Solid phase activity coefficients

There are many types of phase diagrams in addition to the two cases presented here these are summarized in detail by Zief and Wilcox (op. cit., p. 21). Solid-liquid phase equilibria must be determined experimentally for most binaiy and multicomponent systems. Predictive methods are based mostly on ideal phase behavior and have limited accuracy near eutectics. A predic tive technique based on extracting liquid-phase activity coefficients from vapor-liquid equilib-... 

The equilibrium conditions given by eqs. (4.15) and (4.16) can in general be expressed through the activity coefficients. Using a solid-liquid phase equilibrium as an example we obtain... 

In this case the equations are greatly simplified and the ratio of the slopes of the two phase boundaries at xA =1 is given by the activity coefficients of B at infinite dilution in the liquid and solid phases [11] ... 

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. 

Equation 1 implies that solubility is independent of solvent type, and is only a function of the equilibrium temperature and characteristic properties of the solid phase. In real systems the effect of non-ideality in the liquid phase can significantly impact the solubility. This effect can be correlated using an activity coefficient (y) to account for the non-ideal liquid phase interactions between the dissolved solute and solvent molecules. Eq. 1. then becomes [7,8] ... 

The non-random two-liquid segment activity coefficient model is a recent development of Chen and Song at Aspen Technology, Inc., [1], It is derived from the polymer NRTL model of Chen [26], which in turn is developed from the original NRTL model of Renon and Prausznitz [27]. The NRTL-SAC model is proposed in support of pharmaceutical and fine chemicals process and product design, for the qualitative tasks of solvent selection and the first approximation of phase equilibrium behavior in vapour liquid and liquid systems, where dissolved or solid phase pharmaceutical solutes are present. The application of NRTL-SAC is demonstrated here with a case study on the active pharmaceutical intermediate Cimetidine, and the design of a suitable crystallization process. 

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. 

Usually, however, the distribution coefficients determined experimentally are not equal to the ratios of the solubility product because the ratio of the activity coefficients of the constituents in the solid phase cannot be assumed to be equal to 1. Actually observed D values show that activity coefficients in the solid phase may differ markedly from 1. Let us consider, for example, the coprecipitation of MnC03 in calcite. Assuming that the ratio of the activity coefficients in the aqueous solution is close to unity, the equilibrium distribution may be formulated as (cf. Eq. A.6.11)... 

It may be noted that, since the distribution coefficient is smaller than unity, the solid phase becomes depleted in strontium relative to the concentration in the aqueous solution. The small value of D may be interpreted in terms of a high activity coefficient of strontium in the solid phase, /srco3 38. If the strontium were in equilibrium with strontianite, [Sr2+] 10 3-2 M, that is, its concentration would be more than six times larger than at saturation with Cao.996Sro.oo4C03(s). This is an illustration of the consequence of solid solution formation where with Xcaco3 /caC03 -1 ... 

Once the composition of the aqueous solution phase has been determined, the activity of an electrolyte having the same chemical formula as the assumed precipitate can be calculated (11,12). This calculation may utilize either mean ionic activity coefficients and total concentrations of the ions in the electrolyte, or single-ion activity coefficients and free-species concentrations of the ions in the electrolyte (11). If the latter approach is used, the computed electrolyte activity is termed an ion-activity product (12). Regardless of which approach is adopted, the calculated electrolyte activity is compared to the solubility product constant of the assumed precipitate as a test for the existence of the solid phase. If the calculated ion-activity product is smaller than the candidate solubility product constant, the corresponding solid phase is concluded not to have formed in the time period of the solubility measurements. Ihis judgment must be tempered, of course, in light of the precision with which both electrolyte activities and solubility product constants can be determined (12). 

This permits provisional calculation of the compositional dependence of the equilibrium constant and determination of provisional values of the solid phase activity coefficients (discussed below). The equilibrium constant and activity coefficients are termed provisional because it is not possible to determine if stoichiometric saturation has been established without independent knowledge of the compositional dependence of the equilibrium constant, such as would be provided from independent thermodynamic measurements. Using the provisional activity coefficient data we may compare the observed solid solution-aqueous solution compositions with those calculated at equilibrium. Agreement between the calculated and observed values confirms, within the experimental data uncertainties, the establishment of equilibrium. The true solid solution thermodynamic properties are then defined to be equal to the provisional values. 

