Weibull Cumulative Distribution Function for Modeling the Isothermal Kinetics of the Titanium-oxo-alkoxy Cluster Growth

Abstract

The isothermal kinetics of the titanium-oxo-alkoxy cluster (TOAC) growth obtained in the controlled hydrolysis–condensation reaction of Ti(OPri)4 with H2O in n-propanol solution was investigated. Kinetic curves of the TOAC growth were measured at five different temperatures: 298, 303, 308, 313, and 318 K. It was established that the kinetic curves of the TOAC growth could be mathematically modeled by the cumulative two-parameter Weibull probability distribution function of the growth time (t) in a wide range of the degree of growth (0.05 ≤ DG ≤ 0.95). It was found that the dependencies of the shape parameter (β) and scale parameter (η) on temperature (T) can be described by polynomial functions. The activation energy of TOAC growth (Ea,DG), as well as the pre-exponential factor (ln ADG) for different degrees of growth, were calculated using Friedman’s isoconversional method. On the basis of knowing Ea,DG(DG) dependence, the procedure for calculating isothermal dependence of Ea vs growth time was described. The isothermal probability distribution density functions of Ea for the TOAC growth process were calculated. It was established that the conversion kinetic curves of the TOAC growth process could be fully described by a model of the infinite number of parallel first-order growth reactions with time-varying value of the pre-exponential factor and activation energies which are distributed in accordance with a specific function of Ea.