First-Order Consecutive Reactions

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. 

This consists of two consecutive irreversible first-order (or pseudo-first-order) reactions. The differential rate equations are... 

Generalization of Scheme X to any number of consecutive irreversible first-order reactions is obviously possible, although the equations quickly become very cumbersome. However, Eqs. (3-42) and (3-44) reveal patterns in their form, and West-man and DeLury have developed a systematic symbolism that allows the equations to be written down without integration. 

Consecutive reactions involving one first-order reaction and one second-order reaction, or two second-order reactions, are very difficult problems. Chien has obtained closed-form integral solutions for many of the possible kinetic schemes, but the results are too complex for straightforward application of the equations. Chien recommends that the kineticist follow the concentration of the initial reactant A, and from this information rate constant k, can be estimated. Then families of curves plotted for the various kinetic schemes, making use of an abscissa scale that is a function of c kit, are compared with concentration-time data for an intermediate or product, seeking a match that will identify the kinetic scheme and possibly lead to additional rate constant estimates. 

One further system will be solved by the transform method. Scheme XV constitutes two consecutive reversible first-order reactions. 

In the presence of excess bromine, the first-order rate coefficient was 10.3 x 10" 5, but kinetic studies here were complicated due to rapid reaction of bromine with the 2,4,6-tribromophenol to give the intermediate (LXII), with SO3 replaced by Br) which slowly decomposed with a rate coefficient k2 to give two products. By analysis in terms of two consecutive first-order reactions, the values of k2 and k2 were determined as 9.2 x 10"5 and 3.75 x 10"4 and the latter rate being the faster means that two moles of bromine were consumed for every mole of sulphonate undergoing substitution in fact, more than two moles were consumed, the reason for this being undetermined. 

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

M. Gui and S.C. Rutan, Determination of initial concentration of analyte by kinetic detection of the intermediate product in consecutive first-order reactions using an extended Kalman filter. Anal. Chim. Acta, 66 (1994) 1513-1519. 

It will be of interest to present mathematically the picture of the course of consecutive reactions. In the simplest case the substance A considered in the present example undergoes a first-order reaction to yield C the reverse reactions are neglected. The reaction occurring in two first-order steps can now be written as ... 

More complicated reactions schemes, including first-order reversible consecutive processes and competitive consecutive reactions, are considered in a textbook by Irwin [89]. Professor Irwin s textbook also includes computer programs written in the BASIC language. These programs can be used to fit data to the models described. 

Consecutive Reactions that are other than First-Order. For consecutive reactions that are not first-order, closed form analytical solutions do not generally exist. This situation is a consequence of the nonlinearity of the set of differential equations involving the time derivatives of the various species concentrations. A few two-member sequences have been analyzed. Unfortunately, the few cases that have been... 

Comparison of the fractional yields of V in mixed and plug flow reactors for the consecutive first-order reactions. A A- V W. (Adapted from Chemical Reaction Engineering, Second Edition, by O. Levenspiel. Copyright 1972. Reprinted by permission of John Wiley and Sons, Inc.)... 

Consecutive Reactions Where an Inter-mediate Is the Desired Product. Consecutive reactions in which an intermediate species (V) is the desired product are often represented as a series of pseudo first-order reactions... 

The consecutive first order reactions have the rate equations dA... 

These consecutive reversible first order reactions, 1 3... 

For multiple reactions, material balances must be made for each stoichiometry. An example is the consecutive reactions, A = B = C, for which problem P4.04.52 develops a closed form solution. Other cases of sets of first order reactions are solvable by Laplace Transform, and of course numerically. 

The solution for consecutive first order reactions has been worked out by Rodiguin (1964). 

Applying the usual steady-state treatment for consecutive first-order reactions kt at 16 torr pressure over the temperature range 597-701 °C is given by 1.8 x 1011 exp(—47,000/Kr) sec Within experimental error, reactions (1) and (2) were homogeneous processes. However, both k2 and k2 were functions of the total pressure in the system. This dependence is shown in Fig. 1. The methyl zinc decomposition is apparently in its second-order region. Therefore, assuming four effective oscillators and a mean temperature of 1050 °K, = Eohs.+i nRT... 

First we construct a kinetic measurement which we then analyse in both of the above ways. The reaction is the set of two consecutive first order reactions with rate constants ki and fe ... 

Write the energy equation for the CSTR of Example 2.6 in which consecutive first order reactions occur with exothermic heats of reaction A, and Aj. 

Two consecutive, first-order reactions take place in a perfectly mixed, isothermal batch reactor. 

Jl An isothermal perfectly mixed batch reactor has consecutive first-order reactions... 

When reversible steps occur in a reaction scheme, distinctions between consecutive and parallel reactions cannot always be made. For example, the consecutive first-order reactions... 

Irreversible Reactions in Series. We first consider consecutive unimolecular-type first-order reactions such as... 

Figure 3.11 Typical concentration-time curves for consecutive first-order reactions.

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. 

These authors used 6 1 and 30 1 alcohol-to-oil molar ratios for both methanol and butanol. As expected, a pseudo first-order reaction was found at large excess of alcohol for both alcohols. At low excess alcohol, however, the butanolysis reaction (30°C) was second-order, but the methanolysis reaction (40°C) was reported to be a combination of a second-order consecutive reaction and a fourth-order shunt reaction. The shunt reaction, in which three methanol molecules simultaneously attack a TG molecule, was adopted to better fit the kinetic data. However, such a reaction is highly unlikely. Nureddini et al. later found that the inclusion of a shunt mechanism was not necessary to fit the kinetic data of the transesterification reaction, and Boocock et al showed that the shunt reaction assumption came as a misinterpretation of the observed kinetics. At low temperatures (20 0°C) the multiphase methanolysis reaction... 

Fig. 1.8. Consecutive first-order reactions with p0 = 0.1 mol dm-3, fcu = 5 x 10-3 s and k2 = 10"2s (a) the exponential decay of precursor reactant concentration, p (b) growth and decay of intermediate concentrations a(t) and b(t). Also shown in (b), as broken curves, are the pseudo-stationary-state loci, a (t) and b (t), given by eqns (1.31) and...

Fig. 1.9. Consecutive first-order reactions and cubic autocatalysis, showing pseudosteady-state predictions for the intermediate concentrations. Initial concentrations and rate constants are given in Table 1.1.

Fig. 1,10. Actual time-dependent concentration of intermediate A for consecutive first-order reactions with cubic autocatalysis showing pseudo-steady-state behaviour, pre-oscillatory evolution, an oscillatory period, and then the return to pseudo-steady-state behaviour.

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