Ind. Eng. Chem. Res. 1997, 36, 3173-3179
3173
Kinetic Modeling of Metal-Catalyzed Reactions of Alkanes Geoffrey C. Bond* Department of Chemistry, Brunel University, Uxbridge UB8 3PH, United Kingdom
Linear alkanes having two to four carbon atoms react with H2 by hydrogenolysis, giving smaller alkanes, or in the case of n-butane also by isomerization to isobutane. The dependence of the rates on H2 pressure is described by rate equations derived from mechanisms involving a partially dehydrogenated reactive intermediate; the precise form of the equation depends on the number of H atoms (zero, one, or two) assumed to take part on the slow step. An equation in which one H atom is used has been applied to the results obtained with Ru, Rh, and Pt catalysts and provides values for the rate constant k and for the equilibrium constants for H2 chemisorption KH and alkane dehydrogenation KA. Both k and KA increase with temperature on Ru and Pt catalysts, leading respectively to values of the true activation energy Et of ∼40-80 kJ mol-1 and of enthalpy changes of 60-90 kJ mol-1. Deficiencies of such rate equations include failure to allow quantitatively for carbon deposition and the assumption that both reactants utilize the same sites. Possible modifications to meet these criticisms will, however, require more complex rate equations, from which significant numerical values of the constants may not be obtainable. 1. Introduction: The Purpose of Kinetic Modeling In what follows, the term “kinetic modeling” is taken to mean the process of accommodating observed variations of the reaction rate with controlled experimental variables such as reactant concentrations and temperature to a theoretical rate expression derived from an assumed mechanism and extracting numerical values of the constants of that expression in order to use them for a better understanding of catalyst function. For many years, the determination of reaction kinetics was the only means of obtaining information on the mechanism, but its use by physical chemists fell into desuetude because (a) alternative and more informative techniques became available (e.g., the use of isotopic tracers) and (b) it was realized that kinetic information could rarely, if ever, lead to an unambiguous mechanistic statement. In our opinion, the pendulum swung too far in the opposite direction, and the consequential neglect of kinetic analysis and the associated exploration of the effect of experimental variables have been detrimental to the development of the subject. The proper way of regarding reaction kinetics is as a necessary test of a mechanistic hypothesis: no mechanism can be considered plausible unless supported by kinetic information. In recent years, we have returned to the study of the kinetics of heterogeneously-catalyzed reactions through an awareness of the wealth of additional and, indeed, essential information that stems from altering reaction conditions. Our work has been concerned with the reactions of alkanes with H2 on supported metal catalysts (1-7) and has been influenced much by that of Hungarian colleagues on reactions of n-hexane and similar alkanes, (7-10) which has shown the complex manner in which product yields vary, in particular, with H2 pressure and with temperature. However, the very complexity of the product mix militates against quantitative analysis of the results, and we have, therefore, so far confined our attention to the reaction of C2-C4 alkanes with H2, where hydrogenolysis and skeletal isomerization are the only two reactions affording gaseous products. “Carbon” deposition almost always occurs to some extent as these reactions proceed (10, * E-mail address:
[email protected]. S0888-5885(96)00671-9 CCC: $14.00
11): it is a major nuisance, limiting the catalyst lifetime in a chemical plant and obscuring the kinetic information needed for mechanistic discussion. How to cope with the problem will be considered later. Although quantitative kinetic work on the lower alkanes has no obvious or immediate use, we justify our attention to it in the following way. (i) It is a challenge to the power of kinetic analysis to extract meaningful information from systems so prone to self-poisoning; (ii) formation of carbon deposits continues to affect the development and use of many catalytic processes such as alkane dehydrogenation and the reforming of CH4 with H2O or CO2; (iii) there is current interest in the conversion of CH4 to higher alkanes through the polymerization of monocarbon species present in carbon deposits (12, 13). Further to i, we have been struck by the fact that, in most catalysis research, the importance of a variable factor in the preparation of a series of catalysts is decided on the basis of rate measurements performed under a