Macroscopic, Deterministic Chemical Kinetics

A macroscopic, deterministic chemical reacting system consists of a number of different species, each with a given concentration (molecules or moles per unit volume). The word macroscopic implies that the concentrations are of the order of Avogadro s number (about 6.02 x 10 ) per liter. The concentrations are constant at a given instant, that is, thermal fluctuations away from the average concentration are negligibly small (more in section 2.3). The kinetics in many cases, but far from all, obeys mass action rate expressions of the type 

Rate coefficients are averages of reaction cross-sections, as measured for example by molecular beam experiments. The a priori calculation of cross-sections from quantum mechanical fundamentals is extraordinarily difficult and has been done to good accuracy only for the simplest trimolecular systems (such as D -I- H2). 

A widely used alternative approach is based on activated complex theory. In its simplest form, two reactants collide and form an activated complex, said to be in equilibrium. One degree of freedom of the complex, a vibration, is allowed to lead to the dissociation of the complex to products, and the rate of that dissociation is taken to be the rate of the reaction. The rate for the forward reaction is 

Reactions among ionic species require explicit consideration of the interactions of the electric charges on the ions. We cite only the primary salt effect formulated by Brpnsted and Bjerrum [1, p. 920]. For a bimolecular reaction between two ions we have 

For reaction between two ions of like charge the rate coefficient increases with increasing ionic strength for reaction between two ions of unlike charge the rate coefficient decreases with increasing ionic strength. 

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A variational formulation of the dynamics of systems of chemical reactions is presented which utilizes a Rayleigh dissipation function to extend the Lagrangian approach to conservative systems to dissipative systems [l]. The formulation is valid arbitrarily far from equilibrium and is based on macroscopic, deterministic chemical kinetics. It thus stands in contrast to variational principles for dissipative systems which are based on studies of fluctuations and are restricted to the near equilibrium regime [z] or, far from equilibrium, to the vicinity of stable stationary states [3]. 

Let us now digress briefly from matters of simulation to introduce the kinds of chemical systems in which nonequilibrium phase transitions can occur. This is most conveniently done at the macroscopic level of deterministic chemical kinetics. Consider a chemical system under open or closed conditions so that it is possible to maintain a time-independent macroscopic state. 

Both deterministic and stochastic models can be defined to describe the kinetics of chemical reactions macroscopically. (Microscopic models are out of the scope of this book.) The usual deterministic model is a subclass of systems of polynomial differential equations. Qualitative dynamic behaviour of the model can be analysed knowing the structure of the reaction network. Exotic phenomena such as oscillatory, multistationary and chaotic behaviour in chemical systems have been studied very extensively in the last fifteen years. These studies certainly have modified the attitude of chemists, and exotic begins to become common . Stochastic models describe both internal and external fluctuations. In general, they are a subclass of Markovian jump processes. Two main areas are particularly emphasised, which prove the importance of stochastic aspects. First, kinetic information may be extracted from noise measurements based upon the fluctuation-dissipation theorem of chemical kinetics second, noise may change the qualitative behaviour of systems, particularly in the vicinity of instability points. 

According to the assumptions on which classical kinetics is based, deterministic models are adequate as long as deviations from the macroscopic average values remain negligible. A number of situations can be listed to argue for relevance of fluctuations in chemical sytems ... 

The equations of motion are obtained in the standard manner from a Lagrangian, the dissipation function, and equations of constraint, which here express conservation of mass. Since "inertial" effects are absent on the macroscopic level of deterministic kinetics, the Lagrangian (at constant temperature and pressure) Is simply the negative of the Gibbs free energy, which is composed of two contributions. The first is the free energy of the internal species of the system the second is due to external sources which control the chemical potentials of some of the internal species and thus allow the system to be driven away from equilibrium. The key to the formulation is the dissipation function, which is written in the standard fashion as a quadratic form in the rates of reaction ty. 

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