Standard chemical kinetics

A careful study of the fluctuation-controlled kinetics performed in recent years has led us to numerous deviations from the results of generally-accepted standard chemical kinetics. To prevent readers from getting lost in details of different formalisms and the ocean of equations presented in this book, we present in this introductory Chapter a brief summary, explain the necessity of developing the fluctuation kinetics and demonstrate its peculiarities compared with techniques presented earlier. We will use here the simplest mathematical formalism and focus on basic ideas which will be discussed later on in full detail. 

The standard chemical kinetics as mean-field theory... 

The above mentioned assumptions of standard chemical kinetics have a certain analog with the mean field theory, i.e., both use the order parameters only and assume validity of the polynomial expansion in this parameter as it is shown in Fig. 1.5. 

The Smoluchowski equation demonstrates the principal feature of the standard chemical kinetics the latter is defined by a coefficient of the relative... 

The next reasonable step in studying our chemical games is to consider ensembles of A s and B s (e.g., topers and policemen), when they are randomly and homogeneously distributed in the reaction volume and are characterized by macroscopic densities of a number of particles. The peculiarity of the A + B -y B reaction is that the solution of a problem with a single A could be extrapolated for an ensemble of A s (in other words, a problem is linear in particles A). As it was said above, it is analytically solvable for Da = 0 but turns out to be essentially many-particle for Db = 0. It is useful to analyze a form of the solution obtained for the particle concentration tia (t) in terms of the basic postulates of standard chemical kinetics (i.e., the mean-field theory). 

The transition from a stable steady-state solution observed at large p to the oscillatory regime assumes the existence of the critical value of the parameter pc, which defines the point of the kinetic phase transition as p > pc, the fluctuations of the order parameter are suppressed and the standard chemical kinetics (the mean-field theory) could be safely used. However, if p < pc, these fluctuations are very large and begin to dominate the process. Strictly speaking, the region p pc at p > pc is also fluctuation-controlled one since here the fluctuations of the order parameter are abnormally high. 

What was said above is illustrated by Fig. 1.29 and Fig. 1.30 corresponding to the cases p > pc and p < pc respectively. To make the presented kinetic curves smooth, in these calculations the transformation rate A — B was taken to be finite. To make results physically more transparent, the effective reaction rate K (t) of the A —> B transformation is also drawn. The standard chemical kinetics would be valid, if the value of K (t) tends to some constant. However, as it is shown in Fig. 1.30, K(t) reveals its own and quite complicated time development namely its oscillations cause the fluctuations in particle densities. The problems of kinetic phase transitions are discussed in detail in the last Chapter of the book. 

Standard chemical kinetics systems with complete reactant mixing... 

The conclusion suggests itself that the decay law n(t) oc t l obtained earlier in terms of standard chemical kinetics (2.1.8) is replaced by a slower decay. 

Therefore, the standard chemical kinetics overestimates the diffusive smoothing out of initial density inhomogeneities as compared to the thermal fluctuation level (2.1.42). 

Its obvious peculiarity as compared with the standard chemical kinetics, equation (2.1.10), is the emergence of the fluctuational second term in r.h.s. The stochastic reaction description by means of equation (2.2.37) permits us to obtain the equation for dispersions crjj which, however, contains higher-order momenta. It leads to the distinctive infinite set of deterministic equations describing various average quantities, characterizing the fluctuational spectrum. 

Keeping in mind the physical meaning of single- and two-particle densities, equations (2.3.58) to (2.3.61), we can rewrite now equations (4.1.13) in a form similar to the standard chemical kinetics (Section 2.1). To do it, let us first transform the integral expression... 

All the above-said demonstrates well that there are arguments for and against applicability of the superposition approximation in the kinetics of bimolecular reactions. Because of the absence of exactly solvable problems, it is computer simulation only which can give a final answer. Note at once some peculiarities of such computer simulations. The largest deviations from the standard chemical kinetics could be expected at long t (large ). Unlike computer simulations of equilibrium phenomena [4] where the particle density is constant, in the kinetics problems particle density n(t) decays in time which puts natural limits on time of reaction. An increase of the standard deviation at small values of N(t) = (N) when calculating the mean concentration in computer simulations compel us to interrupt simulations at the reaction depth r = Io 3, where... 

The simplest class of bimolecular reactions involves only one type of mobile particles A and could result either in particle coagulation (coalescence, fusion) A + A —> A, or annihilation, A + A — 0 (inert product). Their simplicity in conjunction with the simple topology of d = 1 allows us to solve the problem exactly, which makes it very attractive for testing different approximations and computer simulations. In the standard chemical kinetics (i.e., mean-field theory, Section 2.1.1) we expect in d = 2 and 3 for both reactions mentioned trivial behaviour quite similar to the A+B — 0 reaction, i.e., tia( ) oc t-1, as t — oo. For d = 1 in terms of the Smoluchowski theory the joint density obeys respectively the equation (4.1.56) with V2 = and D = 2Da. 