single set of conditions. We have illustrated the limitations of doing this by selecting the temperature and H2 pressure as the most significant variables. Where Arrhenius lines intersect within the range of measurement, relative activities depend on the temperature chosen for the comparison (14); the relative reactivities of the C2-C4 alkanes depend on the H2 pressure at which the rates are measured (15). By ignoring these effects, we are truly building castles on sand, and we must ask the question, what do we really mean by “catalytic activity”? Our kinetic work has been largely motiviated by a desire to answer this question: we seek to establish rate constants, and then true Arrhenius parameters, and adsorption coefficients (equilibrium constants), and then enthalpies of adsorption, under reaction conditions. We wish to know whether it is the kinetic or the thermodynamic parameters that respond to alteration of a catalyst’s structure and composition and, therefore, which chiefly determine the catalytic activity. Our work has thrown into relief a number of problems connected with kinetic modeling of alkane reactions, which it is the purpose of this paper to review. The problems are several and differ in their nature, but the major is the following. We believe that, in order to © 1997 American Chemical Society
3174 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
derive significant numerical values to the constants of a rate expression, it is necessary to constrain their number to not more than about three, whereas a fully elaborated mechanism requires a larger number. We are therefore forced to use oversimplified mechanisms and rate equations, knowing their limitations, in order to have a tractable system with the power of discrimination. As we shall show, even this small number of constants can provide excessive flexibility. 2. Selection of a Rate Expression The first problem encountered in kinetic modeling is the choice of the rate expression to be used; this is no trivial matter. In the field of alkane hydrogenolysis, it is possible to envisage a plethora of possible mechanisms, and a correspondingly large number of candidate equations suggest themselves. We have to decide what criteria to use in selecting that which is most appropriate to our purposes. An objective statistical parameter such as the minimum standard deviation is not necessarily adequate or sufficient. Other criteria need to be kept in mind; these include (i) the physical rationality of the mechanism from which the equation derives, (ii) the plausibility of the values of the best-fit constants, including their temperature dependence, and (iii) the number of constants that the equation contains. 3. The Power Rate Law Formalism In order to provide the necessary background for a discussion of these and other matters, we first review the salient experimental observations and the deductions that can be immediately drawn from them. As H2 pressure is increased at constant temperature, the rate of hydrogenolysis first increases, then passes through a maximum, and finally decreases. By far the most widely-used alkane has been ethane: it has the advantages that it causes little deactivation of the catalyst by carbon deposition and only a single product is formed, but it has the disadvantage that the rate maximum usually occurs at inconveniently low H2/alkane ratios so that only the region of negative order in H2 is observed. For this reason, it has been commonplace to express the kinetic equation as a simple power rate law:
rh ) kPARPH-β
(ES1)
where the order in alkane R is positive and often close to unity, and β can be as large as 3 (16). The inhibiting effect of H2 clearly points to the need for vacant sites, and from the earliest discussions of the mechanism, it has been thought (not quite unanimously) that the alkane needed to lose a certain number of H atoms to the catalyst surface to be activated for reaction: this number could be estimated from the value of β (16). It has been recognized (17) that such an equation is only an approximation, valid only over a limited range of H2 pressures, to one of the Langmuir-Hinshelwood type, containing adsorption coefficients explicitly; obviously the rate must fall to zero, as the H2 pressure does likewise. However, until recently, few attempts have been made to formulate or to use this more precise approach. This further limitation on the value of the power rate law formalism becomes more apparent as soon as propane and n-butane are examined (Figure 1). The position of the rate maximum moves to higher H2 pressures as the chain length increases, and a simple logarithmic function is clearly inadequate to describe
