First-order-reaction coupling

As an example consider a second-order reaction Here the suitable coupling between pex and V, over and above the couphng due to the compressibility of the system (the equation of state triangle in Fig. 15.3), comes from the fact that on decreasing V the number of reactions per unit time increases as V for fixed total number of particles. For an exothermic (endothermic) reaction this effect leads to increased (decreased) production of heat as V decreases, and consequently to temperature and pressure changes, in addition to those due to compression. For first-order reactions coupling can be achieved through the temperature dependence of rate coefficients. 

Sets of first-order rate equations are solvable by Laplace transform (Rodiguin and Rodiguina, Consecutive Chemical Reactions, Van Nostrand, 1964). The methods of linear algebra are applied to large sets of coupled first-order reactions by Wei and Prater Adv. Catal., 1.3, 203 [1962]). Reactions of petroleum fractions are examples of this type. 

The Merrill and Hamrin criterion was derived for a first-order reaction. It should apply reasonably well to other simple reactions, but reactions exist that are quite sensitive to diffusion. Examples include the decomposition of free-radical initiators where a few initial events can cause a large number of propagation reactions, and coupling or cross-linking reactions where a few events can have a large effect on product properties. 

Despite the rapidity of the coupling reactions epimerization may be extensive. It may be substantially reduced if the condensations are carried out using high concentrations of AA OR. This is a consequence of the differing dependencies of epimerization (first-order) versus coupling (second-order) on AA OR concentration. Both processes are sensitive to the stereochemistry of the reactants, and the balance of factors is such that coupling of (S)-AA OMe with A-... 

Even though the governing phenomena of coupled reaction and mass transfer in porous media are principally known since the days of Thiele (1) and Frank-Kamenetskii (2), they are still not frequently used in the modeling of complex organic systems, involving sequences of parallel and consecutive reactions. Simple ad hoc methods, such as evaluation of Thiele modulus and Biot number for first-order reactions are not sufficient for such a network comprising slow and rapid steps with non-linear reaction kinetics. 

It is generally much easier to study unimolecular than bimolecular processes. It should be noted in this connection that some reactions that are bimolecular in solution are unimolecular in crystals. For example, the coupling or disproportionation of two radicals generated in a crystal cage from a single precursor molecule is a first-order reaction of a radical pair, not a second-order reaction of independent radicals. 

The theory of coupled multicomponent first-order reaction and diffusion given by Wei (1962) has been used to successfully model the metal deposition profiles (Agrawal and Wei, 1984 Ware and Wei, 1985a). The... 

As discussed in Section IV, Agrawal and Wei (1984) and Ware and Wei (1985b) have successfully modeled experimental deposit profiles by using the theory of coupled, multicomponent first-order reaction and diffusion. Wei and Wei (1982) employed this theory to evaluate the influence of catalyst properties on the shape of the deposit profile. Agrawal (1980) developed a model for the deactivation of unimodal and bimodal catalysts based on the consecutive reaction path. These approaches represent a more realistic consideration of the HDM reaction mechanism than first-order kinetics and will, accordingly, be discussed in more detail. 

For the nonisothermal catalyst pellet with negligible external mass and heat transfer resistances, i.e., with Sh —> 00 and Nu —> 00 and for a first-order reaction, the dimensionless concentration and temperature are governed by the following couple of boundary value differential equations... 

Fig. 5.4 Concentration profile for a first-order reaction of a solute coupled to its diffusion into an infinite aqueous medium.

Figure 12(a) shows the typical distributions in local current for a first order reaction with different values of v 2 and applied dimensionless overpotentials ° for the coupled anode model, including mass transfer parameter y. The values of °, in the range of 0.5 to 16 typically represent overpotentials in the approximate range of 25 to 800 mV. The total current flows fromX = 0 (anode fed plane) to X= 1 (membrane). Current is much higher at the face of the electrode adjacent to the membrane or free electrolyte solution and decreases towards the current collector. An increase in potential increases the local current density and thereby increases the overall variation in current density throughout the electrode. 

Figure 12(b) shows the local current distribution of first and second order reactions and applied over potentials ° for the coupled anode model without the mass transfer parameter y. The figure also shows the effect of a change in the electrode kinetics, in terms of an increase in the reaction order (with respect to reactant concentration) to 2.0, on the current distribution. Essentially a similar variation in current density distribution is produced, to that of a first order reaction, although the influence of mass transport limitations is more severe in terms of reducing the local current densities. 

