Determination of kinetic parameters

Kinetics of HDS of heavy oils is typically modeled as a power-law type. Reaction order for sulfur removal is commonly reported to be in the range of 1-3, being in general the lowest values for light distillates and the highest values for heavy feeds. 

The value of reaction order depends mainly on the type and concentration of the sulfur compounds present in a petroleum distillate. The reports in the literature about kinetics of HDS of heavy fractions is limited for instance, Ozaki et al. (1975) used a simple power-law kinetic model to interpret HDS kinetic data obtained from a down-flow fixed-bed reactor of atmospheric residue of Kuwait crude oil and found that apparent reaction order undergoes a change as the reaction temperature is increased. At 380°C and 410°C, the reaction orders were reported to be 2.4 and 1.55, respectively. Later, Kam et al. (2008), by using a procedure based on statistics arguments, have adjusted the Ozaki et al. (1975) data to a second reaction order for all temperatures. 

The reaction rate of HDS has also been modeled with the following Langmuir-Hinshelwood type of expression (Korsten and Hoffmann, 1996)  

In the present work, to take into consideration the inhibiting effect of HjS on HDS reaction. Equation 9.16 was used. The value of was obtained by minimizing the error between predicted and experimental data of sulfur removal. The reaction order for hydrogen nQ was fixed to be 0.5 according to several reports in the literature (Ross, 1965 Mederos, 2010). The value of nii = 0.5 is attributed to the dissociation of H2 onto the catalyst sites. Also, the denominator of Equation 9.16 raised to the seeond power is due to reaction on two consecutive sites over the catalyst surface. 

Apart from HDS, another important reaction during hydrotreating of heavy crude oils is hydrodemetallization, particularly the removal of nickel (HDNi) and vanadium (HDV). 

Kinetic parameters for sequential mechanisms can be conveniently determined from the parametric Eq. 3.60. Experimental design consists in a matrix in which initial rate data are gathered at different concentrations of both substrates (a and b) as depicted in Table 3.6. 

If initial rate data are linearized using the double reciprocal plot of Lineweaver-Burke, from Eq. 3.60  

By plotting 1/v versus 1/a at constant b (columns in Table 3.6) values of apparent kinetic parameters Vap and Kap are obtained from the intercepts in the Y and X-axis respectively, as shown in Fig. 3.8 for ordered (I) and random (II) mechanisms. By plotting 1/v versus 1/b at constant a (rows in Table 3.6) values of apparent kinetic parameters Vap and Kap are obtained from the intercepts in the Y and X-axis respectively, as shown in Fig. 3.8 for ordered (I) and random (II) mechanisms. Ordered sequential mechanism can be easily distinguished from random sequential in Lineweaver-Burke plots in the case of ordered mechanism, intercept in the Y-axis (1/Vap ) is a constant (1/V) independent of a (the substrate who binds to the enzyme first), while in the case or random mechanism it depends on a (see Table 3.5). 

Once the mechanism has been identified, the corresponding kinetic constants can be obtained from the expressions in Table 3.5. 

In the case of ordered mechanism, V is directly obtained from the 1/v versus 1/b plot. Straight lines in the 1/v versus 1/a plot will intersect at a point that is easily demonstrated to correspond to 1/Ka. However, all kinetic parameters can be obtained from secondary plots as shown in Fig. 3.91. In the case of random mechanism, as in ordered mechanism, straight lines in the 1/v versus 1/a plot will 

The model parameters and found in Equation 4.3 can be determined using sev- 

Supercritical Fluids Technology in Lipase Catalyzed Processes 

The plot of the reciprocal initial rate wrses reciprocal substrate concentrations 

To overcome the limitation of the Lineweaver-Burk method, another graphic method for and determination has been proposed, where linearization is achieved by the mnltiplic ipn of both sides of Equation 4.23 by [5], as shown 

Another approach is to use a nonlinear regression technique, which produces a weighted least square that maximizes the efficiency of the parameter estimation. This technique does not require linearization and can be used to determine multiparameters. Hernandez and Ruiz (1998) developed an Excel template for the calculation of enzyme kinetic parameters using this technique. 

The kinetic parameters of the reaction rate are the rate constants k and the orders a, b, n) of the reaction in relation to each component. The effect of the temperature is incorporated in the reaction rate and in order to determine it, the activation energy E and the frequency factor ko should be determined. 

There are two methods integral and differential. The integral method has an advantage of having an analytical solution. The differential method has an approximate or numerical solution. For all cases, experimental data obtained in laboratory are necessary, both for batch and continuous systems. 

Selection of the kinetic model a reaction with defined order (integer or fractional), 

Selection of the system batch or continuous (tubular). For both cases, the expression of the rate is substituted, and we obtain a solution of the type  

1 Biochemical Plots Several methods are readily applied to the determination of kinetic parameters and Tmax)- Traditionally, these terms are determined using the classic biochemical plots, particularly those transformed from the well-known Michaelis-Menten plot, for example, Lineweaver-Burk and Eadie-Hofstee plots (Li et ah, 1995 Nakajima et ah, 2002 Nnane et ah, 2003 Yamamoto et ah, 2003). 

