EPC-SAFT

Fig. 2 Solution densities, vapour pressures, salt activity coefficients, and solubility behaviour of KBr/water solutions. Symbols represent experimental data, lines are ePC-SAFT modelling...

It can be observed from Fig. 2 that ePC-SAFT allows for quantitative descriptions of solution densities (a), vapor pressures (b), salt activity coefficients (c), and solubilities (d). 

The ePC-SAFT " (electrolyte PC-SAFT) equation follows similar ideas to those used in the development of the SAFT-VRE equation, in that the DH approach in the primitive model is used to account for the electrostatic interactions. The electrostatic term in references 384 and 385 is given in the general form... 

Experimental data including the acidic species in the vapor phase within the above concentration range are scarce. Only very few publications of VLE data in that range are available [168, 173]. In contrast, numerous vapor pressure curves are accessible in literature. Chemical equilibrium data for the polycondensation and dissociation reaction in that range (>100 wt%) are so far not published [148]. However, a starting point to describe the vapor-Uquid equilibrium at those high concentratirMis is given by an EOS which is based on the fundamentals of the perturbation theory of Barker [212, 213]. Built on this theory, Sadowski et al. [214] have developed the PC-SAFT (Perturbed Chain Statistical Associated Fluid Theory) equation of state. The PC-SAFT EOS and its derivatives offer the ability to be fuUy predictive in combination with quantum mechanically based estimated parameters [215] and can therefore be used for systems without or with very little experimental data. Nevertheless, a model validation should be undertaken. Cameretti et al. [216] adopted the PC-SAFT EOS for electrolyte systems (ePC-SAFT), but the quality for weak electrolytes as phosphoric... 

Ji X, Held C, Sadowski G (2012) Modeling imidazolium-based ionic liquids with ePC-SAFT. Fluid Phase Equilib 335 64—75... 

Here, we describe the application and typical modelling results for a G model (MSA-NRTL) as well as for an EOS (ePC-SAFT). In addition to strong electrolytes which are almost fully dissociated, we also consider some weak electrolytes (acids like HE or ion-paired electrolytes) that do only partially dissociate in aqueous solution. Here, ion pairing is accounted for by an association/dissociation equilibrium between the ion pair and the respective free ions in solution. 

The following chapters will introduce the two models we focus on the (remodel MSA-NRTL and the equation of state ePC-SAFT, both of which composed of different terms for the SR and for the LR contributions in Eq. (1). After that, we will apply the two models to electrolyte solutions in order to describe thermodynamic properties of strong and weak electrolytes. 

In contrast to a G model, ePC-SAFT is formulated in terms of the Helmholtz energy A to also consider density effects of the system. It is again formulated as a sum of LR and SR contributions... 

The fact that both the DH k as well as the MSA F in the Coulom-bic contributions in Eq. (12) (ePC-SAFT) and in Eq. (2) (MSA-NRTL) are expressed in terms of the dielectric constant e makes clear that these contributions follow the approach of so-called primitive models . This... 

By contrast, the rational activity coefficient ij describes the deviation from the infinite-diluted solution. It can be obtained by ePC-SAFT as the ratio of ion fugacity coefficient q>j at the actual concentration related to the one... 

In case of ePC-SAFT, dispersive (short-range) interactions were only considered for water-water, water-ion, and water-ion pairs but not among ions. Within the ion-specific MSA-NRTL approach, attractive short-range forces are also assumed to occur between anions and cations. 

As mentioned before, ion-specific instead of salt-specific parameters are used here for both models. Thus, the ionic parameters determined for an ion are applicable to all electrolytes containing this ion. Obtaining such a universal set of parameters requires a simultaneous regression of several electrolyte solutions, which is described in Refs. 5 and 15 for both models, MSA-NRTL and ePC-SAFT. 

Using ePC-SAFT, liquid densities, vapour pressures [directly obtained by Eq. (16) not included in the parameter estimation], and solute activity coefficients (MIAC) are modelled whereas only activity coefficients will be presented for MSA-NRTL as this quantity is the only one that can be obtained by a G -model. Any deviations from experimental data will be given by absolute relative deviations (ARD) ... 

