Kinetics concurrent first-order reactions

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

Elder [45] has modelled several multiple reaction schemes, including mutually independent concurrent first-order reactions, competitive first-order reactions, mutually independent n-th order reactions, and mutually independent Avrami-Erofeev models with n = 2 or 3. The criteria identified for recognizing the occurrence of multiple reactions were (i) the apparent order of reaction, n, varies with the method of calculation, and (ii) the kinetic parameters, A and vary with the extent of reaction, a. 

Reactions described by other kinetic routes may be treated in similar fashions. Although, for reasons already explained in Sect. 4.1, mass transfer effects will not influence the selectivity of two concurrent reactions arising from the same reactant, heat transfer between fluid and solid does have an affect. Thus for the first-order reactions... 

The degradation follows an apparent first order reaction with an initial rate (up to 2000 hours exposure in a QUV apparatus) that is faster than the subsequent rate. An assessment of the kinetic data in terms of proposed reaction mechanisms and concurrent changes in properties such as dynamic mechanical behavior can serve as a preliminary basis for evaluation of the material s ability to retain useful properties for time periods consistent with certain design requirements for solar energy system applications. 

The pyridine-catalysed lead tetraacetate oxidation of benzyl alcohols shows a first-order dependence in Pb(OAc)4, pyridine and benzyl alcohol concentration. An even larger primary hydrogen kinetic isotope effect of 5.26 and a Hammett p value of —1.7 led Baneijee and Shanker187 to propose that benzaldehyde is formed by the two concurrent pathways shown in Schemes 40 and 41. Scheme 40 describes the hydride transfer mechanism consistent with the negative p value. In the slow step of the reaction, labilization of the Pb—O bond resulting from the coordination of pyridine occurs as the Ca—H bond is broken. The loss of Pb(OAc)2 completes the reaction with transfer of +OAc to an anion. 

Since the propensity to form adducts in chemistry is high and these adducts undergo a variety of reactions, the rate law (1.98) is quite common. This is particularly true in enzyme kinetics. In reality, these reaction schemes give biphasic first-order plots but because the first step is usually more rapid, for example between A and B in (1.101) we do not normally, nor do we need to, examine this step in the first instance. The value of A", in (1.107) obtained kinetically can sometimes be checked directly by examining the rapid preequilibrium before reaction to produce D occurs. In the reactions of Cu(I) proteins with excited Cr and Ru polypyridine complexes, it is considered that (a) and (b) schemes may be operating concurrently. 

The reaction was found to be first order with respect to amines and acrylamides and no base catalysis was observed. The reaction is believed to occur in a single step in which the addition of amine to Cp of acrylamide and proton transfer from amine to Ca of acrylamide take place concurrently with a four-membered cyclic transition-state structure. The Hammett (px) and Brpnsted (/3X) coefficients are rather small, suggesting an early transition state (TS). The sign and magnitude of the cross-interaction constant, pxy(= —0.26), is comparable to those found in the bond formation processes in the. S n2 and addition reactions. The normal kinetic isotope effect ( h/ d > 1.0) and relatively low A and large negative Avalues are also consistent with the mechanism proposed.192... 

In order to avoid the restrictions to complicated adsorptive reactions in the MOC3D, Selim et al. (1990) developed a simulation system based on the multireaction model (MRM) and multireaction transport model (MRTM). The MRM model includes concurrent and concurrent-consecutive retention processes of the nonlinear kinetic type. It accounts for equilibrium (Freundlich) sorption and irreversible reactions. The processes considered are based on linear (first order) and nonlinear kinetic reactions. The MRM model assumes that the solute in the soil environment is present in the soil solution and in several phases representing retention by various soil... 

Since the use of equilibrium (Freundlich) type with n > 1 is uncommon, we also attempted the kinetic reversible approach given by equation 12.2 to describe the effluent results from the Bs-I column. The use of equation 12.2 alone represents a fully reversible S04 sorption of the n-th order reaction where kj to k2 are the associated rates coefficients (Ir1). Again, a linear form of the kinetic equation is derived if m = 1. As shown in Figure 12.7, we obtained a good fit of the Bs-I effluent data for the linear kinetic curve with r2 = 0.967. The values of the reaction coefficients kj to k2, which provided the best fit of the effluent data, were 3.42 and 1.43 h with standard errors of 0.328 and 0.339 h 1, respectively (see Table 12.3). Efforts to achieve improved predictions using nonlinear (m different from 1) kinetics was not successful (figures not shown). We also attempted to incorporate irreversible (or slowly reversible) reaction as a sink term (see equation 12.5) concurrently with first-order kinetics. A value of kIIT = 0.0456 h 1 was our best estimate, which did not yield improved predictions of the effluent results as shown in Figure 12.7. 

The greatest ratio of the relative, apparent rate constant pvim/ im( = 390) is observed for the reaction system where the hydroidiobic interaction is expected to be the greatest (long acyl-chain substrate in low alcohol medium). The transformation of the kinetic behavior from the pseudo-first order to tte Midiaelis-Menten type also paralleled the increasing hydroj obic interaction. The acceleration behavior mentioned above is concurrent mth this transformation and explicable by the increa d hydro diobic interaction between long-chain substrates and the partially acylated polymer (see also Table 4—3). 

Table 4.7 shows a summary of atom-plus-molecule recombination reactions whose kinetics have been studied in discharge-flow systems the list is not exhaustive only selected data are tabulated. Since these recombination reactions are first-order in [A], only relative concentrations need be measured in order to obtain values for the rate constants. This fact, and the elimination of concurrent atom recombination processes by use of the fixed observation point method, account for the relatively large number of detailed rate studies in the literature. The superficially simple atom + O2 reactions are, for various reasons, more complex than the atom -)- NO reactions. The three reactions H -H NO -f M, O -j- NO -1- M and Cl -1- NO + M possess the following simple mechanism in the 1 torr total pressure region... 

Since more than one of these dissolution processes might occur in the coal extraction experiments, it is necessary to allow for concurrent chemical reactions when constructing a rate equation. Since reactions are either first or second order, a kinetic expression having concurrent reactions of first and second order must be derived. 

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