Figure 1. Dependence of the rate of hydrogenolysis on H2 pressure at 608 ( 1 K using Pt/Al2O3 (5): O, C2H6; 0, C3H8; 4, n-C4H10. PA ) 0.071 atm. Curves calculated by eq ES5B. Scheme 1. Mechanism of the Hydrogenolysis of Ethane (27) chemisorption of H2 chemisorption of C2H6
H2 + 2* h 2H* C2H6 + (7 - m)* h C2Hm* + (6 - m)H* slow step C2Hm* + B f CHp* + CHq* B is either * or H* or H2: the values of p and q depend on which is selected. KA ) θAθH6-m/PAθ*7-m KH ) θH2/θ*2PH
KH KA k
the rate variation with H2 pressure over the whole range; a more complex equation is required. There are qualitative indications in the older literature to suggest that H2 inhibition decreases with increasing temperature, but once again, quantitative use of this information has only recently been made. 4. Principal Features of Alkane Hydrogenolysis We may summarize what we can safely infer about the mechanism of alkane hydrogenolysis by reference to Scheme 1; this shows the essential steps for the reaction of ethane, but the changes needed for higher alkanes are easily introduced. Conversion of the monocarbon species is considered fast and nonrate-limiting. If we suppose the slow step to involve an H atom, the rate equation is first written as
rh ) kθAθH
(i)
The rate maximum occurs when θA and θH are equal (providing their sum remains constant), and the fact that this happens at low PH/PA ratios in the case of ethane implies that the reactive form of the alkane is easily rehydrogenated to an inactive form as PH and, therefore, θH are increased, or conversely, that the dehydrogenation is not greatly favored. As the length of the carbon chain is increased, dehydrogenation to the active state becomes easier, as shown by the relevant thermodynamic parameter (18), so that the position of the rate maximum moves to progressively higher H2
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3175
pressures. Thus, at H2 pressures of less than that giving the maximum rate, the rate is limited by the availability of H atoms, while above the rate maximum the limiting factor is the concentration of the dehydrogenated species θA. (iv) The rate maxima move to higher H2 pressures as the temperature is raised (Figure 2), because the chemisorption of H2 being exothermic, θH decreases at fixed H2 pressure, and more vacant sites become available to facilitate the formation of the dehydrogenated intermediate. Thus, the concentration of the reactive intermediate increases as the temperature is raised, by reason of the endothermic preequilibrium by which it is formed. This contrasts with the classic Langmuir-Hinshelwood mechanism, where the concentrations of the adsorbed species decrease with rising temperature, because they are formed exothermically. The consequence, for alkane reactions, is that the apparent activation energy Ea is greater than the true activation energy Et, contrary to the Langmuir-Hinshelwood model (18). This viewpoint has been used to account for the variation in apparent activation energy with chain length and the very high values (sometimes approaching 400 kJ mol-1) that are encountered in some hydrocarbon transformations (19). 5. Development of Rate Equations Using Langmuir-Hinshelwood Methodology The quantitative expression of this mechanistic concept allows, however, a considerable number of variations and alternatives, each of which generates its own rate equation. We may start illustrating our problems by referring to one of the earliest proposed mechanisms (20), which assumed a reversible dehydrogenation of the alkane but neglected to include an H2 adsorptiondesorption equilibrium, postulating that the slow step involved only gaseous H2. The resulting rate equation (coded ES2 in the paper by Shang and Kenney (21) takes the form
rh ) kyPH/(y + PHγ)
(ES2)
and it can give perfectly acceptable fits to experimental data (5, 21) (Figure 3). Its use is not, however, recommended, as the omission of reversible H2 adsorption is unreasonable, although it is not impossible that at reaction temperatures it is so small as to be negligible. More recent treatments have remedied this defect; those advanced by Gudkov et al. (22) invoked C-C bond-breaking steps of partially dehydrogenated species without the assistance of H atoms or H2 molecules, but it was necessary to propose different species (viz. C2H5* and C2H2*) depending on the range of H2 pressure used. This seemed arbitrary, and their approach was generalized by Cunningham et al. (5, 23, 24), who took their equation for the C2H2* intermediate but instead of using their value of 2.5 for the H2 pressure exponent in the denominator substituted a variable parameter δ; viz.,
rh ) kPAPH2/(PA + k2PHδ)2
Figure 3. Dependence of the rate of hydrogenolysis on H2 pressure at 609 K using PtRe/Al2O3 (5): experimental points modeled by rate expressions ES5B (s), ES2 (‚‚‚), and ES3 (---).