Little attention is paid usually to the manifestation of many-particle effects not at asymptotically long, but relatively short times, e.g., during the transient process when a random particle distribution changes for the quasisteady state and the reaction rate, in terms of standard chemical kinetics [100], equation (4.1.61), has to obey the following equation (for d = 3, in dimensionless units)... 

Fig. 5.18. The non-stationary part of the reaction rate K(t) for the transient kinetics of the A + A 0 reaction (curves 1 to 3) and A + B —> 0 reaction (curve 4), d = 3 (random initial distribution) [101]. The broken line shows prediction of the standard chemical kinetics, equation (5.3.10). The initial concentrations are given.

This asymptotic decay law means that at long time reaction is described formally by the third-order kinetics [68, 102, 103] which is very unusual for the standard chemical kinetics ... 

Despite the fact that formalism of the standard chemical kinetics (Chapter 2) was widely and successfully used in interpreting actual experimental data [70], it is not well justified theoretically in fact, in its derivation the solution of a pair problem with non-screened potential U (r) = — e2/(er) is used. However, in the statistical physics of a system of charged particles the so-called Coulomb catastrophes [75] have been known for a long time and they have arisen just because of the neglect of the essentially many-particle charge screening effects. An attempt [76] to use the screened Coulomb interaction characterized by the phenomenological parameter - the Debye radius Rd [75] does not solve the problem since K(oo) has been still traditionally calculated in the same pair approximation. 

In this Section we consider the case when charge screening could be principally non-equilibrium and thus equations of the standard chemical kinetics are no longer valid [24, 78]. To demonstrate it, let us assume that particles of one kind, say A, are immobile (Da = 0). This situation was considered... 

Despite the fact that from a principal point of view a problem of concentration oscillations could be considered as solved [4], satisfactory theoretical descriptions of experimentally well-studied particular reactions are practically absent. Due to very complicated reaction mechanism (in order to describe the Belousov-Zhabotinsky reaction even in terms of standard chemical kinetics several tens of concentration equations for intermediate products should be written down and solved numerically [4, 9, 10]) these equations contain large number of ill-defined parameters - reaction rates [10]. 

As it follows from the above-said, nowadays any study of the autowave processes in chemical systems could be done on the level of the basic models only. As a rule, they do not reproduce real systems, like the Belousov-Zhabotinsky reaction in an implicit way but their solutions allow to study experimentally observed general kinetic phenomena. A choice of models is defined practically uniquely by the mathematical formalism of standard chemical kinetics (Section 2.1), generally accepted and based on the law of mass action, i.e., reaction rates are proportional just to products of reactant concentrations. 

Staying within a class of mono- and bimolecular reactions, we thus can apply to them safely the technique of many-point densities developed in Chapter 5. To establish a new criterion insuring the self-organisation, we consider below the autowave processes (if any) occurring in the simplest systems -the Lotka and Lotka-Volterra models [22-24] (Section 2.1.1). It should be reminded only that standard chemical kinetics denies their ability to selforganisation either due to the absence of undamped oscillations (the Lotka model) or since these oscillations are unstable (the Lotka-Volterra model). 

Deviation from standard chemical kinetics described in (Section 2.1.1) can happen only if the reaction rate K (t) reveals its own non-monotonous time dependence. Since K(t) is a functional of the correlation functions, it means that these functions have to possess their own motion, practically independent on the time development of concentrations. The correlation functions characterize the intermediate order in the particle distribution in a spatially-homogeneous system. Change of such an intermediate order could be interpreted as a series of structural transitions. 

Statement 1) a stable stationary solution of a complete set of equations of the Lotka model holds. At long t the reaction rate K(t) strives for the stationary value. Time development of concentrations obeys standard chemical kinetics, Section 2.1.1. 

This theory, as originated from the early work of Smoluchowski [20], nowadays has numerous applications in several branches of chemistry, such as colloidal chemistry, aerosol dynamics, catalysis and the physical chemistry of solutions as well as in the physics and chemistry of the condensed state [21-24]. Until recently, its branch called standard chemical kinetics [12, 15, 16] based on the law of mass action seemed to be quite a complete and universal theory. However, because of their entirely phenomenological character, theories of this kind always operate with the reaction rates K which are postulated to be time-independent parameters. 

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