Langmuir 2000, 16, 2389-2393
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Assessment of Adsorbent and Catalyst Heterogeneity by Dynamic Isotope Tracing Marilyn M.-L. Ho and Richard K. Herz* Chemical Engineering Program, and Department of Mechanical and Aerospace Engineering, University of California, San Diego, San Diego, California 92093-0411 Received November 17, 1998. In Final Form: October 28, 1999
Introduction Surface heterogeneity of adsorbents and solid catalysts affects their adsorption performance and catalytic activity and selectivity, respectively. Direct calorimetric measurements1-3 and indirect determinations from isotherm measurements4-6 and thermal desorption7,8 generally reveal a decrease in adsorption energy with increasing coverage. Three causes for such behavior are (a) a distribution of adsorption energies over a heterogeneous surface,9-11 (b) direct and indirect interactions between adsorbed molecules,12-15 and (c) surface reconstruction.1,15,16 More than one cause may be present in a given system, and their effects are difficult to separate. The purpose of this work is to demonstrate the potential of isotope tracing to study the first cause independently of the latter two. Dynamic isotope tracing of a steady-state system involves changing the isotope fraction of a species in the inlet fluid while other conditions remain constant.17-19 The technique is also known as steady-state isotopic transient kinetic analysis.20 The isotope fraction is the ratio between the amount of one selected isotope of the species to the total amount of that species. The isotope fraction of the species and any reaction products leaving the system are measured along with the response of an inert tracer added to the feed. Differences between the kinetics of the isotopes are assumed negligible. Applied to the single-component adsorption system of Figure 1, the dynamic response of the isotope fraction in the exit * To whom correspondence should be addressed. E-mail: herz@ ucsd.edu. Phone: 858-534-6540. Fax: 858-534-4543. (1) Yeo, Y. Y.; Wartnaby, C. E.; King, D. A. Science 1995, 268, 1731. (2) Wartnaby, C. E.; Stuck, A.; Yee, Y. Y.; King, D. A. J. Phys. Chem. 1996, 100, 12483. (3) Cardona-Martinez, N.; Dumesic, J. A. Adv. Catal. 1992, 38, 149. (4) Guinn, K. V.; Rhoades, D. S.; Herz, R. K Surf. Sci. 1997, 393, 47. (5) Ertl, G.; Neumann, M.; Streit, K. M. Surf. Sci. 1977, 64, 393. (6) Rothstien, D. P.; Wu, B.-G.; Lee, V.; Madey, R. J. Colloid Interface Sci. 1985, 106, 399. (7) Falconer, J. L.; Madix, R. J. J. Catal. 1977, 48, 262. (8) Taylor, J. L.; Weinberg, W. H. Surf. Sci. 1978, 78, 259. (9) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: San Diego, CA, 1992. (10) Toth, J. Acta Chim. Hung. 1992, 129, 39. (11) Do, D. D.; Do, H. D. Chem. Eng. Sci. 1997, 52, 297. (12) Persson, B. N. J.; Ryberg, R. Phys. Rev. B: Condens. Matter 1981, 24, 6954. (13) Lauterbach, J.; Boyle, R. W.; Schick, M.; Mitchell, W. J.; Meng, B.; Weinberg, W. H. Surf. Sci. 1996, 350, 32. (14) Quinones, I.; Guiochon, G. Langmuir 1996, 12, 5433. (15) Yeo, Y. Y.; Vattuone, L.; King, D. A. J. Chem. Phys. 1996, 104, 3810. (16) Behm, R. J.; Theil, P. A.; Norton, P. R.; Ertl, G. J. Chem. Phys. 1983, 78, 7437. (17) Le Cardinal, G.; Walter, E.; Bertrand, P.; Zoulalian, A.; Gelus, M. Chem. Eng. Sci. 1977, 32, 733. (18) Happel, J. Chem. Eng. Sci. 1978, 33, 1567. (19) Le Cardinal, G. Chem. Eng. Sci. 1978, 33, 1568. (20) Ali, S. H.; Goodwin, J. G., Jr. J. Catal. 1998, 176, 3.