(ES2A)
They also noted that recognition of other possible intermediates (C2H4*, C2H3*) led to a family of equations of the form
rh ) kPAPHκ/(PA + k2PHκ+1/2)2
Figure 2. Dependence of the rate of n-C4H10 hydrogenolysis on H2 pressure at various temperatures using Pt/Al2O3 (5): curves calculated by eq ES5B.
(ES3)
in which κ is half the number of H atoms lost in forming the reactive intermediate. These equations were ap-
plied (5, 23, 24) to the results for the hydrogenolysis of propane and of n-butane on typical Pt/Al2O3 and PtRe/Al2O3 reforming catalysts: eqs ES2 and ES2A both gave satisfactory fits, but eq ES3 was worse (see Figure 3). The extensive work of Leclercq et al. on alkane hydrogenolysis using supported Pt catalysts initially employed the eq ES2 (25), but a later revision (26) provided for H2 chemisorption, while retaining molec
3176 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 Table 1. Rate Equations Derived from the Mechanism Shown in Scheme 1 (27) B
rh
code
* H* H2
kyG6/D2 kyG7/D2 kyPH/D
y ) KAPA D ) y + G 6 + G7
G6 ) (KHPH)a
ES5B ES5 G7 ) (KHPH)a+1/2 a ) (n - m)/2
Table 2. Dependence of the Values of the Constants for the Hydrogenolysis of Ethane on Ru/Al2O3 at 473 K on the Form of the Rate Equation Used (21) rhb
ka
KA
KH/atm-1
kyPH/(y + PHa) kPA./(1 + bPHa-1) kyPH/[y + PHa(1 + (KHPH)0.5)] kyPH/D
1.44 × 10-4 8.16 × 10-5 1.52 × 10-4
1.06 × 10-3 3.24 × 10-2 1.45 × 101
1.1 × 107
1.53 × 10-4
2.08 × 107
7.31 × 103
a
In units of mol gRu of y and D.
-1
s-1
atm-1. b
See Table 1 for definitions
ular H2 in the slow step. The resulting equation, however, contained five adjustable constants as well as the rate constant, and not surprisingly these workers were unable to evaluate all of them. A further development was introduced by Kristyan and Szamosi (27), who allowed the slow C-C bondbreaking of the dehydrogenated species to involve either an H atom or an H2 molecule or an empty site (see Scheme 1). This generated the set of equations shown in Table 1. Their results for ethane hydrogenolysis on Ni and Pd catalysts (27) were stated to fit the third equation best, but the measurements were quite scattered, and no detailed comparison of the equations was made. It was, however, pointed out that a critical test of the models was whether the curves of rate vs H2 pressure obtained using different ethane pressures intersected or not: if they did not (as theirs did not), the third model was indicated. These equations were further extended by Shang and Kenney (21); their eqs ES4 and ES5 closely resemble the third equation of Kristyan and Szamosi, but their analogues (eqs ES4A, B, and ES5A-C) could not be distinguished in quality when applied to the experimental results for ethane hydrogenolysis on a Ru/Al2O3 catalyst. Equation ES4 was marginally preferred to eq ES5 and its analogues, which specified more precisely the number of sites required in each step. (The reader should note there are numerous errors in the rate expressions as formulated in Table 2 of ref 21.) It is, however, on the modification of eq ES5 to bring an H atom into the slow step (viz., eq ES5B) that our own work has focused; it has the same form as the second of the Kristyan-Szamosi equations above (see Table 1). 6. Significance of the Constants in Rate Equations While it is impossible to eliminate any of the equations suggested by Kristyan and Szamosi (27) (Table 1) or their modifications by Shang and Kenney (21) on the grounds of physical implausibility, it is possible to discuss the relative merits of the best-fit constants that each generates. A selection of those constants for the four equations tested on their results for ethane hydrogenolysis by Shang and Kenney (21) are shown in Table 2. The major and surprising outcome is the great difference between the numerical values of the constants that the various equations afford. Forgetting eqs ES2 and ES3 as mechanistically unsatisfactory, the values
