(451h) Predicting the Temperature Dependence of Sucrose and ?-Glycine Aqueous Solubility from Thermodynamic Data Measured at a Single Temperature

Solubility is a fundamental thermodynamic property in the context of chemical engineering and, more specifically, crystallisation. In general, the solubility of a solid in liquid varies with temperature and, in a practical sense, it is crucial that the temperature dependence is known to allow design, development and optimisation of efficient crystallisation processes. In principle, the solubility can be measured directly; however, in practice, measurements can be challenging, particularly at extremes of temperature and composition. In addition, direct measurements can be time consuming, labour intensive and require a significant amount of solid which may not always be available [1]; thus, alternative approaches mitigating these issues are attractive.

Thermodynamics provides a framework from which models for solubility can be derived. In terms of estimating solubility over a broad range of temperature these models are applied in different ways. At one extreme, models are regressed against a number of solubility measurements and interpolated/extrapolated to ‘predict’ the solubility at other temperatures [2]. Alternatively, model parameters are evaluated through measurement of thermodynamic quantities aside from solubility. The most common thermodynamic models for solubility require pure solute melting enthalpy, melting temperature, solute heat capacity and solute activity in solution [1] [2] [3]. However, this approach is limited for systems where the melting properties are not experimentally accessible, due to thermal decomposition or polymorph transition.

In this work, we present a novel approach to determine the temperature dependence of solubility based on Equation 1 [4] (see supplementary file). By approximating the temperature and composition dependence of a and b (Equation 2) with a 1^st-order Taylor Series expansion, we derive an approximate equation for the temperature dependence of solubility, given in Equation 3 (see supplementary file).

Crucially, the 1^st-order Taylor series expansion makes it possible to evaluate coefficients from data measured at a single temperature and we show that the value of each expansion coefficients in Equation 3 (see supplementary file) is related to combinations of experimentally accessible thermodynamic quantities (e.g. solvent activity, enthalpy of dilution and solution, as well as pure solute and solution heat capacity), none of which include pure solute melting properties. Once evaluated, knowledge of the expansion coefficients allows numerical integration of Equation 3 (see supplementary file) and estimation of solubility at any temperature, if the solubility at the chosen reference temperature is known.

We apply this modelling approach to two well-studied systems – α-glycine-water and sucrose-water – for which a substantial amount thermodynamic data is available at 25^oC. In addition, both systems have well defined solubility curves across a broad range of temperature. For each system, we show that, for the temperature range over which the solubility is satisfactorily known (i.e. 270 – 330 K for glycine and 270 – 370 K for sucrose) our approach predicts the solubility in agreement with direct measurements. Results for a-glycine-water are shown in Figure 1 (see supplementary file), where open circles indicate direct solubility measurements and blue lines indicate solubility predictions using Equation 3 (see supplementary file) (please note: blue dashed and solid lines correspond to solvent activity modelled by the Scatchard-Hildebrand and Scatchard-Hildebrand-Flory-Huggins excess Gibbs free energy models respectively).

We test the robustness of solubility predictions by estimating the impact of uncertainty in experimental data used to estimate expansion coefficients. In addition, given that certain expansion coefficients are related to partial derivates of thermodynamic properties that cannot be measured directly, we estimate the impact of models used to correlate experimental data. We show that solubility predictions far from the chosen reference temperature (298.15 K) are sensitive to both factors and the relevance of this is discussed.

The primary advance achieved with this work is the development novel approach to predict solubility as a function of temperature that, in theory, relies thermodynamic data measured at a single temperature only – mitigating the need to perform challenging direct solubility measurements at extremes of temperature and/or composition. In addition, the approach by-passes limitations imposed by conventional thermodynamic models that rely on accurate measurement of solute melting properties (melting temperature and enthalpy). This has implications for chemical engineering fields that rely on accurate and knowledge of solubility (i.e. crystallisation), providing an alternative framework to approach the prediction of solubility as a function of temperature.

References:

[1] - F. L. Nordstrom and A. C. Rasmuson, “Prediction of solubility curves and melting properties of organic and pharmaceutical compounds,” European Journal of Pharmaceutical Science, vol. 36, pp. 330-344, 2009.

[2] - M. Svard and A. C. Rasmuson, “(Solid + liquid) solubility of organic compounds in organic solvents - Correlation and extrapolation,” J. Chem. Thermodynamics, vol. 76, pp. 124-133, 2014.

[3] - F. L. Nordstrom and A. C. Rasmuson, “Determination of the activity of a molecular solute in saturated solution,” J. Chem. Thermodynamics, vol. 40, pp. 1684-1692, 2008.

[4] - A. T. Wiliamson, “The exact calculation of heats of solution from solubility data,” Trans. Faraday Soc., vol. 40, pp. 421-436, 1944