Figure 1. Schematic representing fluid containing a single adsorbing species in contact with a surface composed of n types of sites which differ in adsorption energy and abundance. Rates of adsorption and desorption (arrows) over a single type of site are equal to each other but differ from the rates over other sites. Direct exchange between site types is assumed negligible. The response of the isotope fraction in the fluid phase y1(t) is measured as the adsorbate isotope fraction in the feed fluid u1(t) is changed. The average isotope fraction in the adsorbed layer yave(t) can be determined from the measurements.
flow contains information only about surface heterogeneity because the total coverage remains constant, and thus, adsorbate-adsorbate interactions and adsorption-induced effects on the surface remain constant. Most applications of this technique have been to reaction systems such as CO hydrogenation to methane.20-22 CO adsorbs to form carbonaceous species, followed by hydrogenation of these species to form methane. The fraction of 13CO in the CO feed is changed and the 13C fraction in the methane product is measured. Data are analyzed in order to determine the activity distribution of the carbonaceous species.23,24 Several factors make CO methanation easier to analyze than the adsorption experiment of Figure 1. The primary factor is that hydrogenation of the carbonaceous species is essentially irreversible such that the methane formed does not exchange carbon isotopes with surface species. This means that there is no coupling of isotope fractions between sites. In contrast, in an adsorption-only system, sites are coupled to each other because molecules can desorb from one type of site and then readsorb on any other type of site. Only a few adsorption systems have been studied with this technique. Sushchikh et al.25 measured the isotope fraction in CO adsorbed over Ir(111) with infrared spectroscopy and extracted the single exchange frequency for this uniform surface, which equals the first-order desorption rate constant. Bajusz and Goodwin26 studied N2 adsorption over LiX zeolite and determined the adsorption capacity and the total N2 surface residence time in the sample bed. Szedlacsek et al.27 proposed that isotope tracing could be used to obtain the exchange frequency as a function of average isotope fraction in the adsorbed layer. Any variation in exchange frequency with average isotope fraction demonstrates surface heterogeneity. However, this information is not complete since the average isotope fraction at any time is a function of the history of the surface during the experiment. (21) Biloen, P.; Helle, J. N.; Van den Berg, F. G. A.; Sachtler, W. M. H. J. Catal. 1983, 81, 450. (22) Hoost, T. E.; Goodwin, J. G. J. Catal. 1992, 137, 22. (23) de Pontes, M.; Yokomizo, G. H.; Bell, A. T. J. Catal. 1987, 104, 147. (24) Hoost, T. E.; Goodwin, J. G. J. Catal. 1992, 134, 678. (25) Sushchikh, M.; Lauterbach, J.; Weinberg, W. H. Surf. Sci. 1997, 393, 135. (26) Bajusz, I.-G.; Goodwin, J. G., Jr. Langmuir 1997, 13, 6550. (27) Szedlacsek, P.; Efstathiou, A. M.; Verykios, X. E. Appl. Catal. A-Gen. 1997, 151, 59.
10.1021/la9816048 CCC: $19.00 © 2000 American Chemical Society Published on Web 01/08/2000
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Notes
The present work demonstrates a method to obtain the abundance distribution of exchange frequencies in the absence of coverage variation effects. The presence of such a frequency distribution is a direct qualitative indicator of surface heterogeneity, and exchange frequency distributions can be related quantitatively to adsorption energy distributions in most cases. For example, over sites with first-order kinetics with nonactivated adsorption, the logarithm of the isotope exchange frequency is directly proportional to the heat of adsorption. This relationship holds approximately for more complex systems in the limit of saturation coverage under which the experiments are performed.
