figure 6. Experimental and theoretical values of CO TPD from a Cu/ ZnO/Al2O3 catalyst. Solid lines, experimental data; dashed lines, theoretical results of eq 41; dotted-dashed lines, theoretical results of eq 45; parameters see Tables 1 and 2.
independently. So ∆SQ ) -116 J mol-1 K-1 is applied on the basis of our previous work.13 The best fit was obtained in the following way: the target function Q was minimized using 50 points for each trace evenly distributed from 200 to 400 K:
TPD
microcalorimetry
63.5a 75.0a -116
65.9 76.1 -116
Fitted value.
difference between the results from eqs 40 and 41. Our previous studies showed that ka > 10-5 Pa-1 s-1 is indeed fulfilled;27,31 that is, the measurement was performed under free re-adsorption conditions. The values of max and δ are quite close to the microcalorimetrically derived results, indicating not only the reliability of the two types of measurements but also the efficiency of the surface heterogeneity model in the description of the broad TPD spectra. The reason for negative min has been explained in our previous work.13 The adsorption energy in this example agrees with the value of d derived from the Tp in vacuum setups (47-59 kJ/mol calculated from the Redhead equation25) and can help to explain Tp ) 345 K of CO from a Cu catalyst reported by Hadden et al.,33 who derived d ) 95 kJ/mol with the Redhead equation, which was questioned by Vollmer et al.32 Considering Nm ) 249 µmol and V˙ ) 25 NmL/min,33 ka ) 10-5 to 2 × 10-4 Pa-1 s-1 for CO adsorption on Cu catalysts at 303 K.31 For this peak, NmRTaka/V˙ > 10, so it should be calculated in the free re-adsorption mode, which yields qdiff p ) 56 kJ/mol according to eq 42 (with β ) 5 K/min33 and ∆SQ ) -120 J/mol/K for CO adsorption on Cu catalysts. In order to check the effect of mass transfer in the experiment, simulations coupling desorption/re-adsorption, external mass transfer and internal mass transfer were carried out with the above kinetics. The volume of the catalyst bed in the TPD reactor was separated into an inter-particle domain and an intraparticle one. The continuity equations are
∂pCO ) - D∇2pCO - u∇pCO ∂t
for inter-particle domain
(44)
∂pCO NmRT ∂Θ ) - D∇2pCO ∂t V ∂t for intra-particle domain (45)
where y is the effluent mole fraction, m ) 4 is the number of traces, n ) 50 is the number of data points, the subscript “exp’’ refers to the experimental data, “cal’’ refers to the theoretically calculated value, i refers to the ith point, and j refers to the jth cal is calculated with eqs 39 and 41. Thus, max and δ trace. yi,j are fitted, as listed in Table 2. The exact calculation with eq 40 shows that when ka > 10-5 Pa-1 s-1, there is no obvious
u is the linear velocity of gas flow along the axial direction, determined by the flow rate and external-particle porosity; D is calculated from Chapman-Enskog formula34 for the externalparticle diffusion or according to Knudsen diffusion for the internal-particle diffusion: its order of magnitude is 10-5 for the former and 10-6 for the latter. Equation 41 is applied to calculate ∂Θ/∂t.
m
Q)
n
∑ ∑ j)1 i)1
cal (yi,j
-
exp 2 yi,j )
6006 J. Phys. Chem. C, Vol. 111, No. 16, 2007
Xia et al.
TABLE 3: Kinetic Parameters for H2 + 2* ) 2H* on a Cu/Al2O3 Sample dissociative adsorption energy barrier (kJ/mol)
a: 23.6a
pre-exponential factor
Aa: 1.6 × 10-2 Pa-1 s-1
a
associative desorption d,min: 56.0 d,max: 72.8a Ad: 8.8 × 109 s-1 a
Fitted value.
Equation 45 is solved by means of the finite element method with the same kinetic parameters listed in Table 2 and the experimental parameters in Table 1. The simulated curves are also presented in Figure 6, proving that there is no obvious effect by mass transfer under these experimental conditions. 4.2. TPD of H2 from a Cu/Al2O3 Catalyst. TPD of H2 from Cu catalysts had been intensively studied by Muhler and coworkers.29,35,36 Temperature-programmed adsorption (TPA) of H2 on and TPD of H2 from several Cu catalysts was investigated in previous studies carried out in our laboratory, where the kinetic parameters corresponding to the desorption peak positions were obtained on the basis of the Langmuir model. The adsorption of H2 on these Cu catalysts was found to be strongly activated, and the re-adsorption during TPD was found to be negligible.35,37 Following these studies, the surface heterogeneity is derived from the width of the TPD peaks in this contribution.
