Determination of the activity of a molecular solute in saturated solution

doi:10.1016/j.jct.2008.06.016

The Journal of Chemical Thermodynamics

Volume 40, Issue 12, December 2008, Pages 1684-1692

https://doi.org/10.1016/j.jct.2008.06.016 Get rights and content

Abstract

Prediction of the solubility of a solid molecular compound in a solvent, as well as, estimation of the solution activity coefficient from experimental solubility data both require estimation of the activity of the solute in the saturated solution. The activity of the solute in the saturated solution is often defined using the pure melt at the same temperature as the thermodynamic reference. In chemical engineering literature also the activity of the solid is usually defined on the same reference state. However, far below the melting temperature, the properties of this reference state cannot be determined experimentally, and different simplifications and approximations are normally adopted. In the present work, a novel method is presented to determine the activity of the solute in the saturated solution (=ideal solubility) and the heat capacity difference between the pure supercooled melt and solid. The approach is based on rigorous thermodynamics, using standard experimental thermodynamic data at the melting temperature of the pure compound and solubility measurements in different solvents at various temperatures. The method is illustrated using data for ortho-, meta-, and para-hydroxybenzoic acid, salicylamide and paracetamol. The results show that complete neglect of the heat capacity terms may lead to estimations of the activity that are incorrect by a factor of 12. Other commonly used simplifications may lead to estimations that are only one-third of the correct value.

Introduction

The solubility of organic compounds in various solvents is of significant importance to the industry in the design of manufacturing processes and for the application of the compound and prediction of its behavior. The solubility depends on solute and solvent properties as often described by the relation $a_{eq}^{solute} = γ_{eq} x_{eq} = a,$ where $a_{eq}^{solute}$ is the activity of the solute in the saturated solution, x_eq is the molar solubility concentration, and γ_eq is the activity coefficient in the solution at equilibrium. a is given by $a = \exp ((μ_{Solid} - μ_{solute}^{0}) / RT)$ , i.e. a represents the difference in chemical potential between the solid phase, μ_Solid, and the thermodynamic reference state, $μ_{solute}^{0}$ , in the definition of the activity coefficient. In the chemical engineering literature on phase equilibria, the thermodynamic reference state for the solute in the solution is often defined as the pure compound as a supercooled melt at the temperature in question, by which the activity coefficient is defined within a Raoult’s law framework. The reference state for the solid phase is usually assumed to equal that of the solute in the solution: $μ_{Solid}^{0} = μ_{solute}^{0}$ , and a is called the activity of the solid phase, obviously exhibiting values different from unity. In the chemistry literature, often the activity of the solid phase is set to unity, i.e. the reference state for the solid phase is the solid phase itself: $μ_{Solid} = μ_{Solid}^{0}$ . However, then $μ_{Solid}^{0} \neq μ_{solute}^{0}$ and we still need to provide an estimate for a above. In the present work we will adopt the chemical engineering terminology and denote a the activity of the solid phase. In this case, the activity of the solid equals the activity of the solute in the saturated solution.

Current and former research efforts in predicting solubility have primarily focused on resolving the influence of the solvent on solubility by prediction of the activity coefficient, e.g. via solubility parameter methods [1], [2] and group contribution methods, e.g. UNIFAC [3]. However, the accuracy in the prediction of solubility is, equally dependent on the estimation of the solid-state activity, and this has received much less attention.

