Apparent reaction orders

A plot illustrating that the order with respect to [H+] varies with [H+] during the reaction in which water is substituted for chloride ion in the complex (H20)5CrCl2+, Eq. (1-16). The ordinate shown is the (normalized) reaction rate, as in Eq. (1-18), and the abscissa is [H+], Both quantities are displayed on logarithmic scales. The apparent reaction orders with respect to [H+] are the limiting slopes, -1 at low [H+] and 0 at high [H+],... 

The apparent reaction order of carbon-catalyzed methane decomposition reaction was determined to be 0.6 0.1 for AC (lignite) and 0.5 0.1 for CB (BP2000) catalysts. Thus, the rate equation for carbon-catalyzed decomposition of methane can be written as follows ... 

Figure 3 shows the results of varying the CO pressure. The maximum activity appears to lie near 600 psi for benzaldehyde reduction. Figure 3 is an attempt to elucidate an apparent reaction order with respect to the arithmetically averaged CO pressure. At pressures less than 400 psi, the order is nearly first order. The situation at higher pressures is not clear however, it is reasonable to speculate that the dominant aspects of the kinetics shift at these pressures. The data suggest the shift is to zero-order dependance. 

In this work we attempt to measure kinetics data in a time short compared with the response time of the catalyst stoichiometry. An alternative is to measure kinetics in a true steady state, i.e., to increase the line-out time at each reactor condition until hysteresis is eliminated. The resulting apparent reaction orders and activation energies would be appropriate for an industrial mathematical model of reactor behavior. 

The above findings also provide the basis for an interesting computation relating to the apparent reaction order for nucleation. We may consider the case where we assume that all of the protein is exclusively... 

From this type of analysis, one would conclude that t must be approximately 28 for a 10% reduction in protomer to cause a 95% reduction in the nucleus concentration. This is a rather startling apparent reaction order even assuming infinite cooperativity between protomers. It is recalled that Hofrichter et al. (1974) found from a similar analysis of the rate of nucleation of human hemoglobin S (HbS) at 30 C that the apparent reaction order for the nucleation of HbS aggregation was about 32. Of course, such analyses are not fully justifiable because one may not assume ideality in the solution properties of biopolymers at high concentrations, particularly at 200 mg/ml in the case of hemoglobin. The computation for the case of tubulin polymerization does, nonetheless, emphasize that nucleation would be an especially cooperative event if only tubulin, and not ring structures, played the active role in nuclei formation. 

One would conclude that / must approximately equal 28 for this process Hofrichter et al found a similar behavior in nucleation of human hemoglobin S (HbS) the apparent reaction order for the nucleation of HbS aggregation was about 32 (See Hemoglogin S Polymerization). Of course, such analyses are not fully justifiable, because one cannot assume ideality in the solution properties at high protein concentrations (See Molecular Crowding). 

The preceding sections have shown the complexity of solvent effects in the solvolysis of acyl chlorides, and how ambiguities in the role of the solvent, particularly in its apparent reaction order, critically affect the assignment of detailed mechanism. It is the intention in this brief section to point to some of... 

This shows that the half-life increases in direct proportion to the initial pressure. This would indicate an apparent reaction order of zero. Expressed loosely, the reaction appears to be faster at low pressures than at high pressures. This is illustrated by the curves in Fig. 10. 

Unlike the double layer effect on the transfer coefficient, which presents a maximum at the pzc, the double layer effect on the reaction order is zero at the pzc. Inspection of Fig. 3 shows that the derivative (9A02/9 In [O] )e adopts positive and negative values, respectively, at both sides of the pzc, so that the apparent reaction order may be smaller or larger than the true order depending on the electode potential in the rational scale. 

Again the apparent reaction order is 2. Of course if Ak is kept fixed, the apparent reaction is n, for, with only u - >,... 

The derivation of a similarly atypical rate expression is required for the simulation of the electrochemical behavior encountered in electrohydrodimerization studies. In these studies, the variation of the bulk concentration of the olefin (e.g., ethyl cinnamate, diethyl furmarate) reveals that there is a concentration dependence to the reaction order associated with the dimerization of the electrogenerated radical ion [33]. This variation in apparent reaction order with concentration can only be attributed to a two-step mechanism [25] involving two independent rate or equilibrium processes. A mechanism that meets this criterion and appears to fit the electrochemical data is the preequilibrium mechanism [36] in which the electrogenerated radical ions first engage in an equilibrium dimerization before the rate-determining ring closure of the dimer takes place. Symbolically, this mechanism may be written ... 

