Based on a concept commonly applied in mathematical programming known as duality, Mitsos and Barton [1] presented a consistent formulation for Gibbs free energy minimization as a dual extremum principle (DEP) in thermodynamics to identify points that satisfy the necessary and sufficient conditions for thermodynamic stability. Following their work, Pereira et al. [2],[3] extended the DEP framework to the volume-composition space, using the Helmholtz free energy and the relationship between optimality conditions and pressure. This approach is known as the HELD (Helmholtz free energy Lagrangian dual) algorithm and allows for the phase equilibrium problem to be solved rigorously without assumptions of the number of phases or initial guesses of the phase compositions. However, the original formulations do not account for ionic species or chemical reactions in any of the phases at equilibria. I will discuss the introduction of additional constraints in the minimization of the Gibbs free energy, such as macroscopic electroneutrality and suitable relationships between molar amounts and extents of reaction using transformed coordinates [4], that restrict the solution space to compositions that are at chemical equilibria. The extended algorithm and is tested in combination with the SAFT-γ Mie group-contribution equation of state [5] to model complex non-ionic and ionic reactive systems with industrial relevance. For instance, dissolved salts can significantly affect the phase equilibria of mixed-solvent solutions. Depending on the nature of ions and organic compounds present phenomena such as salting-out and salting-in lead to dramatic changes in solubility. In the same context, one of the more intriguing examples of water’s effects on biological molecules is the zwitterion formation of naturally occurring amino acids. α-amino acids exist as zwitterions in aqueous solutions, with the neutral form not present to any measurable degree; they contain both positive and negative charged groups but with an overall neutral charge (no net charge). At high or low pH values they behave as weak acids or weak bases, respectively, leading to the formation of charged species with a remarkable effects on the activity of the solutions. I will present results of the solubility of organic solvents (e.g., alcohols, ketones, and organic acids) in water and to the corresponding changes in their phase behaviour on the addition of mono and divalent alkali halide salts, and will describe the thermodynamic behaviour of α-amino acids (e.g., glycine, alanine) as zwitterions, predicting the distribution of species as a function of pH.
[1] Mitsos, A. and Barton, P.I., 2007. A dual extremum principle in thermodynamics. AIChE Journal, 53(8), pp.2131-2147.
[2] Pereira, F.E., Jackson, G., Galindo, A. and Adjiman, C.S., 2010. A duality-based optimisation approach for the reliable solution of (P, T) phase equilibrium in volume-composition space. Fluid Phase Equilibria, 299(1), pp.1-23
[3] Pereira, F.E., Jackson, G., Galindo, A. and Adjiman, C.S., 2012. The HELD algorithm for multicomponent, multiphase equilibrium calculations with generic equations of state. Computers & chemical engineering, 36, pp.99-118
[4] Ung, S., & Doherty, M. F. (1995). Theory of phase equilibria in multireaction systems. Chemical Engineering Science, 50(20), 3201-3216.
[5] Eriksen, D.K., Lazarou, G., Galindo, A., Jackson, G., Adjiman, C.S. and Haslam, A.J., 2016. Development of intermolecular potential models for electrolyte solutions using an electrolyte SAFT-VR Mie equation of state. Molecular Physics, 114(18), pp.2724-2749.