Binks and Ridd164 have made a complete kinetic study of the reaction of indole with several diazotized amines (p-nitroaniline, p-chloroaniline, sulfanilic acid, and aniline). Only the reaction with p-nitrodiazonium salt exhibits a simple kinetic form (pseudo first-order reaction) in the other cases the kinetics appear to be due to the superposition of two reactions, a normal azo-coupling reaction and an autocatalytic side reaction that removes diazonium ions, but does not form azo compounds. 

Were it not for the coupling terms, kbAcD and k fAcA, Eq. 4.34 would have the same form as Eq. 1.55 (neglecting its constant term on the right side), with an exponential-decay solution typical of first-order reactions (Eqs. 1.56 and 4.19). Because the coupling terms are linear in the Ac, however, it is always possible to find a solution to Eq. 4.34 by postulating that a pair of time constants, r, and r2, exists such that the Ac still show an exponential time dependence ( rel axation ) 19... 

The first-order reaction rate constant for the isomerization of peroxynitrous acid to nitrate is 4.5 s 1 at 37°C therefore, at pH 7.4 and at 37°C the half-life of the peroxynitrite/peroxynitrous acid couple (let both these species be referred to as peroxynitrite for the sake of brevity) is less than 1 s. The reaction mechanism of peroxynitrite decomposition was a subject of controversy. Primarily proposed was that peroxynitrous acid decomposes by homolysis, producing two strong oxidants hydroxyl radical and nitrous dioxide (B15) ... 

Actually, 2-pentene exists in cis- and trans-configurations, but this complication is ignored here as relatively unimportant.] Although more complex than the prototype 5.33, this network includes the essential features of coupled parallel first-order reactions and can account for the curves in Figure 5.9. 

Golikeri, S. V., and Luss, D., Aggregation of many coupled consecutive first order reactions. Chem. Eng. Sci. 29,845 (1974). 

The major part of this article will be devoted to a particular class of reaction systems—namely, monomolecular systems. A reaction system of (n) molecular species is called monomolecular if the coupling between each pair of species is by first order reactions only. These linear systems are satisfactory representations for many rate processes over the entire range of reaction and are linear approximations for most systems in a sufficiently small range. They play a role in the chemical kinetics of complex systems somewhat analogous to the role played by the equation of state of a perfect gas in classical thermodynamics. Consequently, an understanding of their behavior is a prerequisite for the study of more general systems. 

Two subclasses of monomolecular systems will be discussed reversible and irreversible monomolecular systems. A reaction system will be called reversible monomolecular if the coupling between species is by reversible first order reactions only. A typical example of a reversible monomolecular system is... 

Contrary to the authors of (3), we found that Am (III) oxidation is not a first order reaction with respect to Am (III) ions. While the oxidation rate does accelerate at low concentrations of Am (III), this acceleration can probably be attributed to reaction (5), which is thermodynamically possible, as it may be calculated from the formal oxidation potentials of Am (VI)/Am (V) and Am (V)/ Am (III) couples (9). + 2+... 

F.A.Masten and J.L.Franklin, A General Theory of Coupled Sets of First Order Reactions, Journal of the American Chemical Society, 72, 3337-3341(1950). 

The direct coupling of P430 with MV was further supported by the acceleration ofthe decay rate ofthe former that is observed with increasing MV concentration. The kinetic plots showed that P430 decay was exponential at all MV concentrations examined. The decay ty, vs. MV concentration fitted a pseudo-first-order reaction, giving an estimated rate constant of 9.6-10 M - at 22 °C. ... 

The basic oscillatory couple exhibiting a limit cycle is F6P-FDP. Moreover, oscillations in GLU are also observed. The end product is glyceraldehyde phosphate (GAP). The first step from GLU to F6P is a first-order reaction and in the second step, an activated form of phosphofructokinase acts as an enzyme. FDP activates this second step. 

A denotes the starting material that is oxidised (or reduced) at the electrode (e.g. Ru(bpy)3+ oxidation or MV2+ reduction on a Pt electrode) and it is subsequently reformed by a catalyst in a pseudo first order reaction that concomitantly forms a product P. P, in our case, is 02 (or H2). On a planer electrode, from the limiting current in the case of pure catalytic control (i.e. at remote enough a potential after the peak in a potential sweep experiment or at t - in an experiment where the potential is fixed sufficiently past the E° of the redox couple), kca, can be calculated using the expression,... 

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