because of this constraint. The remaining unknowns are n exchange frequencies κi and (n - 1) abundance fractions φ i. Negligible spatial gradients are specified and will be obtained in porous adsorbent particles when the diffusion time constant is much less than the exchange time constant. Under the conservative assumption that all sites have the maximum measurable exchange frequency κmax, the criterion is
System Equations
where dp is the particle diameter and De is the effective isotope diffusivity in the adsorbent. For example, for κmax ) 10 s-1, intraparticle gradients will be negligible with isotope gas pressure >10-3 atm over 0.2 mm diameter particles of 100% dispersed, 0.1 wt % Pt/γ-Al2O3, with density ) 1 g/cm3 and De ) 10-2 cm2/s. For porous materials in which intraparticle gradients cannot be eliminated, a cell can be used in which the gradients are measured.28 For the results discussed below, the input is a positive unit step in isotope fraction at t ) 0 flowing into a wellmixed cell. Measurement of the inert response and linear system theory can be used to analyze cases with different inputs and fluid flow patterns. For the specified case, the responses are:
At steady-state adsorption equilibrium, the adsorption rate rai and desorption rate rdi on each site type are equal to each other and to the isotope exchange rate ri (mol/s). Each ri can be divided by the moles of adsorbate covering that site type in order to obtain an exchange frequency, κi (s-1),
κi )
ri Va Ca θi φi
(1)
where Va is the volume of the adsorbent, Ca is the adsorption capacity of the adsorbent (mole/liter of adsorbent), θi is the fractional coverage of site type i, and φi is the fraction of the surface sites that are of type i. The κi are intensive parameters independent of sample size and capacity. The system equations for the fluid phase (eq 2) and the surface sites (eq 3) are
dy1 dt
)-
(
n+1
Qf Vf
VaCa +
)
κiθiφi ∑ i)2
()
VfCf
VaCa
dyi ) κiy1 - κiyi dt
VfCf
( ) ( ) ( )( ) dp 6
n+1
y1(t) ) 1 +
∑ j)1
n+1
yi(t) ) 1 +
∑ j)1
2
b1(j)
Cf 1 1 , De Ca κmax
eλ(j)t
(5)
for the fluid response
λ(j)
(6)
bi(j)
eλ(j)t λ(j) for the surface responses, i ) 2 to n + 1 (7)
y1 +
n+1
∑ i)2
(κiθiφiyi) +
() Qf Vf
for i ) 2 to n + 1
where λ(j) is the jth eigenvalue of the matrix [A] and bi(j) is the ith element of the jth eigenvector. The average isotope fraction in the adsorbed layer is
u1 (2) (3)
where Qf is the volumetric flow rate of fluid (liter/second), Vf is the volume of the fluid inside the adsorption cell, and Cf is the concentration of adsorbing species in the fluid (mole/liter of fluid). The yi for a surface site type i is the fraction of adsorbed molecules covering the site that have the isotopic label being measured. This gives the following in vector-matrix form:
dy b ) [A]y b + [B]u b dt
n+1
yave(t) ) 1 +
(4)
The system is linear with respect to the isotope fractions because the exchange process between any two pools of labeled species is directly proportional to the difference in the isotope fractions of the two pools, regardless of any nonlinearity in adsorption and desorption kinetics. Temperature and isotope concentration must be chosen to make the fractional coverage on all site types closely approach one, thus effectively eliminating the fractional coverage θi as unknowns. Exchange frequency abundance fraction distributions can only be obtained at full coverage
∑ j)1
bave(j)
eλ(j)t
(8)
λ(j)
where n+1
bave(j) )
φibi(j) ∑ i)2
(9)
Although individual surface responses cannot be measured in an experiment, the average isotope fraction in the adsorbed layer yaveexp(t) can be obtained from the experimental fluid response y1exp(t),
yaveexp(t) )
( ) ( )∫ QfCf QfCf tVaCa VaCa
t
0
y1exp(t) dt -
( )
VfCf y exp(t) (10) VaCa 1
where the groups on the right represent, respectively, the total amount of isotope that has flowed into the adsorption (28) Cannestra, A. F.; Nett, L.; Herz, R. K. J. Catal. 1997, 172, 346.
