Activity coefficients calculation, from solubilities

The values of some total ion activity coefficients calculated from different models can be checked when a solid composed of the same ions is equilibrated with a solution without substantially changing its composition. Under these circumstances the activity coefficient product can be determined by dividing the thermodynamic solubility product by the observed equilibrium concentration product. 

The most important aspect of the simulation is that the thermodynamic data of the chemicals be modeled correctly. It is necessary to decide what equation of state to use for the vapor phase (ideal gas, Redlich-Kwong-Soave, Peng-Robinson, etc.) and what model to use for liquid activity coefficients [ideal solutions, solubility parameters, Wilson equation, nonrandom two liquid (NRTL), UNIFAC, etc.]. See Sec. 4, Thermodynamics. It is necessary to consider mixtures of chemicals, and the interaction parameters must be predictable. The best case is to determine them from data, and the next-best case is to use correlations based on the molecular weight, structure, and normal boiling point. To validate the model, the computer results of vapor-liquid equilibria could be checked against experimental data to ensure their validity before the data are used in more complicated computer calculations. 

In the multimedia models used in this series of volumes, an air-water partition coefficient KAW or Henry s law constant (H) is required and is calculated from the ratio of the pure substance vapor pressure and aqueous solubility. This method is widely used for hydrophobic chemicals but is inappropriate for water-miscible chemicals for which no solubility can be measured. Examples are the lower alcohols, acids, amines and ketones. There are reported calculated or pseudo-solubilities that have been derived from QSPR correlations with molecular descriptors for alcohols, aldehydes and amines (by Leahy 1986 Kamlet et al. 1987, 1988 and Nirmalakhandan and Speece 1988a,b). The obvious option is to input the H or KAW directly. If the chemical s activity coefficient y in water is known, then H can be estimated as vwyP[>where vw is the molar volume of water and Pf is the liquid vapor pressure. Since H can be regarded as P[IC[, where Cjs is the solubility, it is apparent that (l/vwy) is a pseudo-solubility. Correlations and measurements of y are available in the physical-chemical literature. For example, if y is 5.0, the pseudo-solubility is 11100 mol/m3 since the molar volume of water vw is 18 x 10-6 m3/mol or 18 cm3/mol. Chemicals with y less than about 20 are usually miscible in water. If the liquid vapor pressure in this case is 1000 Pa, H will be 1000/11100 or 0.090 Pa m3/mol and KAW will be H/RT or 3.6 x 10 5 at 25°C. Alternatively, if H or KAW is known, C[ can be calculated. It is possible to apply existing models to hydrophilic chemicals if this pseudo-solubility is calculated from the activity coefficient or from a known H (i.e., Cjs, P[/H or P[ or KAW RT). This approach is used here. In the fugacity model illustrations all pseudo-solubilities are so designated and should not be regarded as real, experimentally accessible quantities. 

Very few generalized computer-based techniques for calculating chemical equilibria in electrolyte systems have been reported. Crerar (47) describes a method for calculating multicomponent equilibria based on equilibrium constants and activity coefficients estimated from the Debye Huckel equation. It is not clear, however, if this technique has beep applied in general to the solubility of minerals and solids. A second generalized approach has been developed by OIL Systems, Inc. (48). It also operates on specified equilibrium constants and incorporates activity coefficient corrections for ions, non-electrolytes and water. This technique has been applied to a variety of electrolyte equilibrium problems including vapor-liquid equilibria and solubility of solids. 

Manson and Chin 151) reported that the addition of filler to an epoxy binder reduces the epoxy s permeability coefficient (P), as well as the solubility of water in the resin (S) and that the reduction is stronger than expected from theory 1 2). Diffusion coefficients calculated from P and S for the unfilled resin were found to be somewhat higher than those for filled resin. The difference seems to be due to the formation of ordered layers, up to 4 pm thick, around every filler particle. The layers form because of residual stresses caused by the difference between the binder and filler coefficients of thermal expansion. The effective activation energy for water to penetrate into these materials, calculated in the 0-100 °C temperature range, is 54.3 kJ/mol151). 

It is important to emphasize that the alcohols used as cosurfactants have high solubility in the oil phase because they are present there as both single molecules and aggregates. Therefore, in calculating X0o, one must consider the self-association of alcohol in the oil phase. The activity coefficients arise because of the hard-sphere interactions among the droplets. Therefore, in the dispersed phase, which is free of droplets, the activity coefficients are equal to unity, whereas, in the continuous phase, which contains the droplets, the activity coefficients differ from unity. 

