Sequence of First-order Reactions

The preceding scheme can be extended to the general case of an infinite sequence of first-order reactions. If we represent such an infinite sequence by the set of stoichiometric equations 

This set of equations can now be rewritten in more useful symbolic form if we let the symbol D represent differentiation with respect to time, that is, D = d/dt. 

By proceeding in this manner with successive differentiations and eliminations it can be shown that for the ith species we obtain an equation of the form 

If any two of the fc s are equal, then the solution is slightly different. If, for example, h = 2, then the solution has the form 

This expression for the distribution of the instantaneous concentrations of product molecules C may be recognized as a Poisson distribution. Each component Cy will reach a maximum concentration C,-,niax under these conditions at a time 

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Any sequence of first-order reactions can be solved analytically, although the algebra can become tedious if the number of reactions is large. The ODEs that correspond to Equation (2.20) are... 

The graphs in Fig. 12 (41, 41a) represent product selectivity plots in which the relative concentrations of products of HDS are plotted against the percentage conversion of the starting material. The lines represent the calculated product distributions for sequences of first-order reactions having... 

Many reactions like the skeletal isomerizations, hydrogenation-dehydrogenation can be described by sequence of first order reactions (see eg. reaction sequence 2, Table 1.). The expressions for the reversible rates with and without activation energy corrections are given by equations (14) and (16) respectively. 

Unfortunately, there exists no general theory that does for a generad sequence of elementary steps what has been done here for the simple sequence of first-order reactions. Yet the general ideas are clear. While exceptions to the validity of the steady-state approximation are known, they are rare and the steady-state approximation can be considered as the most important general technique of applied chemical kinetics. The treatment of long sequences becomes a simple problem as will now be shown. 

The examples of DDEs that we have considered so far have all been linear. Linear systems allow a considerable amount of analysis and even exact solution, on occasion, but few real systems are linear. We now turn to some examples that involve nonlinear DDEs, starting with two familiar examples, the cross-shaped phase diagram and the Oregonator. In these two models we see how, much like in the case of the sequence of first-order reactions treated above, one can reduce the number of variables by introducing a delay. In the nonlinear case, however, the choice of the delay time is far more difficult than in linear models. 

Derivation of Kinetic Equations for Linear Sequences of First-Order Reactions... 

For a sequence of first order reactions the relaxation times are clearly independent of reactant concentrations and the equations apply equally to the interpretation of large transients. The effects of changing the concentration, for instance of the ligand in the pseudo first order system, will be discussed later. Without such additional diagnostics, which are available in the case of concentration dependent systems, the four rate constants can only be estimated by numerical fitting procedures. If signals in terms of absolute concentrations for A, R and [AR] are available, the equilibrium constants can be evaluated and serve as a useful restriction for the numerical solutions. If the two relaxations are uncoupled, t, T2, then we can simplify from equations (6.2.20) ... 

We consider that one of the essential parts of physical chemistry is some awareness of nuclear chemistry. While the periodic chart poses as a list of stable elements, there are hints of irregularity by the absence of elements no. 43 (technetium, Tc) and no. 61 (promethium. Pm). Modem students are also aware of unstable elements beyond no. 92 (uranium, U). Since nuclear reactions seem to follow a sequence of first-order reactions, we take some time to mention a few of the mechanisms that are occurring in the first-order processes. 

If each type of sulfur compound is removed by a reaction that was first-order with respect to sulfur concentration, the first-order reaction rate would gradually, and continually, decrease as the more reactive sulfur compounds in the mix became depleted. The more stable sulfur species would remain and the residuum would contain the more difficult-to-remove sulfur compounds. This sequence of events will, presumably lead to an apparent second-order rate equation which is, in fact, a compilation of many consecutive first-order reactions of continually decreasing rate constant. Indeed, the desulfurization of model sulfur-containing compounds exhibits first-order kinetics, and the concept that the residuum consists of a series of first-order reactions of decreasing rate constant leading to an overall second-order effect has been found to be acceptable. 

In general there is no exact solution to any sequence of consecutive higher-order equations. The reason for this is that the differential equations are no longer linear equations (as they were in the case of first-order reactions), and nonlinear equations do not have exact solutions except in very particular cases. However, two exact methods are available for studying some aspects of these systems, and there is one more commonly used... 

This method is based on the principle of constant fractional life (usually called half-life ), which applies to a species undergoing reaction in such a way that, after any time interval, a constant fraction of the amount left unreacted at the end of the previous interval has reacted (or a constant fraction remains unreacted), irrespective of the initial concentration. This property is associated with first-order or pseudo-first-order reactions—such as radioactive decay—for which half-lives are often quoted as a measure of reaction rate. This property of constant fractional life also applies to more complex reactions—such as successive and parallel reaction-sequences—involving first-order reactions. The initial concentration of a species reacting with constant fractional life is directly proportional to the amount of product formed at any given time. 