Notes
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Figure 2. Solid line: target exchange frequency distribution used to compute simulated experimental isotope response. This distribution is the “unknown” distribution that is to be determined by the analysis method. Bars: model distribution obtained by the analysis method.
Figure 4. Illustration of increasing the system temperature in order to shift a sample’s exchange frequency distribution across the experimental window. For a system with first-order adsorption-desorption kinetics, the change from a to b corresponds to an increase of 25 K. Figure 3. Results obtained with a significant fraction of the sample’s exchange frequency distribution outside of the experimental window.
cell, the total amount that has flowed out, and the amount that is present in the fluid phase within the cell. Procedure In both methods investigated, the values of (Qf/Vf), (VaCa/ (VfCf)), and a “target” distribution of exchange frequencies are specified, then a simulated experimental response y1exp(t) is computed. This simulated experimental response is analyzed in order to determine a “model” distribution of exchange frequencies that results in a similar response. A method is sought that minimizes the differences between the target and model distributions. In actual experiments, the target distribution would be the unknown to be determined. The first method investigated was an attempt to take advantage of the system’s linearity. A model set of eigenvalues was assumed followed by linear-regression fitting of eq 6 to the simulated experimental response y1exp(t) to determine the coefficients [b1(j)/λ(j)]. Each coefficient is the first element of eigenvector j divided by eigenvalue λ(j). The remaining n coefficients [bave(j)/λ(j)] were obtained from a linear-regression fit of eq 8 to the simulated experimental surface-average response yaveexp(t). That only the weighted average of the surface elements in each eigenvector, bave(j), can be determined is a result of the inherent loss of information in the experiment: only the fluid response and the average surface response can be measured, rather than the fluid and individual surface responses. With this loss of information, the frequency-domain transfer function formed from the eigenvalues and weighting coefficients cannot reliably be inverse-transformed into a correct state-space representation of the experiment. The representation obtained contains direct inputs into
Figure 5. Points are data from Figure 3b of ref 26 and are responses measured over LiX zeolite at 303 K. F(t) for nonadsorbing Ar is the normalized concentration response of Ar. F(t) ) y1(t) for 15N2. Curves are simulations obtained by fitting the exchange model to the data. Up- and downstream mixing cells in the simulation each had volumes of 0.89 cm3. The total volume (Vf) of the sample cell was 0.50 cm3 and was modeled as 10 mixing cells in series. The surface-to-gas capacity ratio (VaCa/(VfCf)) obtained was 7.33.
the surface sites, i.e., nonzero elements u2 ... un+1, whereas, physically, an input can only be applied through the inlet fluid element u1. The successful method was a direct optimization procedure. The first step is to fix the limits of the “experimental window" that defines the range of exchange frequencies about which information can be obtained. Here, the largest exchange frequency is set to equal the inverse of the mean residence time of inert fluid in the
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Notes
Figure 6. Exchange frequency distribution obtained by the fit of the exchange model to the data shown in Figure 5. The exchange frequencies specified were 0.04, 0.4, 4, 40, and 400 s-1.
