The Orthogonal Characteristic System

Thansfobmation op the Rate Constant Matrix into A Symmetric Matrix 

In this appendix, we shall prove, for w-component reversible monomolecu-lar reaction systems, that (1) the characteristic roots are real numbers and (2) there are n independent characteristic directions (vectors). It is sufficient to show that such reversible monomolecular reaction systems can always be transformed into an equivalent new reaction system (to be mathematically precise, a similar system) with new coordinate axes such that 

A symmetric matrix has two important properties (1) all of its characteristic roots are real and (2) it has n independent orthogonal characteristic directions (vectors) 72). Thus, the desired conclusions follow immediately when the proof is obtained that the reaction system can be transformed into a similar system with a symmetrical rate constant matrix. This proof also provides the tools for proving the characteristic roots are nonpositive (negative or zero). Furthermore, it also provides the transformation to an orthogonal characteristic system of coordinates useful for consistency checks and for obtaining the inverse of the matrix X. 

Clearly, if ka = ka for all values of i and j, the matrix K is already symmetrical and the above properties follow immediately. This is a very special case, however, where each forward rate constant equals its corresponding backwards rate constant the equilibrium composition will then be equimolar. 

The principle of detailed balancing from statistical mechanics provides a means for converting any asymmetric rate constant matrix K into a sym- 

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Fig. 37. A two component system illustrating the transformation of the orthogonal characteristics system. Fig. (37a) shows a composition vector in both the A and B systems of coordinates. The A system of coordinates is considered to be fixed to the background. The composition vector and the B system of coordinates are considered to be attached to a rubber sheet to which a stretch and shear is applied to obtain the orthogonal B system shown in Fig. (37b).

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