Fig. 30. Apparent reaction order plot for oxygen desorption from Ir(110) as function of coverage (124).

The apparent HDM reaction orders greater than unity have been attributed to the presence of more than one class of metal compounds reacting with different rates (Oleck and Sherry, 1977 Cecil et al., 1968). Just as in hydrodesulfurization, the simultaneous occurrence of several first-order reactions with different rates can lead to an apparent reaction order greater than unity (de Bruijn, 1976). Wei and Hung (1980) theoretically demonstrated conditions whereby two first-order reactions give rise to apparent second-order kinetics. 

Figure 8.2. Rates of manganese oxide reductive dissolution by 1.00 x 10 4 M oxalate as a function of pH. Reactions were performed in 5.0 x 10 2 M NaCl using either acetate (O) or constant -Pco2 (P) buffers. ([MnOx]0 is 4.81 x 10 5 M.) Numerical values are apparent reaction orders with respect to [H+], [From Stone (1987a), with permission.]...

The reaction rate is close to first order with respect to P at the higher temperatures but the apparent reaction order decreases with decreasing temperature. It was found that the kinetic data could be described by the rate expression... 

The difference in the apparent reaction orders using the two techniques is probably due to the fact that the concentration profiles of reactants and intermediates depend upon the time scale. Thus, differences can be expected for concentration dependent competing mechanisms. In general terms where A represents a diactivated olefin, the major mechanism of dimerization in solvents of low water content was proposed to be given by eqns (121)-(122)... 

As seen in Table 2.1, the overall order of an elementary step and the order or orders with respect to its reactant or reactants are given by the molecularity and stoichiometry and are always integers and constant. For a multistep reaction, in contrast, the reaction order as the exponent of a concentration, or the sum of the exponents of all concentrations, in an empirical power-law rate equation may well be fractional and vary with composition. Such apparent reaction orders are useful for characterization of reactions and as a first step in the search for a mechanism (see Chapter 7). However, no mechanism produces as its rate equation a power law with fractional exponents (except orders of one half or integer multiples of one half in some specific instances, see Sections 5.6, 9.3, 10.3, and 10.4). Within a limited range of conditions in which it was fitted to available experimental results, an empirical rate equation with fractional exponents may provide a good approximation to actual kinetics, but it cannot be relied upon for any extrapolation or in scale-up. In essence, fractional reaction orders are an admission of ignorance. 

Reactions in which a step competes with an earlier reversible step are the simplest that may show features not found in single-step reactions The rate equation may not be a power law, the apparent reaction order may be fractional and vary with conversion, and the activation energy may be negative (lower rate at higher temperature). 

The simplest one-plus equation with the established apparent reaction orders and first order in catalyst is... 

The effect just described produces an apparent reaction order in H30+ that is less than one and may even become negative at very high acid concentrations. The opposite behavior, an apparent order higher than one, is also found in a large... 

If the amounts of the catalyst ingredients (cobalt, phosphine, and base) are increased by the same factor without change in pressure and H2-to-CO ratio, the increase in phosphine concentration uncompensated by an increase in CO pressure shifts ligand exchange equilibrium toward HCo(CO)3Ph. Without base, this shift is at the expense of the more active HCo(CO)4, producing an apparent reaction order lower than one in "catalyst." In the presence of base, the shift is mostly at the expense of the inactive Co(CO)4, resulting in an apparent order higher than one. 

As has already been pointed out, any rate equation containing the concentration of the free catalyst is of little practical use if that concentration is unknown, is difficult or impossible to measure, and may vary with conversion, as is the case if a significant fraction of the total catalyst material is present in the form of intermediates of the reaction. This is often true in catalysis by enzymes or other trace-level catalysts. To be sure, the equations in terms of free-catalyst concentration remain correct. However, unless practically all the catalyst material is present as free catalyst, they no longer reflect the actual reaction orders. This is because the concentrations of the participants affect the rate not only directly as expressed explicitly in those equations, but also indirectly and implicitly through their effect on the free-catalyst concentration As the reactant concentration decreases, so do those of the intermediates in turn, this produces an increase in the free-catalyst concentration that boost the rate and, thereby, decreases the apparent reaction order. To reflect this facet correctly, what is needed are rate equations in terms of the total amount of catalyst material, a quantity that is constant and known. 

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