Parallel reactions limiting cases

In the case of parallel reactions, the fastest reaction will set or control the overall change. In all rate determining cases, the relative speed of the reactions will change with the temperature. This is caused by different energies of activation among the steps in the sequence. This is just one more reason for limiting rate predictions from measurements within the studied domain to avoid extrapolation. 

Looking back over the steps required to derive (5.290), it is immediately apparent that the same method can be applied to treat any reaction scheme for which only one reaction rate function is finite. The method has thus been extended by Baldyga (1994) to treat competitive-consecutive (see (5.181)) and parallel (see (5.211)) reactions in the limiting case where k -> oo.118 For both reaction systems, the conditional moments are formulated in terms of 72(X> and can be written as... 

Finally, when chemical kinetics contrasts with equilibrium, the parallel scheme is not trivial, since one of the products can be favored in the early stages of the batch cycle by faster kinetics and hindered in the later stages by unfavorable equilibrium. Such a case is shown in Fig. 2.4 for parallel reactions of A to Pi via an equilibrium limited reaction and to P2 via an irreversible reaction. 

When the catalyst is immobilized within the pores of an inert membrane (Figure 25.13b), the catalytic and separation functions are engineered in a very compact fashion. In classical reactors, the reaction conversion is often limited by the diffusion of reactants into the pores of the catalyst or catalyst carrier pellets. If the catalyst is inside the pores of the membrane, the combination of the open pore path and transmembrane pressure provides easier access for the reactants to the catalyst. Two contactor configurations—forced-flow mode or opposing reactant mode—can be used with these catalytic membranes, which do not necessarily need to be permselective. It is estimated that a membrane catalyst could be 10 times more active than in the form of pellets, provided that the membrane thickness and porous texture, as well as the quantity and location of the catalyst in the membrane, are adapted to the kinetics of the reaction. For biphasic applications (gas/catalyst), the porous texture of the membrane must favor gas-wall (catalyst) interactions to ensure a maximum contact of the reactant with the catalyst surface. In the case of catalytic consecutive-parallel reaction systems, such as the selective oxidation of hydrocarbons, the gas-gas molecular interactions must be limited because they are nonselective and lead to a total oxidation of reactants and products. For these reasons, small-pore mesoporous or microporous... 

Perhaps the most important single preliminary observation is the influence of the products on the initial rate. If the products have no influence, we have an immediate simplification of eqns. (84) or (85) and a limited number of possibilities for our rate expression. In this case, the reaction is most likely to be a set of parallel reactions with a rate given by eqn. (84), e.g. 

Although the first-order model has met with widespread success, there are conditions under which this model may fail. For example, a more continuous distribution, rather than the extreme duality in reaction rates assumed by the bicontinuum models, may be a more accurate representation of true conditions. Accordingly, the separation of the sorbent into two parallel domains differing in reaction time may be extended to accommodate any number of domains, each with its own unique rate constant. For example, Boesten et al. (1989) have presented a three-site model. The limiting case would be a continuous distribution of domains and associated rate constants. A model describing this case has been developed by Villermaux (1974), where the site population is represented by the transfer-time distribution (i.e., the rate-constant distribution). However, the number of parameters associated with such a model greatly exceeds our present capability to independently evaluate the processes represented by those parameters. Such models would, therefore, be constrained to operation in a calibration mode. 

The same principles apply to the other reactions in Table I. Thus, for polymerization reactions, the molecular weight of the product can be controlled by selective removal of certain constituents from the discharge. If product B is preferred, then further polymerization can be minimized by rapid removal of B. If, on the other hand, the higher molecular weight product C is required, B is allowed to remain in the discharge and C is selectively removed. It frequently happens that the product yield is limited by reverse and parallel reactions. In the latter case, C can be produced in preference to D by selective removal of C. In the former instance, any yield limitations imposed by equilibrium considerations can be overcome by rapid removal of C so that the equilibrium is displaced almost entirely in favor of products. 

