A Simple Graphical Method to Determine the Order in Catalyst

Abstract A graphical analysis to elucidate the order in catalyst is presented. This analysis uses a normalized time scale, t [cat]T n, to adjust entire reaction profiles constructed with concentration data. The method is fast and simple to perform because it directly uses the concentration data, therefore avoiding the data handling that is usually required to extract rates. Compared to methods that use rates, the normalized time scale analysis requires fewer experiments and minimizes the effects of experimental errors by using information on the entire reaction profile.

Mechanistic studies of catalytic reactions have become more common in academia and industry owing to their value for improving processes and also thanks to the availability of new technology to easily monitor the progress of ar eaction. Then ew reaction monitoring techniques can generate abundant, good-quality data during the entire course of areaction, but very few methods have been developed to exploit these features to extract mechanistic information. [1] Herein, as imple graphical analysis that uses all the reaction profile data to establish the order in catalyst is reported.
Thecurrently available analyses to determine the order in catalyst use rate data. Forinitial rates,two main analyses are performed:i nitial rates [2] against [cat] T n (T = total) and loglog plots of the initial rates against the analytical concentration of the catalyst [3] (Figure 1a). Forr ates directly measured during the course of ar eaction or derived from fitted functions of concentration data, the normalized rate against the concentration of as pecies is used (Figure 1b). [4] Herein, am ethod to determine the order in catalyst without the need for rate data by directly comparing reaction concentration profiles is described (Figure 1c).
Although all the analyses hitherto available for determining the order in catalyst use rate data (differential data), few experimental techniques acquire this kind of data directly, and their use is limited because of the intrinsic characteristics of most reactions.T here are often two alternative ways to obtain rate data from traces of concentration against time (integral data). Them ost common one,w hich is based on initial rates,assumes alinear variation in the concentration of reactants at the beginning of ar eaction. This method only uses the data obtained at low conversions or short reaction times, [2,3] discarding the data from the rest of the reaction and therefore disregarding the associated intrinsic information. Thes econd method fits the concentration data to ap reselected function depending on arbitrary parameters,w hich is further differentiated to mathematically estimate the instantaneous rate at different reaction times. [4] Thea rbitrary preselection of af unction can bias the results,a nd the use of ag eneral mathematical function, such as high-order polynomial functions,can create artifacts in the rate.
Theg raphical analysis presented in this paper plots the raw (or primary) data of [A] against anormalized time scale, t [cat] T n (Figure 1c). Thea djustment of the time scale for experiments with different catalyst loadings makes the direct comparison of concentration profiles possible.T he chosen normalization is theoretically based on the fact that the catalyst concentration is constant during the course of the reaction. Therefore, t [cat] T n becomes one of the parameters of the function that describes the concentration of ar eagent at each time point, independently of the complexity of the function. Effectively,t he time scale method compresses all traces proportionally to the catalyst loading without altering their shape. [5] Thetime normalization is performed by multiplying each time point by the total concentration of catalyst used in each experiment raised to an arbitrary power. This power value should be adjusted until all the corrected conversion curves overlay.T his overlay occurs independently of the complexity of the reaction kinetics or changes in the kinetic regime.T he graphical interrogation of kinetic parameters has been popularized by Blackmond and co-workers in the context of the reaction progress kinetic analysis (RPKA). [4a, 6] This kind of analysis has gained wide acceptance both in the academic and industrial communities because it is simple to apply and leads to an intuitive interpretation of the results. Figure 2a shows how the concentration reaction profiles of as imulated Michaelis-Menten system run with different catalyst loadings change when the scale is normalized for different assumed orders in catalyst. Thedifferences between the normalized concentrations are more pronounced in the late stages of the reaction because the effect of different catalyst loadings in the profiles is accumulative.
Thenormalized time scale method has several advantages over those that involve rate data. Thed irect comparison of concentration profiles saves time and work because it avoids the previously necessary data handling to extract rate data. Thed ata treatment to extract rates from the raw concentration data can depend greatly on the treatment method. By avoiding this treatment, the data is presented in am ore compact way,and the reproducibility is increased.
Moreover,t his analysis method requires fewer experiments with different catalyst loadings because it directly compares several points for the entire reaction profile,instead of comparing single points for each reaction, as is the case when using initial rates.Owing to this multipoint comparison, experiments in which only af ew data points have been collected, which are intractable with the rate analysis method, can be successfully analyzed. Figure 2b shows how the order in catalyst can be elucidated with just two traces with four data points each. This characteristic is especially attractive when in situ techniques are not available,and data therefore have to be collected by consecutive sampling or quenching independent reactions.I ns uch cases,i ti sd ifficult to collect enough points at short reaction times to extract initial rates or to derive af unction with such al ow density of points during the entire course of the reaction.