FIGURE 13.2 Biochemical plots for the enz5me kinetic characterizations of biotransformation, (a) Direct concentration-rate or Michaelis-Menten plot (b), Eadie-Hofstee plot (c), double-reciprocal or Lineweaver-Burk plot. The Michaelis-Menten plot (a), typically exhibiting hyperbolic saturation, is fundamental to the demonstration of the effects of substrate concentration on the rates of metabolism, or metabolite formation. Here, the rates at 1 mM were excluded for the parameter estimation because of the potential for substrate inhibition. Eadie-Hofstee (b) and Lineweaver-Burk (c) plots are frequently used to analyze kinetic data. Eadie-Hofstee plots are preferred for determining the apparent values of and Umax- The data points in Lineweaver-Burk plots tend to be unevenly distributed and thus potentially lead to unreliable reciprocals of lower metabolic rates (1 /V) these lower rates, however, dictate the linear regression curves. In contrast, the data points in Eadie-Hofstee plot are usually homogeneously distributed, and thus tend to be more accurate. 

FIGURE 13.3 Determination of the potential involvements of multiple enz5mes in a biotransformation pathway using the common biochemical plots. As shown by the plots, (a) Michaelis-Menten plot (b) Eadie-Hofstee plot and (c) Lineweaver-Burk plot, at least two enzjmatic components (El and E2) are responsible for the substrate s biotransformation one high affinity and low capacity, and the other low affinity and high capacity. Of the three plots shown, the Eadie-Hofstee plot most apparently demonstrates the biphasic kinetics due to either multiple enzymes or possibly the deviations from Michaelis-Menten kinetics, that is, homotropic cooperation. 

2 Computational Approach Computational nonlinear regression analysis, in which the data points in the Michaelis-Menten plots are directly fitted, is the preferred approach for analysis of metabolic kinetic data. Such an approach should be utilized as often as possible, given its unbiased nature. Regression analyses for the determination of and Ymax, or CLim (or Kjoj Vmax) 0 described below, for the SigmaPlot (Version 9.0, Syst Inc.) software. 

Representative results of such analysis, in which the data in Table 13.3 were analyzed, are provided in Table 13.4. 

Enzyme activity can be obtained by keeping the substrate concentration constant and by varying the enzyme concentration. The following differences in absorption should be obtained after 2 min reaction  

This variation could be expressed in terms of the concentration of NADH  

Since we have 1 mol of NADH for 1 mol of pyruvate, the enzymatic activity can be expressed as 

The substrate concentrations during the experiment and corresponding initial velocities are listed in Table 3.2. 

The inverse of the Michaelis constant Km is obtained at the intersection of the plot with the x-axis. This yields a value of 

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Compared with the bonding groups (mol%) to aromatic ring of PS, the degree of acylation was observed when MA was used. These results was obtained by determination of kinetic parameters of PS with MA and AA under the same reaction conditions. As shown in Table 5, if the initial rate (Wo) and rate constant (K) of the acylation reaction between MA and AA are compared, the MA is almost 10-14 times higher than AA in the presence of BF3-OEt2 catalyst. This fact is due to the stretching structure of MA and the effect of the catalyst. 

The techniques referred to above (Sects. 1—3) may be operated for a sample heated in a constant temperature environment or under conditions of programmed temperature change. Very similar equipment can often be used differences normally reside in the temperature control of the reactant cell. Non-isothermal measurements of mass loss are termed thermogravimetry (TG), absorption or evolution of heat is differential scanning calorimetry (DSC), and measurement of the temperature difference between the sample and an inert reference substance is termed differential thermal analysis (DTA). These techniques can be used singly [33,76,174] or in combination and may include provision for EGA. Applications of non-isothermal measurements have ranged from the rapid qualitative estimation of reaction temperature to the quantitative determination of kinetic parameters [175—177]. The evaluation of kinetic parameters from non-isothermal data is dealt with in detail in Chap. 3.6. 

Thermal analysis has been widely and usefully applied in the solution of technical problems concerned with the commercial exploitation of natural dolomite including, for example, the composition of material in different deposits, the influence of impurities on calcination temperatures, etc. This approach is not, however, suitable for the reliable determination of kinetic parameters for a reversible reaction (Chap. 3, Sect. 6). 

Kaplan, W., and Zhang, S., Determination of Kinetic Parameters of LPCVD Processes from Batch Depositions, Stoichiometric Silicon Nitride Films, Prac. 11th. Int. Conf. on CVD, (K. Spear and G. Cullen, eds.), pp. 381-387, Electrochem. Soc., Pennington, NJ 08534 (1990)... 

Escribano, J. et al., Characterization of monophenolase activity of table beet polyphenol oxidase determination of kinetic parameters on the tyramine/dopamine pair, J. Agric. Food Ghem., 45, 4209, 1997. 

It would certainly be desirable to evaluate catalyst performance and understand size and stmctural effects directly under the conditions of fuel cell operation. However, determination of kinetic parameters in a single-cell fuel cell is associated with a number of limitations. Let us consider some of them. 

