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 reaction 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 equilibria A ⇋ B, C ⇋ D, E ⇋ F. A general method is developed for the spectroscopic-kinetic analysis of linear reactions (s = 3) on the basis of three-dimensional absorbance (A) diagrams (A_λ1 vs. A_λ2 vs. A_λ3). A distorted 'absorbance tetrahedron' can be constructed from the curve running in the absorbance space (called Mauser space). The tetrahedron is generated by tangents and osculating planes belonging to the initial point and endpoint of curve (measured). Planes being parallel to the tetrahedral surfaces and running through the points of curve, can be constructed and brought to intersection with the corresponding sides of tetrahedron. The quantities z_i are introduced with the help of distance relationships on the sides of tetrahedron. The differentiation of z_i with respect to time (ż_i) leads to equations which are linearly dependent on z_i. The solution of these differential equations provides the eigenvalues (r_i) searched. The results obtained are in accordance with Theorem 2 of kinetics (two strictly linear reaction systems having the same number of linearly independent reaction steps cannot be distinguished from each other by purely spectroscopic means). The procedure of evaluation is demonstrated by the investigation of the spontaneous hydrolyses of 4-methylumbelliferyl p-trimethylammonium cinnamate chloride, o-nitrophenylacetate and cinnamoylimidazole in borax buffer (0.1 M; pH = 8.7; 10% acetonitrile temperature 25.0°C).