# Eyring–Polanyi Rate Theory for the Homogeneous Nucleation of Organic Crystals from Solution

## TL;DR

This paper models organic crystal nucleation using chemical reaction kinetics and shows how nucleation rates can be predicted from solubility data.

## Contribution

The paper introduces a novel Eyring–Polanyi framework to predict homogeneous nucleation rates from temperature-dependent solubility data.

## Key findings

- Nucleation rates exhibit Arrhenius behavior with temperature.
- Activation enthalpy is about 12 times the enthalpy of solution.
- Gibbs energy of activation remains relatively constant across systems.

## Abstract

Using elementary concepts of chemical reaction kinetics,
the rates
of primary homogeneous organic crystal nucleation from supersaturated
solutions are modeled as nucleation with first order kinetics from
large solute density fluctuations (LSDFs). Solute density fluctuations
are modeled as diffusively driven many-body collisions of weakly interacting
solvated solute molecules. The first order rate constant is a system-specific
supersaturation-independent rate constant for nucleation in LSDFs.
It is shown for several solute–solvent systems that the temperature-dependence
of this nucleation rate constant exhibits Arrhenius behavior. The
activation enthalpy (ΔH
⧧)
and activation entropy (ΔS
⧧) for homogeneous nucleation is determined from an Eyring–Polanyi
analysis of temperature-dependent nucleation rates. The steps of the
Eyring–Polanyi analysis are described in detail for the homogeneous
nucleation of l-histidine from aqueous solutions. The analysis
is also applied to temperature-dependent homogeneous nucleation rates
of salicylic acid in four different solvents. For all systems, the
supersaturation- and the temperature-dependence of the primary homogeneous
nucleation rates are completely reproduced by reference to temperature-dependent
solubility data through the activation parameters ΔH
⧧ and ΔS
⧧. ΔH
⧧ is for all examined
systems approximately 12 times the enthalpy of solution determined
from solubility data, suggesting that nucleation from LSDFs resembles,
at the molecular level, a reverse dissolution process. Within the
temperature ranges used for measuring nucleation rates, the Gibbs
energy of activation ΔG
⧧ does
not vary strongly, resulting in an inverse correlation between enthalpies
and entropies of activation. The Eyring–Polanyi framework thus
provides, for the first time, a method for semiquantitative predictions
of homogeneous nucleation rates from temperature-dependent solubilities.