It is a function expressing the effect of charge of the ions in a solution. It was introduced by -> Lewis and Randall [iii]. The factor 0.5 was applied for the sake of simplicity since for 1 1 electrolytes I = c (electrolyte). It is an important quantity in all electrostatic theories and calculations (e.g., - Debye-Huckel theory, - Debye-Htickel limiting law, - Debye-Huckel-Onsager theory) used for the estimation of -> activity coefficients, -> dissociation constants, -> solubility products, -> conductivity of -> electrolytes etc., when independently from the nature of ions only their charge is considered which depends on the total amount (concentration) of the ions and their charge number (zj). 

Related Calculations. This illustration outlines the procedure for obtaining coefficients of a liquid-phase activity-coefficient model from mutual solubility data of partially miscible systems. Use of such models to calculate activity coefficients and to make phase-equilibrium calculations is discussed in Section 3. This leads to estimates of phase compositions in liquid-liquid systems from limited experimental data. At ordinary temperature and pressure, it is simple to obtain experimentally the composition of two coexisting phases, and the technical literature is rich in experimental results for a large variety of binary and ternary systems near 25°C (77°F) and atmospheric pressure. Example 1.21 shows how to apply the same procedure with vapor-liquid equilibrium data. 

Solvent activity coefficients at 25°C in Solvents, S, Relative to a Standard State of 1 m Solute in Dimethylformamide (D). Calculation from Solubilities... 

It is generally difficult to determine the ion activity coefficients, and one commonly makes shift with values calculated with semiempirical equations see Section 2.3.2. Anyway, y decreases with an increase in total ionic strength. This means that adding any other electrolyte will decrease the activity coefficients, causing the solubility and the dissociation to increase. For instance, if KNO3 is added to a solution of CaCl2, this affects the latter s dissociation equilibrium. y+ and y decrease and y0 remains at 1, and—since the intrinsic dissociation constant remains unaltered—it thus follows from (2.25) that the concentrations of Ca2+ and CP increase and that of the undissociated salt decreases. 

It should be noted that the octanol-water partition coefficient calculated from the Law of Distribution Between Phases and the experimental water solubility agrees well with our experimetally determined partition coefficient. This result is expected since the molar solubility of dioxin in both octanol and water is sufficiently low at saturation that there is no significant impact on the activity coefficient of dioxin in either phase. Further, solubilities of octanol in water, and water in octanol are so slight that there is no significant difference between dioxin solubilities for the pure solvents compared to mutually saturated solvents. 

To calculate from solubility measurements values of the activity coefficient of thallous iodate in its saturated solution in the presence of potassiiun chloride and of potassium thiocj ate. 

In order to use solubility data for salts of moderate solubility in the calculation of thermodynamic values, one must also have the corresponding activity coefficients. Such data, particularly at the higher concentrations, are exceedingly scarce. Most activity coefficient data which now exist are primarily for dilute solutions and have been derived from electrochemical measurements. This subject is covered elsewhere in this chapter, and some of the activity coefficients derived from this source are listed in the appendices. Some interesting data obtained from other sources, particularly from freezing point measurements, are now beginning to appear. ... 

Activity coefficients calculated by these methods agree fairly well for systems where the original equations, i.e., Eqs. (3.50) to (3.66), apply. Carlson and Colburn (5) and Colburn, Schoenborn, and Shilling (8) have shown that the van Laar constants cannot be calculated from solubility data for n-butanol-water and isobutanol-water, but the van Laar equations do not satisfactorily describe the activity coefficients obtained from vapor-liquid data in these systems either. 

From Eq. (13), solubility expressed in mole fraction of the solid x, is dependent on the heat of fusion A which can be related to the sublimation pressure P " of the solid, and the melting temperature Pm (exactly triple-point temperature Prr) of the solid. Assuming an ideal solution, with an activity coefficient /i of unity, the solubility of a solid in a liquid can be calculated. In the present case, the magnitude of separation of two species will depend principally on the difference in their melting temperatures (APm = Pmi — 7m2). Modifying the solvent will produce a nonideal solution with activity coefficients different from unity. In such a case, separation is also dependent on the difference in activity coefficient of both species (A/ = /i - /2). 