Figure 7 shows the H2 TPA and TPD profiles of a Cu/Al2O3 sample;35,37 the profiles were evaluated together with eq 30. In order to reduce the flexibility of the fitting, diff is set to be d 58.0 kJ/mol, which is obtained from eq 38, and Nm ) 45.0 µmol H is applied for the 0.2 g sample.37 The relationship between Aa and Ad is restricted by eq 4, with ∆SQ ) -127 J mol-1 K-1. This value is obtained from the entropy of bridgeadsorbed hydrogen atoms on Cu, which is equal to 1.7 J mol-1 K-1 derived from the vibration frequencies of the adsorbates (1040 cm-1 for symmetric stretching, 1150 cm-1 for wagging,38 and 1226 cm-1 for asymmetric stretching39). Correspondingly, only Ad, d,max, and a are needed to be fitted, and therefore the fitting is rigorous. The kinetic parameters for the best fits are listed in Table 3. The theoretical results calculated from these parameters are shown in Figure 7, with a good agreement with the TPD spectra, indicating the efficiency of the models derived in this work. The error of peak area in the TPA spectra may originate from a shift of the baseline of the mass spectrometer during the measurement. A much lower a is obtained compared with H2 adsorption on Cu crystals (higher than 40 kJ/mol40-44). This is considered reasonable for the supported polycrystalline Cu particles exposing rough defect-rich Cu surfaces. Simulation
TABLE 4: Approach to Θ - T Function in TPD under Different Conditions surface condition
a
no re-adsorption
free re-adsorption
homogeneous
analytical solution
heterogeneous,b no surface diffusion heterogeneous,b surface diffusion equilibrium
integral
first-order: analytical solution second-order: ODEa differential-integral equation
ODE
ODE
Ordinary differential equation. b Using uniform energy distribution.
TABLE 5: Comparison Between the Linearized Relations Derived from TPD Spectra Under Different Conditions condition
y-axisa
type,b
Langmuir no re-adsorption
Langmuir type, free re-adsorption
ln
ln
Tp2 β
Tp2 β
slope d
-∆H
intercept
ln
1 + ln d (first-order) RAd
ln
1 + ln d (second-order) RAdθp
N m Ta ∆SQ + ln Q + ln (-∆H) (first-order) R p V˙ NmTa(1 - θ0/2) ∆SQ + ln(-∆H) (second-order) + ln R 2pQV˙ θ 3
0
type,c
Temkin no re-adsorption
ln
Tp β
diff d
diff
ln
d 1 (first-order) + ln δ Ad(1 - Θp)
ln
d 1 (second-order) + ln δ AdΘp(1 - Θp)
diff
Temkin type, free re-adsorption
ln
Tp β
qdiff p
diff
qp NmRTa ∆SQ + ln Q + ln (first-order) R δ p V˙ NmRTa(1 - Θp) qdiff p ∆SQ + ln + ln (second-order) R δ 2pQV˙ Θ p
a The x axis is always 1/RT . b Refers to an energetically homogeneous surface. c Refers to an energetically heterogeneous surface with uniform p distribution of adsorption energies, δ . RT, Θ not close to 0 and 1.
Re-adsorption and Surface Heterogeneity
J. Phys. Chem. C, Vol. 111, No. 16, 2007 6007 Ad but also of surface heterogeneity and experimental conditions, as summarized in Table 5. Taking the effect of surface heterogeneity into account, TPD experiments can be applied more efficiently beyond the well-known Langmuir model to analyze the energy distribution of polycrystalline irregular catalyst surfaces. Nomenclature CSTR ODE PFR E1 W Aa Ad d D h ∆H ka kB Kdiff
Figure 7. H2 TPA (upper plot) and TPD spectra (lower plot) for a Cu/Al2O3 sample and theoretical evaluations. Experimental conditions: sample mass 0.2 g, Cu surface area 18.4 m2/g, flow rate 20 NmL/ min for TPA and 100 NmL/min for TPD. Solid lines, experimental data;37 dashed lines, theoretical results of eq 30 using the parameters given in Table 3.