In the following, we will give some examples of modelling results obtained with MSA-NRTL and ePC-SAFT. 

Section 3.2.1 describes fiilly dissociated electrolytes. For electrolytes that do not completely dissociate into the respective ions, a chemical-reaction mechanism is implemented in the ePC-SAFT framework (Sec. 3.2.2). Modelling of systems that can form multiple ion pairs is described in Sec. 3.2.3. Finally, we will discuss the experimental behaviour of strong and weak acids and present a respective model strategy (Sec. 3.2.4). Whereas so far activity coefficients of 19 electrolyte systems have been modelled by the MSA-NRTL, the properties of more than 120 systems have been studied with ePC-SAFT. The latter contains not only activity coefficients but also solution densities, which are important quantities for both process design and validation of model parameters. 

Solution densities are important system properties for the design of industrial apparatus. Furthermore, they are typically used for model-parameter estimations and validation of model consistencies. As an example, the liquid densities of six caesium-salt solutions are shown in Fig. 4. All solution densities are presented as density differences Ap between the densities of the aqueous salt solution and pure water at the same temperature. As to be seen, the experimental data can be described with high accuracy even at high salt concentrations of up to 6 mol/kg. Obviously, ePC-SAFT is a powerful model for the description of electrolyte solution densities (which in general cannot be described by a model). This holds true as well for the prediction of these data at different temperatures. 

Fig. 4. Liquid densities of aqueous solutions of six caesium salts related to the density of pure water at 20° C as function of salt molaity. The lines represent calculations with ePC-SAFT. The circles represent experimental data. (From Ref. 15, Elsevier, reprinted with permission.)...

Fig. 5. Influence of salts on the activity coefficient of water at 30°C. The symbols represent experimental data from isopiestic or vapour pressure measurements. The lines are predictions with ePC-SAFT. (a) Influence of cations on the activity coefficient of water. Circles Lil, squares Nal, triangles KI. (b) Influence of anions on the activity coefficient of water. Circles Nal, squares NaBr, triangles NaCl. Largest anion and smallest cation cause the lowest water activity coefficient. (From Ref. 15 Elsevier, reprinted with permission.)...

Fig. 7. Mean ionic activity coefficients of LiAc, NaAc, and KAc. The symbols represent experimental data from Lobo et al at 25°C, the lines are modelling results KAc — stars and dashed lines NaAc — circles and full lines LiAc — triangles and thin lines. Experimental MIAC values increase in the order LiAc < NaAc < KAc. (a) Calculations were performed with the classical ePC-SAFT, neglecting ion pairing. Modelled MIAC values increase in the order LiAc > NaAc > KAc. (b) Calculations were performed with the ePC-SAFT, accounting for ion pairing. Modelled MIAC values increase in the experimentally validated order LiAc < NaAc < KAc. (From Ref. 21, Elsevier, reprinted with permission.)...

As experimental MIAC data show a similar characteristic compared to the strong electrolytes treated in Sec. 3.2.1, the classical ePC-SAFT approach can be applied to model liquid densities, WAC, and MIAC of those systems at 25°C. Because of the fact that the parameters reflect the properties of the hydrated ions, the H" " parameters directly represent the HgO " ion. 

Fig. 8. Solution densities and MIAC of ZnBt2 and Znl2 in water. The symbols represent experimental data, the lines show the calculations performed with ePC-SAFT Znl2 — circles ZnBr2 — squares, (a) MIAC at 25°C. Experimental data from Lobo etal (b) Solution densities at 67.46°C (ZnBr2) and at 19.50°C (Znl2) presented as density differences A/o between salt solution and pure water at the same temperature. Experimental data from Wimby and Berntsson for ZnBr2 and from D Ans et al for Znl2. (From Ref. 21, Elsevier, reprinted with permission.)...

Fig. 9. Activity coefficients in halide-acid solutions at 25°C. Lines are modelled with ePC-SAFT, symbols are experimental data HF — circles and dashed-dotted lines HCl — squares and thin lines HBr — triangles and full lines HI — stars and dashed lines, (a) Mean ionic activity coefficients of the hydrogen halides. Experimental data from Lobo et (b) Activity coefficients of water. Experimental data from Lobo ct (From Ref 21, Elsevier, reprinted with permission.)...

