Monomolecular systems

Straight-Line Reaction Paths. For a three-component reversible monomolecular system only two straight-line reaction paths exist both can be observed experimentally. Normally, the slow straight-line reaction path is estimated as the tangent to any curved-line reaction path at the equilibrium composition. This path is subsequently determined more precisely in the laboratory. The locus of the second, or fast, straight-line reaction path is then calculated (1). 

Since we are considering a 2D system, we note that this is a monomolecular system and all of the centers of mass of the molecules are in the same plane (even if individual nuclei in the molecule are not). S Q), introduced earlier in this chapter, therefore only depends on the component of Q projected onto the scattering plane, Qparaiiei Just as Warren did, the case of lamellar systems like graphite and mica can now be considered, assuming that the scattering system has random orientations of the crystalhtes about an axis normal to the basal (or 2D) plane. Because the magnitude of the parallel component of the structure factor 5 (Q) is the relevant quantity... 

Thus, K has a complete set of eigenvectors that is mutually orthogonal with respect to [,], and has real eigenvalues. This is the essential result for the Wei and Prater (1962) analysis of reaction pathways in monomolecular systems. The latter... 

While Eq. (70) holds in general for monomolecular systems, it holds only in a neighborhood of equilibrium for the general case considered by Krambeck (1994) ... 

Consider first a discrete monomolecular system in which every species present can transform to any other one by a first-order reaction. In such a system, the kinetic equation at all possible compositions is given by Eq. (70), which is repeated here ... 

Krambeck, F. J., Accessible composition domains for monomolecular systems. Chem. Eng. Sci. 39, 1181 (1984b). 

Ozawa, Y., The structure of a lumpable monomolecular system for reversible chemical reactions. Ind. Eng. Chem. Fundament. 12,191 (1973). 

The major part of this article will be devoted to a particular class of reaction systems—namely, monomolecular systems. A reaction system of (n) molecular species is called monomolecular if the coupling between each pair of species is by first order reactions only. These linear systems are satisfactory representations for many rate processes over the entire range of reaction and are linear approximations for most systems in a sufficiently small range. They play a role in the chemical kinetics of complex systems somewhat analogous to the role played by the equation of state of a perfect gas in classical thermodynamics. Consequently, an understanding of their behavior is a prerequisite for the study of more general systems. 

Two subclasses of monomolecular systems will be discussed reversible and irreversible monomolecular systems. A reaction system will be called reversible monomolecular if the coupling between species is by reversible first order reactions only. A typical example of a reversible monomolecular system is... 

The type of approach to be used and its advantage over the conventional approach is illustrated in Section II,A by a brief discussion of the problem of determining the value of the rate constants from experimental data for reversible monomolecular systems. 

Our discussion of monomolecular systems will also provide structural information about an important class of nonlinear reaction systems, which we shall call pseudomonomolecular systems. Pseudomonomolecular systems are reaction systems in which the rates of change of the various species are given by first order mass action terms, each multiplied by the same function of composition and time. For example, the rate equations for a typical three component reversible pseudomonomolecular system are... 

A. The Rate Equations for Reversible Monomolecular Systems 1. The General Solution... 

In the conventional treatment of the kinetics of monomolecular systems, the explicit relations of the rate constants kji to the set of constants (c, X) are obtained only in special cases consequently, even assuming that the constants (c, X) are satisfactorily obtained, the calculation of the values of the rate constants from them is not possible, in general, for the conventional treatment. Although the values of the constants (c, X) are sufficient for determining the composition as a function of time, the rate constants ky,-are more useful quantities since they are the ones more directly related to basic mechanisms. 

The n-component monomolecular system may be treated in exactly the same manner except that an n-dimensional composition space is used. Although n-dimensional spaces with n > 3 cannot be simply put into pictures, a geometrical language still aids our ability to solve problems using the concepts, language, and techniques of two and three dimensional systems. The set of Eqs. (5) reduces to a single equation identical to Eq. (11) except that a is now the column matrix or vector in n-dimensional space given by... 

In addition to the pure component vectors, most of the other composition vectors are also rotated by the matrix K. For reversible n-component monomolecular systems, however, there always exists n independent directions in the composition space such that vectors in these directions will undergo only a change in length under the action of K (see Appendix I for proof). These will be called characteristic directions. Let a/ be any vector in the jth characteristic direction, then... 

Degeneracy in the Values of the Characteristic Roots. For reversible monomolecular systems there are always n independent characteristic directions (see Appendix I for proof). Nevertheless, different unit characteristic vectors may have the same characteristic root. For any two characteristic species with the same value of the characteristic roots, (X,/X,) = 1 and Eq. (58) becomes... 

Conversely, the composition sequence for any initial composition may be determined from the relative matrix as for the monomolecular system. 

A monomolecular system may be defined in terms of the matrix T< instead of the matrix K. For infinitesimal U,... 

These four statements are also sufficient to guarantee that this kinetic system is consistent with the second law of thermodynamics, i.e., that the Gibbs free energy of the system decreases as the reaction proceeds to equilibrium for isothermal, isobaric systems. Statements (1) through (4) may be taken as the axiomatic formulation of monomolecular systems and the properties that we have discussed in the above sections are consequences of them. Further discussions of the equilibrium point and the convergence to it will be found in Section VII. 

We have seen that all w-component reversible monomolecular systems have n — 1 straight line reaction paths and n — 1 decay constants Xy. 

III. The Determination of the Values of the Rate Constants for Typical Reversible Monomolecular Systems Using the Characteristic Directions... 

We shall use special examples to show that the following new features may be exhibited by irreversible monomolecular systems ... 

Some of the new characteristic features of irreversible monomolecular systems not shown by reversible systems may be demonstrated by the three component system... 

When ih/h) —> 1, we have h —> h, Xi —> X2, and Xi —> X2 hence, there are only two characteristic directions and the system is not equivalent to a completely uncoupled system. Monomolecular systems that have too few independent characteristic directions can always be expressed as an equivalent system in which the only coupling is by sequences of irreversible steps. For example, a seven component system with only four independent characteristic directions might be equivalent to... 

For reversible monomolecular systems, the rate constant matrix K is transformed into the diagonal rate constant matrix A by the transformation... 

For reversible monomolecular systems, the left characteristic vector that corresponds to the right characteristic vector Xo is... 

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