First-Order Parallel Reactions

ILLUSTRATION 5.3 DETERMINATION OF RELATIVE RATE CONSTANTS FOR COMPETITIVE PARALLEL SECOND-ORDER REACTIONS (FIRST-ORDER IN EACH SPECIES)... 

The experimentally observed pseudo-first order rate constant k is increased in the presence of DNA (18,19). This enhanced reactivity is a result of the formation of physical BaPDE-DNA complexes the dependence of k on DNA concentration coincides with the binding isotherm for the formation of site I physical intercalative complexes (20). Typically, over 90% of the BaPDE molecules are converted to tetraols, while only a minor fraction bind covalently to the DNA bases (18,21-23). The dependence of k on temperature (21,24), pH (21,23-25), salt concentration (16,20,21,25), and concentration of different buffers (23) has been investigated. In 5 mM sodium cacodylate buffer solutions the formation of tetraols and covalent adducts appear to be parallel pseudo-first order reactions characterized by the same rate constant k, but different ratios of products (21,24). Similar results are obtained with other buffers (23). The formation of carbonium ions by specific and general acid catalysis has been assumed to be the rate-determining step for both tetraol and covalent adduct formation (21,24). 

Meehan and Bond (23) on the other hand, have taken an opposite view, namely that k3I1(C) k3I(C), while k3II(T) k3I(T). Thus, in this view, the hydrolysis occurs at external binding sites, while covalent binding occurs at intercalation sites. Furthermore, they reject the common intermediate model (Equation 2) on the basis of their belief that the rates of reaction for tetraol formation and adduct formation and the ratio of the products should be the same in such a model. While these rates of reaction are the same and the product ratios are observed to be different, this is fully consistent for a set of parallel pseudo-first order reactions involving a common intermediate (29) as pointed out above. Thus, the data of Meehan and Bond does not demonstrate the validity of the two-domain model (23). 

The conclusion, therefore, is that when parallel competing first-order reactions occur in isothermal pellets with large pores the intrinsic selectivity is unaffected. However, in Izurge pellets with small pores, the selectivity reduces to the square root of the value for the unimpeded reaction. Thus for large 0... 

Scheme 4.2 Parallel (competitive) first-order unidirectional reactions of a single reactant.

Based on the investigation of reaction, heat transfer and mass transfer of the KD306-type sulfur-resisting methanation catalyst [9-11], the non-isothermal one-dimensional and two-dimensional reaction-diffusion models for the key components have been established, and solved using an orthogonal collocation method in this paper. The scope is to study the catalyst intraparticle reaction-diffusion processes that involve parallel, non-first order, equilibrium-restrained reactions. 

Reactions were studied under the pseudo first-order condition of [substrate] much greater than [initial dihydroflavin]. Under these conditions, the reactions are characterized by a burst in the production of Flox followed by a much slower rate of Flox formation until completion of reaction. The initial burst is provided by the competition between parallel pseudo first-order Reactions a and b of Scheme 3. These convert dihydroflavin and carbonyl compound to an equilibrium mixture of carbinolamine and imine (Reaction a), and to Flox and alcohol (Reaction b), respectively. The slower production of Flox, following the initial burst, occurs by the conversion of carbinolamine back to reduced flavin and substrate and, more importantly, by the disproportionation of product Flox with carbinolamine (Reaction c followed by d). Reactions c and d constitute an autocatalysis by oxidized flavin of the conversion of carbinolamine back to starting dihydroflavin and substrate. In the course of these studies, the contribution of acid-base catalysis to the reactions of Scheme 3 were determined. The significant feature to be pointed out here is that carbinolamine does not undergo an elimination reaction to yield Flox and lactic acid (Equation 25). The carbinolamine (N(5)-covalent adduct) is formed in a... 

It is in the area of approximate lumping that the distinction between the meaning of lumping, as used in this article, and of overall kinetics, possibly becomes a fuzzy one. For instance, the analysis of many parallel irreversible first-order reactions, to be briefly discussed in the next section, has been called lumping in the literature, though it is not in any sense an exact (Wei-Kuo) lumping, nor an approximate one One chooses to consider only one lump, as given later by Eq. (97) but the qualitative kinetic behavior of the lump in no way resembles that of the... 

Models of parallel pseudo first-order reactions consider the case when two interactions with different rate constants proceed simultaneously. Such situations can be attributed to different kinds of receptor sites or to different states of the analyte [8,11]. In the first case the model can describe heterogeneity of the sensor surface the second may concern a macromolecular analyte that can be present in various conformations, protonation states, etc. Besides two sets of rate constants, the models also require specification of proportion p between the two fractions of the receptor or analyte. For the model considering two kinds of receptors, the following equations are obtained ... 

Fig. 4 Kinetics and sensorgram according to the model of two parallel pseudo first-order reactions attributed to two kinds of receptors (Eq. 19). Parameters ao = 1-5 jxM, /S = 1 nM cm, p = 0.8, fcai = 10 s fcai = 0-001 s , fca2 = 8 x 10 M s , fcd2 = 0.04 s ...

A simple model of two parallel independent first-order reactions was found to closely describe the decomposition of an oil shale kerogen from the Rundle deposit in Australia. The model was based on the fractions of mobile and rigid components in the kerogen, inferred from the NMR data. A major aspect of the model was that the macromolecular material in the oil shale kerogen had a bimodal distribution of cross-link density. During heating, a... 

