Parallel and Independent Reactions

The Diels Alder reactions of maleic anhydride with 1,3-cyclohexadiene, as well the parallel reaction network in which maleic anhydride competes to react simultaneously with isoprene and 1,3-cyclohexadiene [84], were also investigated in subcritical propane under the above reaction conditions (80 °C and 90-152 bar). The reaction selectivities of the parallel Diels-Alder reaction network diverged from those of the independent reactions as the reaction pressure decreased. In contrast, the same selectivities were obtained in both parallel and independent reactions carried out in conventional solvents (hexane, ethyl acetate, chloroform) [84]. 

Let us suppose that a single-reactor system in which a series of parallel and independent reactions take place, is supplied with raw material of composition A, B, C,. ..,K. 

In general, for parallel and independent reactions, the selectivity to the products that are formed in the slow reactions will be increased relative to the products formed in the fast reactions, if an external transport resistance is present For series reactions, the selectivity to the intermediate product usually will be decreased if an external transport resistance is present. 

Note that the interference of chemical reactions may alter the effective rate constant of the secondary reaction and break the independence principle of elementary chemical reactions. Moreover, under these conditions non-spontaneous reactions may proceed, which are eliminated in parallel and consecutive reactions. 

In this chapter we discuss reactor selection and general mole balances for multiple reactions. There are three basic types of multiple reactions series, parallel, and independent. In parallel reactions (also called competing reactions) the reactant is consumed by two different reaction pathways to form different products ... 

Electrochemical dissolution of electrons and electron thermoemission may be regarded as two parallel and independent processes. It is seen from Fig. 12 that electrochemical dissolution ig preferred from the thermodynamic viewpoint, because electrons are transferred to a lower level. The mechanism of the process is, however, dictated not only by the thermodynamic factor, but mostly by the activation energy for one or the other pathway of reaction. Electrochemical dissolution demands reorganization of the solvent, while thermoemission does not. From Table 1 and Fig. 11 it follows that electrochemical dissolution is observed in liquid ammonia and hexamethylphosphotriamide solutions (transfer coefficient a = 0.5-0.75). 

Scheme 3.1 Parallel, sequential (consecutive) and independent reactions...

Rate laws derived for reaction schemes consisting of any number and combination of consecutive, parallel and independent first-order reaction steps can be integrated in closed form to give a sum of exponential terms, so that the total absorbance A of a solution in which these reactions proceed can also be expressed as a sum of exponential terms (Equation 3.9), but see last paragraph of this Section. 

There are four basic types of multiple reactions series, parallel, complex, and independent. These types of multiple reactions can occur by themselves, in pairs, or all together. V en there is a combination of parallel and series reactions. they are often referred to as complex reactions. 

The thermodynamic probability of the formation of products of many parallel and subsequent reactions is calculated by taking into consideration general and simultaneous equilibria among them. This is not possible for the Fischer-Tropsch reaction, because the number of reactions is theoretically unlimited. Therefore, for calculation purposes, it is assumed that the reactions are independent. The Fischer-Tropsch synthesis is strongly exothermic, and the removal of heat represents an important problem in the technology of this process. In order to facilitate a comparison of thermodynamic data for various reactions, such data are given with respect to one mole of carbon (Figure 13.4). 

The multiple reaction algorithm can be applied to parallel reactions, series reactions, complex reactions, and independent reactions. The availability of software packages (ODE solvers) makes it much easier to solve problems using moles or molar flow rates Fj rather than conversion. For liquid systems, concentration is usually the preferred variable used in the mole balance equations. 

In contrast to consecutive reactions, with parallel competitive reactions it is possible to measure not only the initial rate of isolated reactions, but also the initial rate of reactions in a coupled system. This makes it possible to obtain not only the form of the rate equations and the values of the adsorption coefficients, but also the values of the rate constants in two independent ways. For this reason, the study of mutual influencing of the reactions of this type is centered on the analysis of initial rate data of the single and coupled reactions, rather than on the confrontation of data on single reactions with intergal curves, as is usual with consecutive reactions. 