of KA range over a factor of 106 and KH over 104. Moreover, the ratio KA/KH changes from 10-7 to 104 when eq ES5 is substituted for ES4, and KA changes by about 107 when eq ES5B is used in place of eq ES5. We may try to introduce common sense as a guide to the selection of the “best” equation. Since the rate maxima occur at values of PA/PH of very approximately unity (in our own work, with Ru/Al2O3, PA ) 0.04 atm and rmax occurs at PH values less than 0.02 atm (28)), we think that KA and KH should be of the same order of magnitude; eq ES5B gives this result. To be precise, for this equation, the rate maximum occurs when 2aKAPA is equal to (KHPH)a+0.5. We have also obtained best-fit values of the constants for eq ES5B for the reactions of propane and for n-butane, as well as for ethane, on a Ru/Al2O3 catalyst, pretreated in different ways, at 418K (2, 28), and on Pt/Al2O3 at ∼608 K (5). In addition, the values for the reactions of propane and of n-butane have been obtained using Rh/Al2O3, Rh/SiO2, and Rh/TiO2 at 433 K (6) and Pt/SiO2 (EUROPT-1) at 533 K (29). The values found after the first high-temperature reduction (HTR1, ∼750 K) are summarized in Table 3. The following conclusions emerge. (1) Where all three alkanes have been studied (5, 28), the rate constant k increases with the alkane chain length, as does also KA: a positive correlation between these two constants has been noted previously, and possible causes have been discussed (5, 18). (2) For Ru/Al2O3 (2), for Rh except Rh/TiO2 (6), and for Pt/SiO2 (29), KH is almost independent of the alkane used; for Pt/Al2O3 (5), it appears to decrease as the alkane chain length increases. Note that the nature of the optimization routine used means that unique values of the constants cannot be obtained and that, therefore, differences of less than a factor of 2 are not considered significant (except for the rate constant, which in effect is a scaling factor). (3) The highest values of KH are shown by Rh, but the lower values of all constants given by Rh/TiO2 are a consequence of the HTR1 taking the catalyst partially into the “strong metal-support interaction” (SMSI) state (6). Moderate values of KH are shown by Ru catalysts (2, 3), and the lower values given by Pt catalysts (5, 29) reflect the higher temperature at which it is necessary to use them. (4) Many of the reported values of a (Table 3) do not correspond closely to integral numbers of H atoms. This may be because shapes of the computed curves are not highly sensitive to the value of a used, the reported values therefore not having great precision; alternatively, there may be two or more species of comparable reactivity so that the values of a are in effect averages. A fuller discussion of the probable significance of the constants’ values will be found in the cited references. Of the various catalysts we have studied, only the 6.3% Pt/SiO2 (EUROPT-1) effects significant isomerization of n-butane to isobutane (29). The selectivity for this process Si decreases with increasing H2 pressure and with lowering of the reaction tempeature, i.e., with increasing θH. The rate variation with H2 pressure is also well described by eq ES5B (Figure 4), but it appears that the active intermediate is more dehydrogenated, and the relevant KA smaller, than for the corresponding hydrogenolysis intermediate. This neatly accounts for the way in which Si changes with the operating variables and the high apparent activation energy (>200 kJ mol-1). The value of KH is close to that for hydrogenolysis, as expected. From this short discussion, it will appear that the values of the constants provided by fitting eq ES5B to
