An Inflection Point in the Observable Growth Curve for General-order Reaction Process with Power-law Mixing Rule

Abstract

Theoretical treatment of composite or reaction system of food materials requires the mathematical modelling of mixing rule for observables such as thermal, electrical and rheological ones. Typical models are of series, parallel and random type, and are generalized here to a power-law model. In the previous report, the present author developed the kinetics for a general observable ο in the first-order reaction process (R→P) where ο increases with reaction order x≡ [P] / ( [R] + [P] ) through a power-law type mixing rule ο^ν = (1-x) ο^ν_R+xο^ν_P with ο_R<ο_P, and provided a formula for an inflection point (IP) in the observable growth curve. In this article was given a similar argument for IP in the ο-growth curve for a genaral n-th order reaction process dx/dt=K_n (1-x) ⁿ, and the following consequences were obtained. For ν≥1, the ο-t curve is convex everywhere (ο<ο_P) for any positive n. When n is restricted to be a positive integer, the ο-t curve can possess IP in the region -1<ν<1 with smaller values of n (=1, 2…), and so for ν≤-1 only with n=1. For ν<1, the value of observable at IP is given by the formula ο^*= ( (1-ν) / (1- (1-n) ν) ) ^1/νο_P, and the growth rate do/dt depends on ο_R/ο_P for n≠1. The results for the power-law rule coincide in the ν→0 limit with those obtained for a logarithmic rule (a random model) . An inverse observable ο^-1 decreases with a concavity property for any positive integer n.