Continuous-deterministic reaction kinetics

Consider the simple reversible reaction between two chemical compounds A and B, 

The reaction rates and —rs can be defined as the change in the number of moles of A and B, respectively, per unit volume per unit time. By definition 

In this simple case of a reversible reaction, only one of the two concentrations is an independent variable, say Ca, with Cb completely determined by Ca and the initial conditions. 

The ordinary differential equation, Eq. 13.5, can be written in terms of numbers of molecules, multiplying both sides by the constant volume. 

Equation 13.7 can be solved with initial conditions iV,o = iV, (t = 0). It is convenient to introduce a new variable, the conversion of A, defined as 

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On the other hand, if the numbers of reacting molecules are very small, for example in the order of O(10 iV ), then integer numbers of molecules must be modeled along with discrete changes upon reaction. Importantly, the reaction occurrences can no longer be considered deterministic, but probabilistic. In this chapter we present the theory to treat reacting systems away from the thermodynamic limit. We first present a brief overview of continuous-deterministic chemical kinetics models and then discuss the development of stochastic-discrete models. 

In a continuous-flow chemical reactor, the concern is not only with probabilistic transitions among chemical species but also with probabilistic liansitions of each chemical species between the interior and exterior of the reactor. Pippel and Philipp [8] used Markov chains for simulating the dynamics of a chemical system. In their approach, the kinetics of a chemical reaction are treated deterministically and the flow through the system are treated stochastically by means of a Markov chain. Shinnar et al. [9] superimposed the kinetics of the first order chemical reactions on a stochastically modeled mixing process to characterize the performance of a continuous-flow reactor and compared it with that of the corresponding batch reactor. Most stochastic approaches to analysis and modeling of chemical reactions in a flow system have combined deterministic chemical kinetics and stochastic flows. 

A brief coverage of stochastic processes in general, and of stochastic reaction kinetics in particular. Many dynamical systems of scientific and technological significance are not at the thermodynamic limit (systems with very large numbers of particles). Stochasticity then emerges as an important feature of their dynamic behavior. Traditional continuous-deterministic models, such as reaction rate... 

Figure 18.1 Regimes of the problem space for multiscale stochastic simulations of chemical reaction kinetics. The r-axis represents the number of molecules of reacting species, x, and the ) -axis measures the frequency of reaction events, A. The threshold variables demarcate the partitions of modeling formalisms. In area I, the number of molecules is so small and the reaction events are so infrequent that a discrete-stochastic simulation algorithm, like the SSA, is needed. In contrast, in area V, which extends to infinity, the thermodynamic limit assumption becomes vahd and a continuous-deterministic modehng formalism becomes valid. Other areas admit different modehng formalisms, such as ones based on chemical Langevin equations, or probabilistic steady-state assumptions.

Abstract chemical models exhibiting nonlinear phenomena were proposed more than a decade ago. The Brusselator of PRIGOGINE and LEFEVER [54] has oscillatory (limit cycle) solutions, and the SCHLOGL [55] model exhibits bistability, but these models have only two variables and hence cannot have chaotic solutions. At least 3 variables are required for chaos in a continuous system, simply because phase space trajectories cannot cross for a deterministic system. As mentioned in the Introduction, the possibility of chemical chaos was suggested by RUELLE [1] in 1973. In 1976 ROSSLER [56], inspired by LORENZ s [57] study of chaos in a 3 variable model of convection, constructed an abstract 3 variable chemical reaction model that exhibited chaos. This model used as an autocatalytic step a Michaelis-Menten type kinetics, which is a nonlinear approximation discovered in enzymatic studies. Recently more realistic biochemical models [58,59] have also been found to exhibit low dimensional chaos. 

The most complex systems of kinetic equations cannot be solved analytically. In addition, when two of the differential equations of these systems describe processes that occur on drastically different timescales, their numerical integration using methods involving finite increments is unstable and unreliable. These methods are inherently deterministic, since their time evolution is continuous and dictated by the system of differential equations. Alternatively, we can apply stochastic methods to determine the rates of these reactions. These methods are based on the probabihty of a reaction occurring within an ensanble of molecules. This prob-abihstic formulation is a reflection either of the random nature of the coUisions that are responsible for bimolecular reactions or of the random decay of molecules undergoing unimolecular processes. Stochastic methods allow us to study complex reactions without either solving differential equations or supplying closed-form rate equations. The method of Markov chains... 

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