Application of Weibull distribution function for modelling the isothermal kinetics of the titanium-oxo-alkoxy clusters growth

Abstract

The isothermal kinetics of the titanium-oxo-alkoxy clusters (TOAC) growth obtained in the controlled hydrolysis-condensation reaction of Ti(OPr i)4 with H2O in n-propanol solution was investigated. Kinetic curves of the TOAC growth were measured at five different temperatures: 298 K, 303 K, 308 K, 313 K and 318 K. It was established that the kinetic curves of the TOAC growth could be mathematically modelled by the cumulative two-parameter Weibull probability distribution function of the growth time (t). The forms of functional dependencies on temperature for the shape (β) and the scale (η) parameters were determined. It was shown that the isothermal TOAC growth is a complex kinetic reaction and the activation energy of growth (Ea,α) for different degrees of growth (α), was calculated. Based on knowing Ea,α(α) dependence, the procedure for calculating isothermal dependence of activation energy (E a) on growth time, was described. The isothermal probability distribution density functions of Ea for TOAC growth process, were calculated. It was established that the isothermal kinetics of the TOAC growth can be modelled by an infinite number of parallel first-order growth reactions with time-varying value of the pre-exponential factor (ln A) and activation energies which are distributed in accordance with a specific function of E a. © 2012 Elsevier B.V. All rights reserved.