The r-butyl reactant ions are formed in the near vicinity of the electron beam, and they drift to the ion exit slit of the ionization chamber under the influence of the repeller field. Protonated ester molecules (taken as a typical example) are formed along the path of the r-butyl ions by proton-transfer reactions, and after formation the protonated ester ions drift toward the ion exit slit under the influence of the repeller field. In the process of doing this they undergo decomposition reactions which may be looked upon as a set of parallel, competing first-order reactions, that is. 

A similar analysis ean be applied to other reaction mechanism, such as parallel irreversible first-order reaetions A B and A—>C, a triangle of reversible first-order reactions and simple models with nonlinear dependencies. The results obtained for the temporal changes of concentrations can be direetly applied to steady-state PHI models, with the astronomic time replaced by the space time or residence time r. Similar analyses can also be done for CSTRs. 

Step 4 of the thermal treatment process (see Fig. 2) involves desorption, pyrolysis, and char formation. Much Hterature exists on the pyrolysis of coal (qv) and on different pyrolysis models for coal. These models are useful starting points for describing pyrolysis in kilns. For example, the devolatilization of coal is frequently modeled as competing chemical reactions (24). Another approach for modeling devolatilization uses a set of independent, first-order parallel reactions represented by a Gaussian distribution of activation energies (25). 

Next consider the case of parallel first-order and second-order reactions shown in Scheme VIII. 

Thus, if Ca and Cb can both be measured as functions of time, a plot of v/ca vs. Cb allows the rate constants to be estimated. (If it is known that B is also consumed in the first-order reaction, mass balance allows cb to be easily expressed in terms of Ca-) The rate v(Ca) is the tangent to the curve Ca = f(t) at concentration Ca-This can be determined graphically, analytically, or with computer processing of the concentration-time data. Mata-Perez and Perez-Benito show an example of this treatment for parallel uncatalyzed and autocatalyzed reactions. 

Show that the reaction follows first-order kinetics when [A]0 s> [S]o, and show that lAi > varies linearly with l/[A]. The same pattern is seen when maleic anhydride (B) is used instead of A, except that the line in the double reciprocal plot is parallel to the abscissa. It intersects the ordinate at the same point as the y-intercept for A. Why are the slopes for A and B different, and their intercepts the same ... 

Parallel reactions, 58-64, 129 Partitioning ratios, 79 Perturbation (see Chemical relaxation) pH profiles, 139-145 bell-shaped, 141-142 Phosphorous acid, oxidation of, 186-187 Physical methods for kinetics, 22-25 end point reading unknown, 25-28 sample calculation for, first-order,... 

Example 4.9 Find the conversion for a first-order reaction in a composite system that consists of a perfect mixer and a piston flow reactor in parallel. 

Suppose you have two identical PFRs and you want to use them to make as much product as possible. The reaction is pseudo-first-order and the product recovery system requires a minimum conversion of 93.75%. Assume constant density. Do you install the reactors in series or parallel Would it affect your decision if the minimum conversion could be lowered ... 

Suppose there are two parallel, first-order reactions in a steady-state CSTR. Show that neither the fed-batch nor fast-fill-and-hold strategies can achieve a bumpless startup if the reactions have different rate... 

The oxidations of formic acid by Co(III) and V(V) are straightforward, being first-order with respect to both oxidant and substrate and acid-inverse and slightly acid-catalysed respectively. The primary kinetic isotope effects are l.Sj (25°C)forCo(IU)and4.1 (61.5 C°)for V(V). The low value for Co(lII) is analogous to those for Co(IIl) oxidations of secondary alcohols, formaldehyde and m-nitrobenzaldehyde vide supra). A djo/ h20 for the Co(III) oxidation is about 1.0, which is curiously high for an acid-inverse reaction . The mechanisms clearly parallel those for oxidation of alcohols (p. 376) where Rj and R2 become doubly bonded oxygen. 

Note that with Cao = Cco = Cqo = 0, Cbo = 1, ki = k3 = 0 the program models a simple first-order parallel reaction sequence. 

Parallel First-Order Reactions. In many instances, the active drug may degrade through more than one pathway ... 

This technique is readily adaptable for use with the generalized additive physical approach discussed in Section 3.3.3.2. It is applicable to systems that give apparent first-order rate constants. These include not only simple first-order irreversible reactions but also irreversible first-order reactions in parallel and reversible reactions that are first-order in both the forward and reverse directions. The technique provides an example of the advantages that can be obtained by careful planning of kinetics experiments instead of allowing the experimental design to be dictated entirely by laboratory convention and experimental convenience. 

Irreversible First-Order Parallel Reactions. Consider the irreversible decomposition of a reactant A into two sets of products by first-order reactions. 

Reversible First-Order Parallel Reactions. This section extends the analysis developed in the last section to the case where the reactions are reversible. Consider the case where the forward and reverse reactions are all first-order, as indicated by the following mechanistic equations. 

These reactions are competitive parallel reactions that are each first-order in the competitive species. Equation 5.2.61 is applicable. 

Fuguitt and Hawkins have reported the data below as typical of these reactions. The reported data have been modified somewhat for the purposes of this problem. By summing the amounts of dimer, a- and /Tpyronene, and tf//oocimene present in the final mixture, one may determine the total amount of alio-ocimene formed in the initial reaction. These investigators have postulated that all three of the parallel reactions are first-order. Is the data consistent with this hypothesis If so, what are the values of the three first-order rate constants ... 

In the simplest cases of reactive transport, a species sorbs according to a linear isotherm (Chapter 9), or reacts kinetically by a zero-order or first-order rate law. There is a single reacting species, and only one reaction is considered. In these cases, the governing equation (Eqn. 21.1 or 21.2) can be solved analytically or numerically, using methods parallel to those established to solve the groundwater transport problem, as described in the previous chapter (Chapter 20). 

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