cell. The smallest exchange frequency is determined by measurement stability and experiment duration. The simulated experimental window here ranged from 0.003 to 10 s-1, corresponding to an experimental duration of 300 s and an inert residence time of 0.1 s, respectively. A sample’s distribution can be shifted into the experimental window by varying temperature. Next, a set of five or six exchange frequencies κi of a model surface was spaced equidistant from each other on a logarithmic scale in order to span the experimental window. An initial set of abundance fractions φi for the set of model frequencies is proposed, with a set of equal fractions being a reasonable choice. Then the isotope response is computed for the model set of frequencies and abundance fractions. Finally, abundance fractions are varied using the Nelder-Mead simplex algorithm in order to minimize the differences between the model response y1(t) and the target response y1exp(t). The optimal abundance fractions are compared to the target distribution. Results Results are shown for the direct optimization method. The bars in Figure 2 form a histogram showing the abundance distribution of model frequencies that gave the best fit to the response data produced by the target distribution. For this case, the analysis procedure produces a histogram that reproduces the general shape of the target distribution. The histogram shown in Figure 2 is one possible analysis result. Other results were obtained by varying the initial guesses of the abundance fractions and by changing the set of model frequencies. For the case shown in Figure 2, the computed shape of the distribution was essentially invariant with changes in initial abundance fractions and changes in the set of model frequencies. The minimum spacing of the model κi was found to be 1 order of magnitude, i.e., 1 on the logarithmic exchangefrequency scale. While this initially appears to be a significant constraint, the exchange frequencies in many real adsorption systems can be expected to span many orders of magnitude. For first-order desorption kinetics, the exchange frequency of a site type is equal to the desorption rate constant for that site type. Consider a system that has nonactivated adsorption, first-order
adsorption-desorption kinetics, and a system temperature of 350 K. A 1 order of magnitude change in exchange frequency corresponds to a change in desorption activation energy and, thus, a heat of adsorption of 6.7 kJ/mol (1.6 kcal/mol). Heats of adsorption are often found to vary, for all possible reasons, over a much wider range. For example, the heat of adsorption of CO over polycrystalline Pt varies by 100 kJ/mol with CO coverages between 0.01 and 0.9.4 A maximum of six model frequencies can be used to fit experimental data. The highest model exchange frequency should be located at the high-frequency limit of the experimental window. The abundance fraction obtained for this highest model frequency will correspond to the abundance of sites in the target distribution with this and higher exchange frequencies. The smallest model exchange frequency should be located 0.5 or 1 order of magnitude to the left of the left-hand limit of the experimental window. The abundance fraction obtained for this smallest model frequency will correspond to the abundance of sites in the target distribution with this and lower exchange frequencies, as shown in Figure 3. The method gives consistent results for target distributions, on a logarithmic frequency scale, that are (a) linearly sloping upward and downward and for (b) unimodal and (c) bimodal Gaussian distributions. Accurate estimates of bimodal Gaussian distributions can only be obtained when the distribution peaks are separated by at least 2.5 orders of magnitude in exchange frequency. For a system with nonactivated adsorption and first-order desorption kinetics, this peak separation corresponds to an adsorption enthalpy difference of 17 kJ/mol. In cases where the experimental window is too narrow to span the target distribution, the target distribution can be shifted across the experimental window by performing experiments at different temperatures, with Cf adjusted to keep the surface saturated. Such a pair of experiments is illustrated in Figure 4. Consistent estimates of unimodal Gaussian distribution shapes can be obtained when random Gaussian-distributed error added to the data has a standard deviation 2% of the maximum signal. This analysis assumes negligible direct exchange between site types. When this is not true, model distributions returned will overestimate site abundance fractions at higher fluid-surface exchange frequencies. Any such
Notes
skewing of a distribution should be temperature dependent and be identifiable by that behavior. The procedure was used to analyze data reported by Bajusz and Goodwin26 for 15N2 exchange over LiX zeolite. Results of the analysis are shown in Figures 5 and 6. The zeolite adsorbent was located in a small tube, and the Ar inert and 15N2 responses could not be fit by modeling the system as a single mixing cell. The best fit was obtained by modeling the sample bed as 10 mixing cells connected in series, a conventional method for modeling packed beds of particles. Bajusz and Goodwin reported that the total residence time of an average N2 molecule on the surface within the sample bed ranged from 3.3 to 3.8 s. This total residence time is an extensive property that is directly
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proportional to sample weight. Our analysis provides information about the adsorption-desorption kinetics. Figure 6 shows that the exchange frequency distribution peaks at a frequency somewhat higher than 4 s-1. Although this frequency is at the limit of the experimental window for these data, this result indicates that the average residence time of an N2 molecule on an adsorption site during a single adsorption event is 0.25 s or less. The procedure presented here can be used to design experimental systems optimized to obtain such surface kinetic information. LA9816048