Such a sequence is known as series or consecutive reactions. In this case, B is known as an intermediate because it is not the final product. A similar situation is very common in nuclear chemistry where a nuclide decays to a daughter nuclide that is also radioactive and undergoes decay (see Chapter 9). For simpficity, only the case of first-order reactions will be discussed. 

The formulation of Wei and Prater offers many advantages. Its limitations are not as confining as suggested by the single application discussed in this chapter, namely, the analysis of first-order reversible reactions. The method can be extended readily to networks of first-order reactions that are not all reversible. Furthermore, as discussed in Chapter 5, rate equations for the coupled sequences of a network are very frequently of a type that can still be handled by the analysis. They are commonly, though not always, of the form ... 

Displacement of oxalate from [Pt(ox)2] by chloride takes place by a series of first-order reactions in the pH range 5—8. In the proposed reaction sequence (6)... 

Analogy with a chain reaction. The equivalence between a chain of several poles, making a multipole, and the dipole formed by the two ends of the chain % is a well-known property of chains composed of linear relationships. A repre-S L sentative example in physical chemistry is the chain formed by a sequence of first-order chemical reactions. The poles between the ends are the intermediate species and the poles at the ends are the substrate A and product species Z. In such a chain, the overall reaction rates and equilibrium constants correspond to those of the equivalent dipole A - Z. 

An example of a two-stage hydrolysis is that of the sequence shown in Eq. IV-69. The Idnetics, illustrated in Fig. IV-29, is approximately that of successive first-order reactions but complicated by the fact that the intermediate II is ionic [301]... 

The rates of many reactions are not represented by application of the law of mass action on the basis of their overall stoichiometric relations. They appear, rather, to proceed by a sequence of first- and second-order processes involving short-lived intermediates which may be new species or even unstable combinations of the reaclants for 2A -1- B C, the sequence could be A -1- B AB followed by A -1- AB C. 

A comparative study [651] of the relative stabilities of various forms of U03 by DTA methods lists the temperatures of onset of reaction in the sequence a < e < amorphous < 0 < U02.9 < S < 7 (673, 733, 773, 803, 853, 863 and 903 K, respectively). Themal stabilities, as measured by the first-order reaction rate coefficient, magnitudes of E or enthalpies of reaction, increased with increasing structural symmetry. 

Consider a reaction scheme consisting of a sequence of two first-order (or pseudo-first-order) reactions in which intermediate I builds up and later falls. 

Radioactive decay provides splendid examples of first-order sequences of this type. The naturally occurring sequence beginning with and ending with ° Pb has 14 consecutive reactions that generate a or /I particles as by-products. The half-lives in Table 2.1—and the corresponding first-order rate constants, see Equation (1.27)—differ by 21 orders of magnitude. 

Within the strictly chemical realm, sequences of pseudo-first-order reactions are quite common. The usually cited examples are hydrations carried out in water and slow oxidations carried out in air, where one of the reactants... 

The kinetics of hydrogenation transfer is covered by the use of an exchange superoperator assuming a pseudo first-order reaction. Thereby, competing hydrogenations of the substrate to more than one product can also be accommodated. In addition, the consequences of relaxation effects or NOEs can be included into the simulations if desired. Furthermore, it is possible to simulate the consequences of different types of pulse sequences, such as PH-INEPT or INEPT+, which have previously been developed for the transfer of polarization from the parahydrogen-derived protons to heteronuclei such as 13C or 15N. The... 

A metabolite, molecular entity, or some other event/ process that precedes another component in a longer sequence of events or conversions. For example, the isoprenoid metabolite squalene is a precursor of cholesterol and glucose 6-phosphate is a precursor of glycogen, ribose, and pyruvate. See Series First Order Reaction Pulse-Chase Experiments... 

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. 

When this is incorporated into expressions for rates of disappearance of reactant or formation of product, it is clear that the reaction is expected to display first-order behaviour so that, kinetically, the reaction is indistinguishable from a single-step conversion of R into R The rate constant, k0bs, is composite, containing contributions from all the elementary steps, with simplification becoming possible when k2 k-1 or /c i k2. Kinetics alone cannot provide evidence for the existence of a transient intermediate species if all reactions in the sequence are first order. 

A catalytic cycle is a sequence of steps. When one step is much slower than the others, we say that this step is rate determining and we ignore, for kinetics purposes, the other (fast) steps. Nevertheless, sometimes there are two slow steps with similar rates, and sometimes the rate of a specific step changes in the course of the reaction or under different conditions. One common situation is that of two consecutive first-order reactions, as in Eq. (2.44). 

Here and in what follows the subscripts in and out indicate the Chi molecules localized near the inner and outer surfaces of the vesicle membranes. If no more than one pair of Chl+ and A- particles is generated on the inner surface of each vesicle, the recombination by reaction (7) may be described in terms of first-order kinetics. Having in mind that the number of D, A and Chlout molecules considerably exceeds the number of 3Chl and Chl+ particles, one can treat all the remaining rate constants of reaction sequence (5)-(9) also as pseudo first-order ones. In accord with this reaction scheme, the quantum yield of the transfer of the first electron through the vesicle membrane can be expressed as ... 

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