By examining the compositional dependence of the equilibrium constant, the provisional thermodynamic properties of the solid solutions can be determined. Activity coefficients for solid phase components may be derived from an application of the Gibbs-Duhem equation to the measured compositional dependence of the equilibrium constant in binary solid solutions (10). 

It has already been pointed out that the calculated equilibrium constants are known better than the analytical data on which they are based. So we may not attribute the observed difference in provisional equilibrium values and experimental values (Table VI) to uncertainties in the aqueous model. There are, however, uncertainties in estimating 31og K(x)/3x from Figure 1. Slopes estimated from Figure 1 are probably known within 20%. Uncertainties of 20% In 3 log K(x)/3x translate directly to uncertainties of 20% in solid phase activities and activity coefficients. 

Accordingly, sorption has received a tremendous amount of attention and any method or modeling technique which can reliably predict the sorption of a solute will be of great importance to scientists, environmental engineers, and decision makers (references herein and in Chaps. 2 and 3). The present chapter is an attempt to introduce an advanced modeling approach which combines the physical and chemical properties of pollutants, quantitative structure-activity, and structure-property relationships (i. e., QSARs and QSPRs, respectively), and the multicomponent joint toxic effect in order to predict the sorption/desorp-tion coefficients, and to determine the bioavailable fraction and the action of various organic pollutants at the aqueous-solid phase interface. 

We have already seen that, within the range of Nernst s law, the solid/liquid partition coefficient differs from the thermodynamic constant by the ratio of the Henry s law activity coefficients in the two phases—i.e.,... 

The level of impurity uptake can be considered to depend on the thermodynamics of the system as well as on the kinetics of crystal growth and incorporation of units in the growing crystal. The kinetics are mainly affected by the residence time which determines the supersaturation, by the stoichiometry (calcium over sulfate concentration ratio) and by growth retarding impurities. The thermodynamics are related to activity coefficients in the solution and the solid phase, complexation constants, solubility products and dimensions of the foreign ions compared to those of the ions of the host lattice [2,3,4]. 

One problem with the use of pL as a key parameter in both adsorption and absorption is the difficulty in obtaining accurate values for pL for solid SOCs, since they are not experimentally accessible and must be estimated (e.g., see Finizio et al., 1997, and references therein). In addition, as discussed in the preceding section with respect to absorption into a liquid phase, slopes of 1 for plots of log Kp against log pL are only expected if the activity coefficients, y, do not change along a series of compounds. 

We will start our discussion by considering a special case, that is, the situation in which the molecules of a pure compound (gas, liquid, or solid) are partitioned so that its concentration reflects equilibrium between the pure material and aqueous solution. In this case, we refer to the equilibrium concentration (or the saturation concentration) in the aqueous phase as the water solubility or the aqueous solubility of the compound. This concentration will be denoted as Qf. This compound property, which has been determined experimentally for many compounds, tells us the maximum concentration of a given chemical that can be dissolved in pure water at a given temperature. In Section 5.2, we will discuss how the aqueous activity coefficient at saturation, y, , is related to aqueous solubility. We will also examine when we can use yf as the activity coefficient of a compound in diluted aqueous solution, y (which represents a more relevant situation in the environment). 

At 25°C phenanthrene is a solid. Because the free energy contributions of phase change (i.e., melting, or condensation in the case of a gas) to the overall free energy of solution are not affected by salts in the solution, it is the aqueous activity coefficient that is increased as salt concentration increases (Eq. 5-28). Hence, the actual solubility decreases by the same factor (Eq. 5-27). The Kf value of phenanthrene is 0.30 M 1 (Table 5.7). Since 34.2%o salinity corresponds to a total salt concentration of 0.5 M (see text), [salt]tot for 30%o is equal to 0.44 M. Insertion of these values into Eq. 5-28 yields ... 

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