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3177 Table 3. Values of the Constants of Equation ES5B for the Hydrogenolysis of Ethane, Propane, and n-Butane on Various Catalysts
a
catalyst
alkane
T/K
ka
KA/atma-1
KH/atm-1
a
ref
Ru/Al2O3 Ru/Al2O3 Ru/Al2O3 Rh/Al2O3 Rh/SiO2 Rh/TiO2 Pt/Al2O3 Pt/Al2O3 Pt/Al2O3 Pt/SiO2 Pt/SiO2
C2H6 C3H8 n-C4H10 n-C4H10 n-C4H10 n-C4H10 C2H6 C3H8 n-C4H10 C3H8 n-C4H10
418 418 418 430 428 429 607 608 609 533 533
29 80 111 189 151 36 33 45 57 9 ∼11
0.60 17 37 47 30 6.6 10 18 27 0.70 ∼2.3
18 25 27 63 70 18 9 5 2 5.4 ∼6.5
1.85 2.34 1.38 1.75 0.86 1.27 2.25 1.52 0.88 1.41 ∼0.90
2, 28 2, 28 2, 28 6 6 6 5 5 5 29 29
In units of mmol gcat-1 atm-1 h-1.
Figure 4. Dependence of the rates of hydrogenolysis (O) and of isomerization (0) of n-C4H10 on H2 pressure at 533 K using Pt/ SiO2 (27): curves calculated by eq ES5B.
the experimental results for a number of catalysts enjoy a modicum of chemical common sense. For example, the trend of increasing KA as the alkane chain length grows is associated with the rising thermochemical ease of dehydrogenation (18); there is a related but slightly less clear trend for the alkane to reach the necessary extent of dehydrogenation by loss of fewer H atoms. Figure 1 shows how the inhibiting effect of H2 pressure, coupled with the location of the rate maximum, correlates with intuitive expectations based on the constants in Table 3. We have also noted above that the apparent activation energy Ea decreases with increasing chain length (2, 4, 7) and that this is probably in consequence of a decreasing enthalpy change ∆HAQ for the dehydrogenation. Thus the true activation energy Et is given by Q
Et ) Ea - ∆HA
a relation first postulated by Temkin in 1935 (30). By computing the values of k1, KA, and KH at three or more temperatures and applying either the Arrhenius or van’t Hoff equation (Figure 5), we have obtained values of Et in the range 40-80 kJ mol-1 and of ∆H between 60 and 90 kJ mol-1. The measured value of Ea depends,
Figure 5. Arrhenius plot of rate constant k (filled points) and van’t Hoff plot of KA (open points) using values obtained by eq ES5B applied to the results for the hydrogenolysis of C3H8 (∆) and of n-C4H10 (O) on Pt/Al2O3 (5).
however, upon the reactant pressures used, because the contributions that the enthalpy terms make in the Temkin equation depend on the size of the pressure terms in the rate expression (18). A “compensation effect” occurs, essentially because of the use of apparent rather than true activation energies. The theoretical framework outlined above would lead us to expect that k and KA should both increase with a rise in temperature, which they do, and that KH should decrease, which it tends only to do at the higher temperatures. To this extent, therefore, eq ES5B largely fits with expectation; the higher KH values seen at low temperatures (2, 5) may arise from a limitation in the accuracy of optimization process or in the experimental data. 7. Possible Future Refinements to Rate Equations We must, however, restrain our excitement at the apparent usefulness of a single rate equation in describing the results and in enlarging our understanding of the reaction mechanism. Equation ES5B does not fit all the experimental results equally well (6), and if we accept Popper’s dictum, we can only disprove the underlying hypothesis but never wholly substantiate it.