Thenormalized time scale method is especially beneficial compared to analyses that use rates when the measurement of concentrations contains relatively large errors or there are outliers.H uman visual analysis is exceptionally potent in identifying trends that are part of continuous profiles and in minimizing the effect of random experimental errors in single points.F igure 2c exemplifies this feature by using the same data as in Figure 2a;h owever,ar andom error normally distributed with as tandard deviation of 0.05 has been added to the concentration values.E ven with such al arge error associated with the data points,i ti sp ossible to see that the concentration profiles overlay when the time scale is normalized by the total concentration of catalyst raised to the correct power. Conversely,t oc ompensate for the statistical error when initial rates are used to determine the order in catalyst, it would be necessary to use data out of the initial range of concentrations with linear behavior.D espite all these qualities of the method, there are some caveats that should be taken into consideration. To perform Figure 2. The correct order in catalyst is the one that causes all the curves to overlay (in this case first order). [7] the visual analysis,t he normalized abscissa axis should be rescaled to approximately the last value available.The graphs for different orders in catalyst can thus be fairly compared.
Owing to the use of visual analysis,there is no mathematical function to describe the error in the determination of the order.I nstead, ar ange of orders leading to ag ood overlay could be given if necessary.J ust as with other analysis methods,itisnot possible to determine the order in catalyst if the quantity of catalyst is unknown or if it changes in an unknown way during the reaction. This problem is particularly important when there are fast catalyst deactivation processes that are due to the presence of impurities in comparable concentrations to the analytical concentration of the catalyst. Thenormalized time scale method is useful to determine any order in catalyst. Tw owell-known catalytic systems have been chosen to illustrate its potencyi nr eal cases:t he hydrolytic kinetic resolution of terminal epoxides catalyzed by cobalt salen complexes [8] and the Heck coupling reaction using palladacycle catalysts. [9] Theh ydrolytic kinetic resolution of terminal epoxides involves acooperative mechanism where two discrete catalyst molecules interact in the rate-determining step of the reaction. [8] Therefore,t he reaction has as econd-order dependencyo nt he catalyst concentration. Figure 3s hows the results that were obtained by applying the normalized time scale method to the corresponding catalytic network, using the kinetic values that had previously been reported. [8c] Three different traces corresponding to catalyst loadings of 0.85, 0.60, and 0.43 mol %a re shown in analogy to the conditions reported in the literature,a lthough two of them would be enough to determine the second order in catalyst.
Amore challenging case to determine the order in catalyst is the Heck coupling presented in Figure 4. [9] Thef avorable formation of an inactive off-cycle catalytic dimer that is in fast equilibrium with the corresponding active monomer has been reported. Owing to this equilibrium, the monomer concentration is not linearly proportional to the total concentration Figure 3. The normalizedt ime scale method shows that the hydrolysis of terminal epoxides is second order in the Co III salen complex. [7] Figure 4. The normalizedt ime scale method enables the differentiation of small changes in the order in catalyst for different concentrations of catalyst. [7] of catalyst added, and therefore the order in catalyst depends on the catalyst concentration. Theo rder in catalyst can only vary between first order for very small concentrations of palladium and half order for very high concentrations of palladium. [5] As shown in Figure 4a,t he theoretical order in catalyst would be around 0.86 for catalyst concentration from 10 À5 to 210 À5 m and 0.55 for concentrations from 10 À3 to 210 À3 m.F igure 4b shows how the normalized time scale method is used to determine the correct order in catalyst, even for these very similar palladium concentrations.T he order in catalyst in such mechanistic scenarios is ag ood indicator of the distribution of the catalyst between monomeric and dimeric species;o rders close to one indicate that amajor percentage of the catalyst is present as the monomeric species,w hereas orders in catalyst close to 0.5 indicate that most of the catalyst is present as the inactive dimer.
In conclusion, ap ractical and powerful method to elucidate the order in catalyst has been presented. The normalized time scale method uses the concentration of as ubstrate at different time points,t hus circumventing the necessity of measuring or deriving rate data. It fully exploits the potential of currently available in situ spectroscopic techniques.This analysis method is simple and fast, it requires fewer experiments than traditional analyses and can even handle sets of data with reduced numbers of points or large random experimental errors.F or all of these reasons,t he method is expected to attract widespread acceptance and become the preferred option for determining the order in catalyst when using concentration profiles.