Figure 9 shows experimental data and the ePC-SAFT calculations of the components activity coefficients in aqueous systems containing hydrogen halides. The calculations for HCl, HBr, and HI in water were performed with the universal parameter set. The modelled MI AC of the acids are illustrated in Fig. 9(a). The influence of the considered hydrogen halides on the water activity coefficients (WAC) can be seen in Fig. 9(b). 

Until now the calculations showed that both MSA-NRTL and ePC-SAFT are powerful models for the calculation of thermodynamic properties that are required for designing technical processes. In this section, we finally focus on the meaning of the model parameters and how they can be used to interpret experimentally observed phenomena. 

The first parameter which shall be investigated is the hydrated-ion diameter. In case of ePC-SAFT, the size of this parameter is determined by the volume of the bare ion and its surface charge density, both of which influencing the degree of hydration. Thus, it gives an indirect hint to the... 

However, the cation-size series of ePC-SAFT and MSA-NRTL differ in their sequence. Obviously, the adjusted MSA-NRTL parameters do not reflect the series of the Pauling diameters. This is due to the fact that they directly reflect the strength of hydration and that no volumetric properties were used for the respective parameter estimation. Thus, the most strongly hydrated cation (Li+) within the alkali halide series considered here seems to have the largest diameter. In contrast, the ePC-SAFT diameters for the alkali cations were obtained by adjusting them to MIAC and volumetric data (solution densities). They follow the same trend and are in the same order of magnitude as the Pauling diameters are. 

For the cations Li+, Na+, and K+, Collins et al reported X-ray and neutron diffraction measurements of aqueous salt solutions, providing the distance between a central cation and the nearest water oxygen. This directly compares to the ePC-SAFT parameter cr, since this is in fact the diameter of the hydrated cation. Comparing these experimental values aexp to the a parameters given in Table 3 shows an excellent agreement of experimental data and adjusted parameters. 

The second parameter which is used in the electrolyte models is a parameter representing the short-range interactions between water and an ion. In Table 4, anion-water and cation-water parameters are compared for the two models. Both the ePC-SAFT ujk, as well as the MSA-NRTL TW-ion interaction parameters directly reflect the strength of ionic hydration. The higher and the more negative rw-ion are, the more strongly hydrated the ion is (strong interaction with water) within the considered alkali halides. 

Table 3. Comparison of hydrated-ion sizes Experimental values from X-ray and neutron diffraction measurements versus ePC-SAFT and MSA-NRTL parameters.

Table 4. Comparison of water/ion interaction parameters for ePC-SAFT and MSA-NRTL.

Obviously, the smallest cation but the largest anion interact most strongly with water within the considered systems. Analysing the experimental activity coefficients of water, these parameter series can be confirmed (Fig. 5 and accompanying text). Additionally, it is obvious that the cations are much stronger hydrated than the anions, which is also reflected by the ePC-SAFT and MSA-NRTL parameters u/k and t where the general relation is valid ... 

In this study, the ePC-SAFT EOS as well as the MSA-NRTL model were applied to describe thermodynamic properties of numerous aqueous electrolyte solutions. Whereas only activity coefficients are obtained by the G model, volumetric properties can be calculated with an EOS. Ion-specific parameters were used independent of the electrolyte which the ions are part of. The model parameters possess a physical meaning and show reasonable trends within the ion series. Two ion parameters are needed in ePC-SAFT, whereas six parameters are necessary for applying MSA-NRTL. Next to the standard alkali halide electrolyte systems, both models even capture the non-ideal behaviour of solutions containing acetate or hydroxide anions where a reversed MIAC series is experimentally observed. Until now, thermodynamic properties of more than 120 aqueous systems could be successfully modelled with ePC-SAFT. The MSA-NRTL parameter set has also been applied to a couple of systems (so far 19 solutions). Implementing an ion-pairing reaction in ePC-SAFT,... 

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