3178 Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997
There must be an infinite progression of “better” rate expressions, but none is or can be perfect. We stop hunting when the law of diminishing returns starts to apply. It is nevertheless important to review the mechanism which produces such an acceptable rate expression and to examine the validity of the hidden assumptions which it contains. The mechanism shown in Scheme 1 contains the following implications. (1) The total number of “sites” available to the reactants in the forms postulated in the mechanism is always the same and, specifically, is independent of reactant pressures and temperature. This is unlikely to be true because it is very clear that alkanes quickly react with clean metal surfaces to form a “carbonaceous deposit” by loss of more H atoms than is required to form the species reactive in hydrogenolysis and which is more strongly bound to the surface by a greater number of C-M bonds (2, 3, 10, 11). In the case of Pt catalysts, on which most work has been done, the deposit formed below ∼600 K probably contains species having the same number of carbon atoms as the reactant and corresponds to the “Pt-C-H” state proposed by Sa´rka´ny and Paa´l (7, 10). This may be in dynamic equilibrium with the less hydrogenated reactive species; indeed, rates obtained with standard reactant pressures interspersed in a series in which H2 pressure is altered change in a way that suggests this. Thus, high H2/alkane ratios tend to suppress deposit formation, and low ratios encourage it. Quantitative estimates of the coverage by carbonaceous deposit are possible by using Auger (30) or photoelectron spectroscopy (10), but the information has not yet been quantitatively introduced into a formal kinetic treatment. The amount of deposit formed increases with alkane chain length and is often minimal with ethane; with Ru/ Al2O3 catalysts, it also depends on the pretreatment used (1, 2, 28). In Scheme 1, the reactive species is shown as held to the surface by a single C-M bond; it is more likely that the number of such bonds formed equates to the number of H atoms lost. Therefore, perhaps the alkane chemisorption step should be formulated as, for example,
C2H6 + 2(6 - m)* f C2Hm(6 - m)* + (6 - m)H* (ii) One modification of eq ES5, coded eq ES5C, assigns two sites to the intermediate, but it does not appear to have been tested. It is, in any event, doubtful whether it is possible to distinguish between the fits obtained when small changes to the H2 pressure exponents are made. A final likely point of weakness in the mechanism as portrayed in Scheme 1 is the assumption that all sites shown by asterisks are equal; some may be more equal than others. To be specific, it is improbable that the sites at which C-M bonds are formed are the same as those to which H atoms are chemisorbed. H atoms may not be fussy about where they reside, and indeed, they will probably be highly mobile under reaction conditions. The alkane and H2 molecules are thus in competition only in the sense that both require some H-type sites. The alkane in addition needs A-type sites. Thus,
C2H6 + (6 - m)A + (6 - m)H f C2Hm(6 - m)A + (6 - m)HH (iii) 8. Inherent Dilemmas in Kinetic Modeling It is evident that the quantitative incorporation into the rate expression of (i) an active area that is itself a
Figure 6. Hydrogenolysis of n-C4H10 on Rh/TiO2 at 429K: comparison of direct (O) and semilogarithmic (0) plots of rate vs H2 pressure (6).