Figure 2. The elastic (G7) and viscous modulus (G") as a function of the frequency of oscillation at three different UV exposure times. At t = 1660 s, G7 and G" are parallel, indicating the sample has reached its gel point. Note that G is initially smaller than G" throughout the frequency range (t = 1280 s), but becomes larger than G" and independent of frequency towards the end of the reaction (t = 2780 s), indicating that the sample has been transformed from a liquid to a highly cross-linked network.

The system can be operated in the parallel mode, discontinuously (batch-wise) with each reactor as an independent unit, semi-continuously or as a reactor cascade. Both homogeneous and heterogeneous reactions as well as product and catalyst separation and catalyst recycling are possible. 

The constant for reaction (31) for /3-diketones in tautomeric keto (HK) and enol (HE) forms, can be partitioned between Kk and KE (reactions 32 and 33). At equilibrium, equations (34) and (35) hold, and values obtained are given in Table 19. Kinetics showed that the reaction occurs through the enol form by parallel acid-independent and inverse-acid paths (Scheme 15).514 K... 

Ray and Chanda [261] studied bismuth molybdates (prepared by the method of Peacock [250,251]) in an integral flow reactor. At constant W/F = 8 g h mol-1 and a feed ratio isobutene/oxygen = 1/6, a maximum selectivity of 75% was found at 400—450°C. As with propene, the reaction is first order with respect to isobutene and the rate is independent of the oxygen pressure. The reoxidation of the catalyst is very fast. Assuming that the kinetics can be described by three parallel first-order reactions for the production of methacrolein, carbon monoxide and carbon dioxide, rate coefficients were calculated (Table 18). 

The marine facultative anaerobe bacterium Serratia marinoruhm and the yeast Rhodotoruhi rubra both methylate arsenate ion to methylarsonate, but only the latter produces cacodylic acid (258). Human volunteers who ingested 500 fig doses of As as sodium arsenite, sodium methylarsonate, and sodium cacodylate excreted these compounds in their urine (259). Of these three, approximately 75% of the sodium arsenite is methylated, while 13% of methylarsonate is methylated. Rat liver subcellular fractions methylated sodium arsenate in vitro, providing the first direct evidence for possible mammalian methylation independent of symbiotic bacteria (260). Shariatpanahi el al. have reported kinetics studies on arsenic biotransformation by five species of bacteria (261). They found that the As(V)-As(IIl) reduction followed a pattern of two parallel first-order reactions, while the methylation reactions all followed first-order kinetics. Of the five species tested, only the Pseudomonas produced all four metabolites (arsenite, methylarsonate, cacodylate, trimethylarsine) (261). 

Here c(x, t)dx is the concentration of material with index in the slice (x, x + dx) whose rate constant is k(x) K(x, z) describes the interaction of the species. The authors obtain some striking results for uniform systems, as they call those for which K is independent of x (Astarita and Ocone, 1988 Astarita, 1989). Their second-order reaction would imply that each slice reacted with every other, K being a stoichiometric coefficient function. Only if K = S(z -x) would we have a continuum of independent parallel second-order reactions. In spite of the physical objections, the mathematical challenge of setting this up properly remains. Ho and Aris (1987) have shown how not to do it. Astarita and Ocone have shown how to do something a little different and probably more sensible physically. We shall see that it can be done quite generally by having a double-indexed mixture with parallel first-order reactions. The first-order kinetics ensures the individuality of the reactions and the distribution... 

The non-dimensionalization used in this work is perhaps the simplest, but it suffers from the defect that important physical bifurcation parameters are not isolated. The simple cuspoid diagrams are probably not those that would be obtained from experiments, where the residence time is a convenient parameter. Balakotaiah and Luss (1983) considered such a formulation for two parallel or simultaneous reactions the diagrams for the case of sequential reactions are similar, at least when the activation energies are equal. The maximum multiplicity question, however, is independent of the formulation and we conjecture that diagrams with seven steady states could be found in a small region of parameter space, though we have not looked for them. 

A 4-week, single-blind, randomized, parallel-groups, dose-response study was conducted to test botanicals for allergy as measured by a skin patch test (wheal-and-flare reaction). The study was approved by an independent review board, informed consent was obtained before any screening or study measures, and the study was... 

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