A Thermodynamic Approach for Predicting Physical-chemical Properties

2.1 A brief historical overview

Since the beginning of physical chemistry, tens of thousands of articles have been written in order to understand the behavior of fluid mixtures. Although there is no general theory on the properties of fluid mixtures, a large number of theories and models whose area of application is restricted to a particular type of mixture are available [PRA 99]. The Gibbs free energy is a function of the mole fractions of the different constituents of the mixture. A first approach consists therefore of writing a limited expansion of this function and, according to the order of the expansion, the Margules, Redlich–Kister or Van Laar equations appear [HAL 68, PRA 99]. However, although series expansion is very flexible, (1) it does not explain physical and chemical phenomena occurring in a mixture and (2) it has mainly no predictive power since the phase transition of binary mixtures to ternary mixtures requires the adjustment of additional parameters.

In order to construct a theory of liquid mixtures, it is necessary to know two types of information: the structure of liquids (i.e. how the molecules in the liquid are disposed in space) and the intermolecular forces between similar or dissimilar molecules. However, information about the second type of information is insufficient and as a consequence all the theories must make simplifying hypotheses to overcome this drawback [PRA 99]. Theoretical works have concerned liquid mixtures whose molecules are apolar and spherical in shape: for example, the theory independently developed in 1933 by Scatchard and Hildebrand [PRA 99] frequently used for hydrocarbon mixtures by virtue of the quality of these predictions. All these theories have subsequently been extended – in a semi-empirical way in general – to more complexly shaped molecules. Thus Wilson [WIL 64] has developed a model in which the notion of local composition is introduced: the concentration of molecules i around a molecule j is different from the concentration of molecules j around a molecule i. Renon and Prausnitz [REN 68] introduced this concept into Scott’s equation [SCO 56] and obtained the non-random two liquid (NRTL) equation. This equation constitutes an advance insofar as, based on the parameters of binary mixtures, it becomes possible to predict the behavior of ternary mixtures without any additional parameters to adjust.

With the advent of computer resources at the end of the 1960s, the prediction methods of the activity coefficient that have been developed rely both on the concept of local composition as well as on the principle of group contribution. In the group contribution method, the basic idea is that if tens of millions of different chemical compounds have been identified, the number of functional groups constituting these compounds is much narrower: depending on models, this number varies between 30 and 100. Expanding the idea of group contribution to mixtures is thus very attractive: while the number of pure compounds may be high and consequently the number of possible mixtures much more significant by several orders of magnitude, the number of functional groups remains the same. The problem is therefore reduced to the representation of a low number of functional groups and of their interactions. The first fundamental assumption of the group contribution method is to assume that the physical property of a fluid is the sum of contributions due to the functional groups of molecules. This makes it possible to correlate the properties of a large number of fluids based on a small number of parameters that characterize individually the contributions of the groups. The second fundamental hypothesis of a group contribution method is additivity: the contribution of a group in a molecule is independent of that of another group in this molecule. This hypothesis is valid only when the influence of a group in a molecule is not affected by the nature of the other groups within this molecule. Consequently, any one group contribution method is necessarily approximate since the contribution of a particular group in a molecule is not necessarily the same as in another molecule.

Several methods of group contributions have been developed: Analytical Solution of Groups (ASOG) [WIL 62], Universal Quasichemical (UNIQUAC) [ABR 75] and Uniquac Functional Group Activity Coefficient (UNIFAC) [FRE 75]. The UNIQUAC method requires the adjustment of two parameters from the measurements of liquid–vapor equilibrium in binary mixtures, in contrast for higher order mixtures (ternary, quaternary, etc.) it becomes predictive. The ASOG and UNIFAC methods are predictive: the parameters of these methods are identified once and for all: it is no longer necessary to perform experiments in order to calculate the mixture properties of new molecules. These groups’ contribution methods are used every day in the chemical industry, new developments achieved by UNIFAC [WEI 87, LAR 87, GME 98] or ASOG [KOJ 79, TOG 90] in order to extend the application domain of these methods as well as their precision. New methods such as COSMOS-RS [ECK 02, KLA 95], group Contribution Model Solvation [LIN 99] and the Segment Contribution Solvation model [LIN 02], although validated for a number of cases, are still under development.

Models that can be applied to electrolytes are essentially based on the works (1) of Debye and Hückel [DEB 23], which from a simplified description of long-range interactions such as ion–ion have given a satisfactory prediction of salt activities for low concentrations, and (2) of Pitzer [PIT 73] that by generalizing the Debye–Hückel equation has allowed the prediction of salt activities up to saturation. Chen et al. [CHE 82] have extended the concept of local composition developed in the NRTL model [REN 68] to electrolyte solutions by introducing two important parameters: the local electroneutrality and the repulsion between ions of the same charge. Kikic et al. [KIK 91], in order to take into account the effects of salts on the vapor-liquid equilibrium, have added a Debye–Hückel term to the UNIFAC equation. However, in all cases, Robinson and Stokes [ROB 59], then Achard et al. [ACH 92b] and Ben Gaïda et al. [BEN 10] have clearly highlighted the need to define the ion entity in a solution in terms of hydration degree.

2.2 The structure of the thermodynamic model

In the field of chemical engineering, a very large number of models have been developed for non-ideal solutions. In this chapter, only the model developed by Achard [ACH 92a] will be presented as it provides the basis for the present work.

2.2.1 The interactions to be taken into account

Biological, liquid or solid media are complex because of the number of constituents present (sugars, lipids, carbohydrates, salts, organic acids, etc.) which leads to a wide variety of interactions between the species in solution. Three types of main interactions are identified [ACH 92a]:

– Short-range interactions resulting from dispersion forces and from differences in size and in form between molecules. When molecules of a different nature are mixed, their medium is no longer the same as in pure solutions. Molecules are in perpetual movement and by coming into contact, induce modifications of their energy potential. These intermolecular or Van der Waals forces cause deviations from ideality. The other component concerns the geometry (area, volume and shape) of mixed molecules. There is a mixing entropy contribution because there is a transition from an ordered state to a more disordered state.

– Interactions of chemical nature (short-range interactions) come from the chemical properties of groups present in molecules. These properties are of two kinds: they result from association phenomena with the establishment of hydrogen bonds between polar...