Semiempirical Equations for Modeling Solid-State Kinetics Based on a Maxwell−Boltzmann Distribution of Activation Energies: Applications to a Polymorphic Transformation under Crystallization Slurry Conditions and to the Thermal Decomposition of AgMnO4 Crystals

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Abstract
Many solid-state reactions and phase transformations performed under isothermal conditions give rise to asymmetric, sigmoidally shaped conversion−time (x−t) profiles. The mathematical treatment of such curves, as well as their physical interpretation, is often challenging. In this work, the functional form of a Maxwell−Boltzmann (M−B) distribution is used to describe the distribution of activation energies for the reagent solids, which, when coupled with an integrated first-order rate expression, yields a novel semiempirical equation that may offer better success in the modeling of solid-state kinetics. In this approach, the Arrhenius equation is used to relate the distribution of activation energies to a corresponding distribution of rate constants for the individual molecules in the reagent solids. This distribution of molecular rate constants is then correlated to the (observable) reaction time in the derivation of the model equation. In addition to providing a versatile treatment for asymmetric, sigmoidal reaction curves, another key advantage of our equation over other models is that the start time of conversion is uniquely defined at t = 0. We demonstrate the ability of our simple, two-parameter equation to successfully model the experimental x−t data for the polymorphic transformation of a pharmaceutical compound under crystallization slurry (i.e., heterogeneous) conditions. Additionally, we use a modification of this equation to model the kinetics of a historically significant, homogeneous solid-state reaction: the thermal decomposition of AgMnO4 crystals. The potential broad applicability of our statistical (i.e., dispersive) kinetic approach makes it a potentially attractive alternative to existing models/approaches.

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