In the general case, several of the processes previously considered and illustrated in Figure 14.4.1 can contribute simultaneously to the rate of the reaction. For example, A might be reduced in the film at a rate controlled, not by a single process, but jointly by the parallel processes of its diffusion within the film and its cross-reaction with mediator Q. The overall general mathematical treatment is more complicated than for the various limiting cases discussed in Section 14.4.2 and requires a fuller discussion than can be given here (80). The different processes are represented by the characteristic currents described above ... 

The cases dealt with below refer to those below the Schwab inversion point Eq. (11.12) is also to be taken into consideration. The reaction rate constants run approximately parallel to the heights of the energy barriers E, where E is the absolute value of E. We shall call the reactions limited by the adsorption stage (E = E ) as the first-stage reactions and those limited by the desorption stage (E = E ) as the second-stage ones. 

Next, we rewrite the two codes to handle another limiting case, that of parallel reactions of A to form either B or D ... 

The possible solutions of Equ.(4-157) shown in Figure 4-40 are placed between two parallels. These represent the limit cases either that no reaction takes place and the internal temperature always remains equal to the coolant temperature or that the reaction is performed adiabatically. 

Recently, our group developed and validated a reactor model suitable for design calculations in a thin-gap single-pass high-conversion electrochemical cell [23, 24). The model is based on electrolyte plug flow and includes electrochemical kinetics and mass transfer limitations. It has been developed for the case of three consecutive electrochemical reactions, with the key product formed by the second reaction, but can easily be modifled in order to be used for other reaction schemes, such as parallel reactions or solvent oxidation. 

Consider another limiting case, a reaction following a number of parallel routes each involving only one slow step. If we could measure the current corresponding to one of the routes, we would be able to determine, with the aid of the Horiuti formula, the true stoichiometric number of the slow step of this route (since there is only one slow step, the true Pp is determined)... 

A useful analysis of single-scan voltammetric curves was given by Saveant and Vianello " and developed for cases where a chemical reaction of any order precedes, follows, or runs in parallel with an electron transfer step. Limiting cases of pure kinetic or complete diffusion control were considered. In a further paper, these authors extended their treatment to the special cases of monomerization (isomerization) or dimerization as the coupled chemical step, i.e., for processes such as... 

In this section we shall consider first of all reactions of the type nA + mB A B in which the reaction product A B has a narrow range of homogeneity. An example would be the reaction A1 H- Sb = AlSb at 350 °C. In general, we shall not be concerned with determining the chemical diffusion coefficient by a Boltzmann-Matano analysis, since in the limiting case the diffusion profile in the diffusion couple consists of a pure step function. Reactions of this type parallel reactions between ionic crystals as discussed in section 6.2. Two situations must be separately discussed ... 

Design and analysis procedures for the limiting cases of uniform and shell-progressive chemical deactivation coupled with sintering closely parallel those given for the general case and are very similar to those already developed in the previous section on reactions affected by diffusion and chemical deactivation. Likewise, the... 

For the parallel reaction scheme we have analyzed (i.e. any one of N defects can cause the relaxation) we find two relaxation laws, i) The Poisson law, and ii) The Stretched exponential law. Both are probability limit distributions. The universality of the stretched exponential law is tied to this fact, i.e. specific details of the defect motion do not matter, only that = -. This case can arise from activated hopping over, a distribution f(A) of, potential barriers. If t, the time to overcome a barrier of height A is... 

The impedance data have been usually interpreted in terms of the Randles-type equivalent circuit, which consists of the parallel combination of the capacitance Zq of the ITIES and the faradaic impedances of the charge transfer reactions, with the solution resistance in series [15], cf. Fig. 6. While this is a convenient model in many cases, its limitations have to be always considered. First, it is necessary to justify the validity of the basic model assumption that the charging and faradaic currents are additive. Second, the conditions have to be analyzed, under which the measured impedance of the electrochemical cell can represent the impedance of the ITIES. 

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