The first set of experiments was conducted in methanol. The substrate concentration was varied from 15 to 50 mM at a 200 pM concentration of 1 for the determination of kinetic parameters for the transformation of 8 into 9. The catalytic rate constant was determined to be 0.04 min and the Michael constant was determined to be 40 mM at 30°C. The rate constant is comparable to those reported for other dinuclear Cu(ll) complexes with a comparable Cu -Cu distance of 3.5 A, but about one magnitude lower than those observed for complexes with a shorter intermetallic distances (12-14), e.g. 2.9 A (kcat = 0.21 min ) (12) or 3.075 A (kcat = 0.32 min (13). The rate constant Aion for the spontaneous (imcatalyzed) oxidation of 8 into 9 was determined to be 6 x 10" min and corresponds to the oxidation without catalyst under otherwise identical conditions. The rate acceleration (Arca/Aion) deduced from these values is 60,000-fold. 

If a two-electron charge transfer reaction takes place in two separate steps, each being accompanied by transfer of a single electron, the mathematical expression for the determination of kinetic parameters becomes more involved and complicated. 

From the derivations in Appendix B, it is evident that the present faradaic rectification formulations for multiple-electron charge transfer not only enable the determination of kinetic parameters for each step of three-electron charge transfer processes but may also be extended to charge transfer processes involving a higher number of electrons. However, the calculations become highly involved and complicated. 

The values of the rate constants obtained are fairly comparable to those given in the literature, and the present technique appears to be quite reliable for determination of kinetics parameters of fast reactions. 

Polarographic studies of organic compounds are very complicated. Many of the compounds behave as surfactants, most of them exhibit multiple-electron charge transfer, and very few are soluble in water. The measurement of the capacitance of the double layer, the cell resistance, and the impedance at the electrode/solution interface presents many difficulties. To examine the versatility of the FR polarographic technique, a few simple water-soluble compounds have been chosen for the study. The results obtained are somewhat exciting because the FR polarographic studies not only help in the elucidation of the mechanism of the reaction in different stages but also enable the determination of kinetic parameters for each step of reduction. 

Illustrations 5.3 and 5.4 indicate how one utilizes the concepts developed in this section in the determination of kinetic parameters for competitive parallel reactions. 

The method yields unique mechanistic information, it permits the determination of kinetic parameters, it allows the determination of the degree of reversi-... 

Yuasa H, Miyamoto Y, Iga T, Hanano M (1986) Determination of kinetic parameters of a carrier-mediated transport in the perfused intestine by two-dimensional laminar flow model Effects of the unstirred water layer. Biochim Biophys Acta 856 219-230... 

Rottenbacher, L., Behlau, L. and Bauer, W., The application of infrared gas analyzers for the fast determination of kinetic parameters for ethanol production from glucose, ]. Biotech., 2 (1985a) 137-147. 

The behavior of the Cd(II)/Cd(Hg) system in the absence and presence of n-pentanol in noncomplexing media was analyzed using reciprocal derivative and double derivative chronoamperome-try with programmed current (RDCP and RDDCP respectively) [54]. The RDCP and RDDCP are very versatile in the determination of kinetic parameters of electrode processes. 

Schoenemann, E., Hahn, H., and Bracht, A. (1991), Determination of kinetic parameters from non-isothermal conductivity measurements by an integral method, Thermochim. Acta, 185(1), 171-176. 

E. Gileadi, G. Stoner, andJ. O M. Bockris,/. Electrochem. Soc. 113 585 (1966). Calculation of errors in the determination of kinetic parameters arising from potential sweep rates that are too high. 

Before NMR spectroscopy and mass spectrometry revolutionized the structural elucidation of organic molecules, UV spectroscopy was an important technique and was used to identify the key chromophore of an unknown molecule. The importance of UV is much diminished nowadays, but it still retains its place in certain applications, such as the determination of kinetic parameters, (the Michaelis constant) and A cat (the turnover rate of an enzyme, in molecules per second), for a number of enzymic reactions and in the analysis of pharmaceuticals. 

For complex mechanisms such as ECE or other schemes involving at least two electron transfer steps with interposed chemical reactions, double electrodes offer a unique probe for the determination of kinetic parameters. Convection from upstream to downstream electrodes allows the study of fast homogeneous processes. The general reaction scheme for an ECE mechanism can be written... 

This section is devoted to the basic kinetics of interfaces in solids. In Chapter 10 we shall work out some ideas in more detail and introduce atomic models for the determination of kinetic parameters. 

Concerning the determination of kinetic parameters of the voltammograms of quasi-reversible and irreversible electrode processes, Fig. 3.10b shows the existence of different linear zones in a similar way to that observed for planar electrodes (see Fig. 3.6). For practical purposes, it is helpful to use spherical microelectrodes, for which a broader linear region is obtained under steady-state conditions, since the process behaves as more irreversible as the radius decreases. For fully irreversible charge transfers, Eq. (3.74) simplifies to... 

A novel approach has been reported for the determination of kinetic parameters at a fixed value of the frequency based on a new feature called amplitude-based quasi-reversible maximum [33]. This term is related to the quasi-reversible... 

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