A.queous Solubility. SolubiHty of a chemical in water can be calculated rigorously from equiHbrium thermodynamic equations. Because activity coefficient data are often not available from the Hterature or direct experiments, models such as UNIFAC can be used for stmcture—activity estimations (24). Phase-equiHbrium relationships can then be appHed to predict miscibility. Simplified calculations are possible for low miscibiHty however, when there is a high degree of miscibility, the phase-equiHbrium relationships must be solved rigorously. 

Calculation of activity coefficients from mutual solubility data... 

For systems that are only partially miscible in the liquid state, the activity coefficient in the homogeneous region can be calculated from experimental values of the mutual solubility limits. The methods used are described by Reid et al. (1987), Treybal (1963), Brian (1965) and Null (1970). Treybal (1963) has shown that the Van-Laar equation should be used for predicting activity coefficients from mutual solubility limits. 

In principle, Gibbs free energies of transfer for trihalides can be obtained from solubilities in water and in nonaqueous or mixed aqueous solutions. However, there are two major obstacles here. The first is the prevalence of hydrates and solvates. This may complicate the calculation of AGtr(LnX3) values, for application of the standard formula connecting AGt, with solubilities requires that the composition of the solid phase be the same in equilibrium with the two solvent media in question. The other major hurdle is that solubilities of the trichlorides, tribromides, and triiodides in water are so high that knowledge of activity coefficients, which indeed are known to be far from unity 4b), is essential (201). These can, indeed, be measured, but such measurements require much time, care, and patience. 

For the purpose of this case study we will select Isopropyl alcohol as the crystallization solvent and assume that the NRTL-SAC solubility curve for Form A has been confirmed as reasonably accurate in the laboratory. If experimental solubility data is measured in IPA then it can be fitted to a more accurate (but non predictive) thermodynamic model such as NRTL or UNIQUAC at this point, taking care with analysis of the solid phase in equilibrium. As the activity coefficient model only relates to species in the liquid phase we can use the same model with each different set of AHm and Tm data to calculate the solubility of the other polymorphs of Cimetidine, as shown in Figure 21. True polymorphs only differ from each other in the solid phase and are otherwise chemically identical. 

The selected normalization of the NaCl activity coefficient has two particular advantages for solid-liquid equilibria. First, the solubility product is calculated directly from available solubility data no activity-coefficient data are required. 

Figure 19.10. Variation of solubility of AgCl with ionic strength, from which activity coefficients can be calculated. Data from Ref. 3.

On the other hand, micelle formation has sometimes been considered to be a phase separation of the surfactant-rich phase from the dilute aqueous solution of surfactant. The micellar phase and the monomer in solution are regarded to be in phase equilibrium and cmc can be considered to be the solubility of the surfactant. When the activity coefficient of the monomer is assumed to be unity, the free energy of micelle formation, Ag, is calculated by an equation... 

Illustrative Example 5.1 Deriving Liquid Aqueous Solubilities, Aqueous Activity Coefficients, and Excess Free Energies in Aqueous Solution from Experimental Solubility Data Problem Calculate the Cf (L), yf 1 and G of (a) di-n-butyl phthalate, (b) y-1,2,3,4,5,6-hexachlorocyclohexane (y-HCH, lindane), and (c) chloroethene (vinyl chloride) at 25 °C using the data provided in Appendix C. 

Figure 5.6 Illustration of the effect of a completely water-miscible solvent (CMOS, i.e., methanol) on the activity coefficient of organic compounds in water-organic solvent mixtures decadic logarithm of the activity coefficient as a function of the volume fraction of methanol. Note that the data for naphthalene (Dickhut et al., 1989 Fan and Jafvert, 1997) and for the two PCBs (Li and Andren, 1994) have been derived from solubility measurements whereas for the anilins (Jayasinghe etal., 1992), air-water partition constants determined under dilute conditions have been used to calculate y,f.

P 5.1 Calculating Aqueous Activity Coefficients and Excess Free Energies in Aqueous Solution from Experimental Solubility Data... 

Calculate the activity coefficients of (a) n-octane, and (b) aniline in water-saturated (see Table 5.1) u-hexane (%,), toluene (yt), diethylether (yiA), chloroform (y c), n-octanol (y0), and in water from the Kuw values given in Table 7.1. The aqueous solubilities of the two compounds are given in Appendix C. Compare and discuss the results. 

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