shows that the re-adsorption during the TPD measurements with a ) 23.6 kJ/mol and Aa ) 1.6 × 10-2 s-1 is negligible. In general, it should be kept in mind that the precise position of the desorption maximum of a TPD peak may be obscured by experimental noise, which may lead to errors in the adsorption energy and in Ad or ∆SQ. Therefore, the fitting of several TPD peaks is more reliable than the derivation from the linearized plots. In addition, it is recommended to estimate the distribution of a and the extent of re-adsorption properly based on the chemical nature of the adsorbates/adsorbents interactions and on additional adsorption experiments such as static microcalorimetry in order to arrive at the correct model and to obtain a unique fitting of the experimental TPD data. 5. Summary and Conclusions Different conditions concerning re-adsorption and surface heterogeneity for TPD experiments were thoroughly analyzed in this work, as summarized in Table 4. Exact solutions for the TPD from energetically homogeneous surfaces were derived for the nonre-adsorption and free re-adsorption condition, for first-order desorption and for second-order desorption. These solutions are a convenient basis for the simulation and microkinetic analysis of TPD profiles. TPD from heterogeneous surfaces was analyzed with the uniform energy distribution model. Surface diffusion was found to play an important role in the adsorption kinetics on heterogeneous surfaces, and strong heterogeneity broadens the desorption peak and changes its shape. It is shown that, in the plot of ln Tp/β versus 1/RTp, the slope is approximately qdiff (freely occurring re-adsorption) or diff d (absence of re-adsorption) corresponding to the coverage at the desorption peak maximum, while the intercept is a function not only of ∆SQ or
l M Nm p pQ qdiff R ∆SQ t T Ta Tp u V V˙ β a d diff d δ max d,max ν φ Θ θ Θ0 θ0 Θp θp
continuous flow stirred tank reactor ordinary differential equation plug-flow reactor exponential integral function Lambert W function pre-exponential factor for adsorption, Pa-1s-1 pre-exponential factor for desorption, s-1 diameter, m diffusion coefficient, m2/s Planck’s constant adsorption enthalpy, kJ/mol adsorption rate constant, Pa-1 s-1 Boltzmann constant coefficient reflecting the surface diffusion equilibrium, dimensionless length, m molar mass of the adsorptive, kg/mol amount of adsorption sites on the surface, mol partial pressure of adsorptive gas, Pa standard pressure differential heat of adsorption, kJ/mol ideal gas constant standard adsorption entropy, J mol-1 K-1 time, s temperature, K ambient temperature temperature at peak maximum of desorption, K velocity, m/s volume, m3 volume flow rate, m3/s heating rate, K/s -∆H, kJ/mol energy barrier for adsorption, kJ/mol energy barrier for desorption, kJ/mol differential desorption barrier on heterogeneous surface corresponding to Θp width of uniform energy distribution, kJ/mol maximum in uniform energy distribution, kJ/mol maximum d in uniform energy distribution, kJ/mol order of the reaction, dimensionless flow conductance, m3/s mean fractional coverage on an energetically heterogeneous surface, dimensionless fractional coverage on sites with identical energy, dimensionless Θ at the start of TPD θ at the start of TPD Θ(T ) Tp) θ(T ) Tp)
References and Notes (1) Masel, R. Principles of Adsorption and Reaction on Solid Surfaces; John Wiley & Sons: New York, 1996.
