Analysis of linear reaction systems with three linearly independent steps on the basis of the absorbance tetrahedron and the formal integration

Linear reaction systems consist by definition of first-order reaction steps . Linearly independent reactions are independent of reaction order. Each re action mechanism consists of a distinct number (s) of linearly independent reaction steps. Thus, the mechanism A --> B --> C --> D can be described by three linearly independent reactions as it is also true for the multiple e quilibria A reversible arrow B, C reversible arrow D, E reversible arrow F. A general method is developed for the spectroscopic-kinetic analysis of li near reactions (s = 3) on the basis of three-dimensional absorbance (A) dia grams (A(lambda1) vs. A(lambda2) vs. A(lambda3)). A distorted 'absorbance t etrahedron' can be constructed from the curve running in the absorbance spa ce (called Mauser space). The tetrahedron is generated by tangents and oscu lating planes belonging to the initial point and endpoint of curve (measure d). Planes being parallel to the tetrahedral surfaces and running through t he points of curve, can be constructed and brought to intersection with the corresponding sides of tetrahedron. The quantities zi are introduced with the help of distance relationships on the sides of tetrahedron. The differe ntiation of z(i) with respect to time (z(i)) leads to equations which are l inearly dependent on zi. The solution of these differential equations provi des the eigenvalues (r(i)) searched. The results obtained are in accordance with Theorem 2 of kinetics (two strictly linear reaction systems having th e same number of linearly independent reaction steps cannot be distinguishe d from each other by purely spectroscopic means). The procedure of evaluati on is demonstrated by the investigation of the spontaneous hydrolyses of 4- methylumbelliferyl p-trimethylammonium cinnamate chloride, o-nitrophenylace tate and cinnamoylimidazole in borax buffer (0.1 M; pH = 8.7; 10% acetonitr ile; temperature 25.0 degreesC). (C) 2001 Elsevier Science B.V. All rights reserved.