The activity of the solid depends on the enthalpy of fusion at the temperature of interest. However, far away from the melting temperature the reference state of a supercooled melt is in practice experimentally inaccessible. Hence, simplifications and approximations are used. A standard simplification often encountered in the engineering literature is to assume that the enthalpy of fusion is independent of temperature $\ln a = \frac{Δ H^{f} (T_{m})}{R} (\frac{1}{T_{m}} - \frac{1}{T}) .$ An alternative approach has been suggested [1]: $\ln a = \frac{Δ H^{f} (T_{m})}{{RT}_{m}} \ln (\frac{T}{T_{m}})$ and a less simplified approach leads to $\ln a = \frac{Δ H^{f} (T_{m})}{R} [\frac{1}{T_{m}} - \frac{1}{T}] - \frac{Δ C_{p} (T_{m})}{R} [\ln (\frac{T_{m}}{T}) - \frac{T_{m}}{T} + 1],$ where ΔC_p(T_m) is the difference in heat capacity between the melt and the solid at the melting temperature. The simplifications behind equations (2), (3), (4), inevitably result in, more or less, poor estimations of the activity of the solid, and accordingly to that the activity coefficients at equilibrium, calculated from experimental data by equation (1), are of questionable quality.

In the present paper is presented a novel approach to determine the activity of the solid phase that avoids the assumptions behind equations (2), (3), (4). In this approach, the activity of the solid phase is determined from the solubility and temperature dependence of solubility of the compound in different solvents and at different temperatures. Together with data on melting temperature and enthalpy of melting, these solubility data are used within a rigorous thermodynamic framework to determine the activity of the solid phase. For illustration, the method is applied to five organic compounds, viz. para-, meta-, and ortho-hydroxybenzoic acid, salicylamide, and paracetamol.

Section snippets

Experimental

Most of the experimental data required for the present work have been published previously [4], [5], [6], [7], [8]. However, most of the heat capacity data of interest for comparison and for evaluation of the heat capacity of the supercooled melt has been determined here.

Heat capacity measurements at atmospheric pressure were performed using Differential Scanning Calorimetry, TA Instruments, DSC2920, from 283 K, in 5 K increments, to approximately 30 K above the melting temperature of o

Theory

In the usual definition of the activity of the solid phase, a, of a molecular compound, the hypothetical supercooled melt is used as the thermodynamic reference state, and $\ln a = \frac{Δ H^{f} (T_{m})}{R} [\frac{1}{T_{m}} - \frac{1}{T}] - \frac{1}{RT} \int_{T_{m}}^{T} Δ C_{p} d T + \frac{1}{R} \int_{T_{m}}^{T} \frac{Δ C_{p}}{T} d T,$ where ΔH^f(T_m) is the enthalpy of fusion at the melting temperature, and ΔC_p is given by $Δ C_{p} = C_{p} (l) - C_{p} (s) .$ Obviously, a is only dependent on properties of the pure compound, and represents the difference in chemical potential between the solid state and the supercooled melt.

Evaluation

The experimental correlations between $Δ H_{Soln}^{vH}$ and ln(x_eq), equation (16), for the investigated compounds are shown in figure 4 at T = 288.15 K (left) and T = 308.15 K (right).

The diagrams in figure 4 are representative for the compounds at all investigated temperatures. The data are well correlated by a second-order relation of the type: $Δ H_{Soln}^{vH} / (J \cdot {mol}^{- 1}) = α {(\ln (x_{eq}))}^{2} + β \ln (x_{eq}),$ which automatically satisfies the condition that the van’t Hoff enthalpy of solution approaches zero when x_eq approaches unity.

Conclusions

Within a rigorous thermodynamic framework, the activity of the solute in a saturated solution, i.e. the activity of the solid phase, can be determined by combining standard thermodynamic experimental data at the melting temperature with measurements over solubility in different solvents at different temperatures. This allows for accurate determination of the activity coefficients from experimental solubility data, and for determination of the heat capacity of the supercooled melt at room

Acknowledgements

The authors acknowledge the Swedish Research Council and the Industrial Association for Crystallization Research and Technology for financial support.

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Determination of the activity of a molecular solute in saturated solution

Abstract

Introduction

Section snippets

Experimental

Theory

Evaluation

Conclusions

Acknowledgements

Eur. J. Pharm. Sci.

J. Pharm. Sci.

Regular and Related Solutions

Molecular Thermodynamics of Fluid-phase Equilibria

AIChE J.

J. Chem. Eng. Data

J. Chem. Eng. Data