6008 J. Phys. Chem. C, Vol. 111, No. 16, 2007 (2) Chan, C. M.; Aris, R.; Weinberg, W. H. Appl. Surf. Sci. 1978, 1, 360. (3) Chan, C. M.; Weinberg, W. H. Appl. Surf. Sci. 1978, 1, 387. (4) Cvetanovic´, R. J.; Amenomiya, Y. AdV. Catal. 1967, 17, 103. (5) Konvalinka, J. A.; Scholten, J. J. F.; Rasser, J. C. J. Catal. 1977, 48, 365. (6) Sawa, M.; Niwa, M.; Murakami, Y. Zeolites 1990, 10, 307. (7) Joly, J. P.; Perrard, A. Langmuir 2001, 17, 1538. (8) Niemantsverdriet, J. W.; Markert, K.; Wandelt, K. Appl. Surf. Sci. 1998, 31, 211. (9) Winterbottom, W. L. J. Vac. Sci. Technol. 1972, 9, 936. (10) Winterbottom, W. L. Surf. Sci. 1973, 36, 195. (11) Pisani, C.; Rabino, G.; Ricca, F. Surf. Sci. 1974, 41, 277. (12) Forzatti, P.; Borghesi, M.; Pasquon, I.; Tronconi, E. Surf. Sci. 1984, 137, 595. (13) Xia, X.; Litvinov, S.; Muhler, M. Langmuir 2006, 22, 8063. (14) Rudzinski, W.; Borowiecki, T.; Panczyk, T.; Dominko, A. Langmuir 2000, 16, 8037. (15) Seebauer, E. G. Surf. Sci. 1994, 316, 391. (16) Gorte, R. J. Catal. Today 1996, 28, 405. (17) Gorte, R. J. J. Catal. 1982, 75, 164. (18) Demmin, R. A.; Gorte, R. J. J. Catal. 1984, 90, 32. (19) Hinrichsen, O.; Rosowski, F.; Muhler, M.; Ertl, G. Stud. Surf. Sci. Catal. 1997, 109, 389. (20) Hinrichsen, O.; Hornung, A.; Muhler, M. Chem. Eng. Technol. 1999, 22, 1039. (21) Kanervo, J. M.; Keskitalo, T. J.; Slioor, R. I.; Krause, A. O. I. J. Catal. 2006, 238, 382. (22) Cardona-Martinez, N.; Dumesic, J. A. AdV. Catal. 1992, 38, 149. (23) Chambers, A. Modern Vacuum Physics; Chapman & Hall/CRC: Boca Raton, FL, 2005; pp 77-117. (24) Dumesic, J. A.; Rudd, D. F.; Aparicio, L. M.; Rekoske, J. E.; Trevino, A. A. The Microkinetics of Heterogeneous Catalysis; American Chemical Society: Washington, DC, 1993.
Xia et al. (25) Redhead, P. A. Vacuum 1963, 12, 203. (26) Knox, X.; Dadyburjor, D. B. Chem. Eng. Commun. 1981, 11, 99. (27) Xia, X.; Naumann d’Alnoncourt, R.; Strunk, J.; Litvinov, S.; Muhler, M. Appl. Surf. Sci., in press (doi:10.1016/j.apsusc.2006.12.070). (28) Grant, W. A.; Carter, G. Vacuum 1965, 15, 13. (29) Muhler, M.; Nielsen, L. P.; To¨rnqvist, E.; Clausen, B. S.; Topsøe, H. Catal. Lett. 1992, 14, 241. (30) Strunk, J.; Naumann, d’Alnoncourt, R.; Bergmann, M.; Litvinov, S.; Xia, X.; Hinrichsen, O.; Muhler, M. Phys. Chem. Chem. Phys. 2006, 8, 1225. (31) Xia, X.; Naumann d’Alnoncourt, R.; Strunk, J.; Litvinov, S.; Muhler, M. J. Phys. Chem. B 2006, 110, 8409. (32) Vollmer, S.; Witte, G.; Wo¨ll, Ch. Catal. Lett. 2001, 77, 97. (33) Hadden, R. A.; Sakakini, B.; Tabatabaei, J.; Waugh, K. C. Catal. Lett. 1997, 44, 145. (34) Middleman, S. An Introduction to Mass and Heat Transfer: Principles of Analysis and Design; John Wiley & Sons: New York, 1998. (35) Genger, T.; Hinrichsen, O.; Muhler, M. Catal. Lett. 1999, 59, 137. (36) Hinrichsen, O.; Genger, T.; Muhler, M. Chem. Eng. Technol. 2000, 11, 956. (37) Wilmer, H.; Genger, T.; Hinrichsen, O. J. Catal. 2003, 215, 188. (38) McCash, E. M.; Parker, S. F.; Pritchard, J.; Chesters, M. A. Surf. Sci. 1989, 215, 363. (39) Gundersen, K.; Hammer, B.; Jacobsen, K. W.; Nørskov, J. K.; Lin, J. S.; Milman, V. Surf. Sci. 1993, 285, 27. (40) Campbell, J. M.; Campbell, C. T. Surf. Sci. 1991, 259, 1. (41) Anger, G.; Winkler, A.; Rendulic, K. D. Surf. Sci. 1989, 220, 1. (42) Hayden, B. E.; Lamont, C. L. A. Phys. ReV. Lett. 1989, 63, 1823. (43) Rasmussen, P. B.; Holmblad, P. M.; Taylor, P. A.; Chorkendorff, I. Surf. Sci. 1993, 287/288, 79. (44) Sandoval, M. J.; Bell, A. T. J. Catal. 1993, 144, 227.