function of reactant pressures and (ii) differentiated sites for reaction intermediates would each demand at least one additional variable parameter. We have thus far adopted the stance that the greater the number of variables used, the closer will be the fit obtainable with the experimental results: as the great mathematician Cauchy said, “With five constants I can draw an elephant; with six I can make its tail wag”. Clearly, from what has been shown above, a more complex rate expression is not necessary to fit the results: indeed, the worry is that overly simple expressions are so successful. We have therefore not been attracted to undertake further modification of a reaction mechanism, as in Scheme 1, that is self-evidently imperfect, because we can forsee (given sufficient data points of sufficient reliability) we shall inevitably get better fits, but it is not certain whether the constants so engendered will be any more meaningful than those we already have. We believe that the elasticity of the constants’ values, i.e., the rate at which standard deviation changes as the value of any one constant is altered, decreases as the number of constants used increases and, therefore, so does their significance. Further progress in kinetic analysis in this field would seem to hinge on two factors: (i) more accurate experimental results, and (ii) improved optimization routines. Concerning i, measured rates often need correction by some arbitrary procedure to allow for changes in activity, caused by growth in the carbonaceous deposit, while the level of the selected variable is randomly altered (29). Of the lower alkanes, ethane causes the least deactivation, but it is unsatisfactory to use in that the maximum rate resulting from the change in H2 pressure is frequently not accessible. The location of this maximum and the associated change of the slope greatly assists the optimization process and improves the reliability of the computed constants; fewer constants are needed to describe a single monotonic function. Perhaps for this reason also, our attempts to perform optimization on the logarithms of the rate and H2 pressure have not been helpful, as the resulting plot is often linear over much of its length. We were motivated to do this in order to be able to visualize more clearly the differences between the observed and calculated rates over the whole range of rates, which could vary by factors of up to 500 as the H2 pressure was changed (see Figure 6). This procedure also gave greater but
Ind. Eng. Chem. Res., Vol. 36, No. 8, 1997 3179
undue weight to less accurately measurable very low rates. Concerning ii, we have used the Levenburg-Marquardt algorithm, manually altering the value of m until a minimum standard deviation was obtained, yielding the best-fit values of all constants. Normally, each constant was initially set at unity, but the final values depended somewhat on the initial values taken and were not exactly reproducible. Nevertheless, the variations do not obscure the trends shown in Table 3, but some improvement in the speed and accuracy of the computation would be very desirable. Conclusions The comprehensive and quantitative description of each and every unit step occurring in the reactions of alkane on the surfaces of metal catalysts would require such a number of constants that the fitting of the resulting complex rate expression to experimental results would be unlikely to lead to reliable numerical values of these constants. It is already clear that the optimum solution of an equation of the type as eq ES5B does not depend critically on the value of one of the constants (viz., m), and the greater the number of constants, the larger will be the elasticity of their values obtained by the optimization process. While the construction of more complex equations is easy and their solution for selected numerical values of the constants presents no problems, judgement as to their suitability to account for experimental observations is not a straightforward matter; a simple standard deviation test is certainly not adequate. More accurate experimental data will be needed and improved optimization routines, before further progress can be made: but there is an inherent dichotomy between realism and practicality in the modeling of complex chemical processes. Acknowledgment I gratefully acknowledge the European Commission’s support of the work of Dr. J. C. Slaa under Programme SC1*(T91-0681), and the assistance of Mr. A. D. Hooper in checking a number of the calculations. Our interest in kinetic modeling has been stimulated by the distinguished work of Prof. Gilbert Froment. Nomenclature Lower Case Letters a ) n + 1 - m/2 (number of H2 molecules released from alkane to form reactive species) k ) rate constant m ) number of H atoms remaining in reactive form of alkane n ) number of H atoms initially in alkane rh ) rate of hydrogenolysis ri ) rate of isomerization y ) KAPA Upper Case Letters Ea ) apparent activation energy Et ) true activation energy KA ) equilibrium constant for dehydrogenative chemisorption of alkane KH ) equilibrium constant for chemisorption of H2 PA ) pressure of alkane PH ) pressure of H2
Greek Letters R, β ) orders of reaction in eq ES1 γ, δ, κ ) exponents of H2 pressure in eqs ES2, ES2A, and ES3, respectively θA ) surface coverage by reactive form of alkane θH ) surface coverage by H atoms θ* ) surface coverage by vacant sites Other * ) active site A ) active site specific to alkane (for C-M bond formation) H ) active site specific to H atoms (for H-M bond formation)
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Received for review October 21, 1996 Revised manuscript received February 4, 1997 Accepted March 26, 1997X IE960671Z
X Abstract published in Advance ACS Abstracts, June 15, 1997.
