Invariant Manifolds and Extents of Reactions

The present section is concerned with the following basic property of reacting systems The state variables of a closed uniform system remain at all times in a bounded R-dimensional subset of the iV+1 dimensional state space. To derive this property, we shall first discuss the way in which conservation and thermodynamic principles restrict the admissible class of reaction rate functions. 

Any chemical kinetic model should in principle be consistent with the conservation of atomic species, the nonnegativity of concentrations and temperature, and the first and second laws of thermodynamics. The first requirement was considered in the previous section and led to Eq. (1.1.9), for the elements of the stoichiometric matrix. The nonnegativity of concentrations and temperature will be insured by Postulate 1.2.1, formulated below. The first law of thermodynamics is included in the energy equation describing a reacting system. These three requirements will be assumed to be satisfied throughout this work. 

The requirement that the kinetic model obeys the second law of thermodynamics is often inconvenient as it involves expressions for entropy, chemical potentials, etc., which are not available for all systems. Moreover, some empirical kinetic models, which describe chemical systems approximately, may not be consistent with the second law over the entire range of interest. Yet these models, often the only ones available, are quite useful. We shall study, therefore, kinetic models which satisfy the other requirements mentioned above but do not necessarily obey the second law of thermodynamics. Section 1.5 is concerned with kinetic models satisfying the second law of thermodynamics. For a somewhat different formulation of the admissible class of reaction kinetic models see the paper by Wei [40]. 

When only the concentrations are of interest we shall let c = (ci,c y) and denote by, the corresponding Euclidean space and its posi- 

It is convenient to consider an isolated system of unit volume whereby C/, the total internal energy of the system, is also the internal energy per unit volume. The time evolution of this system is described by the equa-tions ,  

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