First-order Reactions General Treatment

First-order Reactions General Treatment. There has been a considerable amount of work done on the solution of particular and general systems of first-order reactions. All such systems are capable of exact, explicit mathematical solutions. If we consider the most general case of a system of s components Ci, C2,. . . , C in which first-order reactions of the following type may take place between any two components 

Then we can write a set of s linear differential equations for the rates of reaction of the components. A typical such equation for the mth component Cm is 

It cun be shown that the most general solution of a coupled set of linear, homogeneous first-order equations, represented by Eq. (III.6A.3), has the form 

Since the order of summation can be interchanged, we can rewrite this equation in the form 

For this last equation to hold true for all values of t, each coefficient of in the eciuation must be zero and in addition the last term must be zero. For each value of m we will thus have s equations of the form 

---

The equation involving t for the general case of a reaction of the nth order as shown above applies to any value of n except n = 1, for this case the treatment leading to exponential equation shown in first-order reaction (In a/(a- x) = kt) must be employed. The equation is applicable for n = 2. Other cases, including those of nonintegral orders, can easily be worked out. The half-life, t0 5, is seen to be inversely proportional to k in all cases, and inversely proportional to the (n - 1) power of the concentration. 

Many special cases are given in Rodigin and Rodigina [12]. The situation of general first-order reaction networks has been considered by Wei and Prater [13] in a particularly elegant and now classical treatment Boudart [S] also has a more abbreviated discussion. 

Complex kinetic schemes cannot be handled easily, and, in general, a multidimensional search problem must be solved, which can be difficult in practice. This general problem has been considered for first-order reaction networks by Wei and Prater [13] in their now-classical treatment. As described in Ex. 1.4-1, their method defines fictitious components, B , that are special linear combinations of the real ones, Aj, such that the rate equations for their decay are uncoupled, and have solutions ... 

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. 

For irreversible first-order reactions in a PFR, Krambeck showed that the asymptotic kinetics Ra(C) = Cf whenever the feed contains a finite amount of unconvertible species. This is also true for reversible first-order reactions. " The case where the feed may or may not contain unconvertibles was treated by Ho and Aris. ° The general treatment of the PCM starts with the expectation that the long-time behavior of the mixture should be governed by the most refractory part of the feed (as will be seen later, this is not always true). To find what goes on at large t, D k) and c/x) near = 0 can be expanded as follows"" ... 

For the general treatment of arbitrary shapes involving a first-order reaction by Murray, see the book by Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Vol. I, Oxford University Press, London (1975). 

Generalized first-order kinetics have been extensively reviewed in relation to teclmical chemical applications [59] and have been discussed in the context of copolymerization [53]. From a theoretical point of view, the general class of coupled kinetic equation (A3.4.138) and equation (A3.4.139) is important, because it allows for a general closed-fomi solution (in matrix fomi) [49]. Important applications include the Pauli master equation for statistical mechanical systems (in particular gas-phase statistical mechanical kinetics) [48] and the investigation of certain simple reaction systems [49, ]. It is the basis of the many-level treatment of... 

An important example for the application of general first-order kinetics in gas-phase reactions is the master equation treatment of the fall-off range of themial unimolecular reactions to describe non-equilibrium effects in the weak collision limit when activation and deactivation cross sections (equation (A3.4.125)) are to be retained in detail [ ]. 

Reactions catalyzed by hydrogen ion or hydroxide ion, when studied at controlled pH, are often described by pseudo-first-order rate constants that include the catalyst concentration or activity. Activation energies determined from Arrhenius plots using the pseudo-first-order rate constants may include contributions other than the activation energy intrinsic to the reaction of interest. This problem was analyzed for a special case by Higuchi et al. the following treatment is drawn from a more general analysis. ... 

The general form of a first-order rate law is rate — i [ A], where A is a reactant in the overall reaction. Mathematical treatment converts this general form into an equation relating concentration and time. For a... 

However, a question arises - could similar approach be applied to chemical reactions At the first stage the general principles of the system s description in terms of the fundamental kinetic equation should be formulated, which incorporates not only macroscopic variables - particle densities, but also their fluctuational characteristics - the correlation functions. A simplified treatment of the fluctuation spectrum, done at the second stage and restricted to the joint correlation functions, leads to the closed set of non-linear integro-differential equations for the order parameter n and the set of joint functions x(r, t). To a full extent such an approach has been realized for the first time by the authors of this book starting from [28], Following an analogy with the gas-liquid systems, we would like to stress that treatment of chemical reactions do not copy that for the condensed state in statistics. The basic equations of these two theories differ considerably in their form and particular techniques used for simplified treatment of the fluctuation spectrum as a rule could not be transferred from one theory to another. 

This scheme represents two independent parallel reactions of the enantiomers. A general treatment for thermal kinetic resolution was given by Kagan and Fiaud [42]. For unimolecular photoreactions first-order equations seem to be appropriate [40]. Accordingly the rates are... 

The pseudo-first order rate coefficients Xjk used here were first introduced by Christiansen [3] as "reaction probabilities" Uj. Equivalent quantities are also standard in generalized treatments of chain reactions [4,5]. 

The influence of longitudinal dispersion on the extent of a first-order catalytic reaction has been studied by Kobayashi and Arai (K14), Furusaki (F13), van Swaay and Zuiderweg (V8), and others. They use the one-dimensional two-phase diffusion model, and show that longitudinal dispersion of the emulsion has little effect when the reaction rate is low. Based on the circulation flow model (Fig. 2) Miyauchi and Morooka (M29) have shown that the mechanism of longitudinal dispersion in a fluidized catalyst bed is a kind of Taylor dispersion (G6, T9). The influence of the emulsion-phase recirculation on the extent of reaction disappears when the term tp defined by Eq. (7-18) (see Section VII) is greater than about 10. For large-diameter beds, where p does not satisfy this restriction, their general treatment includes the contribution of Taylor dispersion for both the reactant gas and the emulsion (M29). 

---