Activity coefficient fitting

In contrast with the individual ion activity coefficients fit the mean activity coefficient ft can be measured, calculation of which can be achieved through eqn. 2.46 as follows ... 

Fig. 3.2. Partial pressures (a), activities (6), and activity coefficients (c) for the system acetone-water at 26 C. [Data of BearOy McVicavy and Fergitaon, J. Phya. Chem, 34, 1310 (1930).J Activity coefficients fitted with two-suffix Van Laar equations Aab log 6.5, Aba log 4.0...

Hctivity Coefficients. Most activity coefficient property estimation methods are generally appHcable only to pure substances. Methods for properties of multicomponent systems are more complex and parameter fits usually rely on less experimental data. The primary group contribution methods of activity coefficient estimation are ASOG and UNIEAC. Of the two, UNIEAC has been fit to more combinations of groups and therefore can be appHed to a wider variety of compounds. Both methods are restricted to organic compounds and water. 

Conductivity measurements yield molar conductivities A (Scm2 mol-1) at salt concentration c (mol L-1). A set of data pairs (Af, c,), is evaluated with the help of non linear fits of equations [89,93,94] consisting of the conductivity equation, Eq. (7), the expression for the association constant, Eq. (3), and an equation for the activity coefficient of the free ions in the solution, Eq.(8) the activity coefficient of the ion pair is neglected at low concentrations. 

Rard also employed Pitzer s electrolyte activity coefficient model to correlate the data. It was found that the quality of the fit depended on the range of molalities that were used. In particular, the fit was very good when the molalities were less than 3 mol/kg. 

In the above two equations, the former value is valid for basic SI units and the latter value for / in moles per cubic decimetre and a in nanometres. The parameter a represents one of the difficulties connected with the Debye-Hiickel approach as its direct determination is not possible and is, in most cases, found as an adjustable parameter for the best fit of experimental data in the Eq. (1.3.29). For common ions the values of effective ion radii vary from 0.3 to 0.5. Analogous to the limiting law, the mean activity coefficient can be expressed by the equation... 

These models are semiempirical and are based on the concept that intermolecular forces will cause nonrandom arrangement of molecules in the mixture. The models account for the arrangement of molecules of different sizes and the preferred orientation of molecules. In each case, the models are fitted to experimental binary vapor-liquid equilibrium data. This gives binary interaction parameters that can be used to predict multicomponent vapor-liquid equilibrium. In the case of the UNIQUAC equation, if experimentally determined vapor-liquid equilibrium data are not available, the Universal Quasi-chemical Functional Group Activity Coefficients (UNIFAC) method can be used to estimate UNIQUAC parameters from the molecular structures of the components in the mixture3. 

Among the possible analytical methods for alkalinity determination, Gran-type potentiometric titration [2] combined with a curve-fitting algorithm is considered a suitable method in seawaters because it does not require a priori knowledge of thermodynamic parameters such as activity coefficients and dissociation constants, which must be known when other analytical methods for alkalinity determination are applied [3-6],... 

In fitting these data, we note that at pH 7.5 selenate is present almost exclusively as the SeO " oxyanion, and the species activity coefficient in the dilute fluid is nearly one. We can, therefore, take the species activity as equal to its dissolved concentration, in mol kg-1. If this had not been the case, we would need to account for the speciation and activity coefficient in determining the value of se04 for each experiment. 

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. 

Solubility modelling with activity coefficient methods is an under-utilized tool in the pharmaceutical sector. Within the last few years there have been several new developments that have increased the capabilities of these techniques. The NRTL-SAC model is a flexible new addition to the predictive armory and new software that facilitates local fitting of UNIFAC groups for Pharmaceutical molecules offers an interesting alternative. Quantum chemistry approaches like COSMO-RS [25] and COSMO-SAC [26] may allow realistic ab-initio calculations to be performed, although computational requirements are still restrictive in many corporate environments. Solubility modelling has an important role to play in the efficient development and fundamental understanding of pharmaceutical crystallization processes. The application of these methods to industrially relevant problems, and the development of new... 

Table 1. Data and Results of Fit for Aqueous Solutions of uni-univalent electrolyte at 298.15°K - Mean Ionic Activity Coefficient Data... 

The interaction parameters are weak, linear functions of temperature, as shown in Table 5, Table 6 and Figure 6. These tables and figure show the results of isothermal fits for activity coefficient data of aqueous NaCl and KBr at various temperatures. The Pitzer equation parameters are, however, strongly dependent on temperature (Silvester and Pitzer, (23)). 

A second type of ternary electrolyte systems is solvent -supercritical molecular solute - salt systems. The concentration of supercritical molecular solutes in these systems is generally very low. Therefore, the salting out effects are essentially effects of the presence of salts on the unsymmetric activity coefficient of molecular solutes at infinite dilution. The interaction parameters for NaCl-C02 binary pair and KCI-CO2 binary pair are shown in Table 8. Water-electrolyte binary parameters were obtained from Table 1. Water-carbon dioxide binary parameters were correlated assuming dissociation of carbon dioxide in water is negligible. It is interesting to note that the Setschenow equation fits only approximately these two systems (Yasunishi and Yoshida, (24)). 

For applications where the ionic strength is as high as 6 M, the ion activity coefficients can be calculated using expressions developed by Bromley (4 ). These expressions retain the first term of equation 9 and additional terms are added, to improve the fit. The expressions are much more complex than equation 9 and require the molalities of the dissolved species to calculate the ion activity coefficients. If all of the molalities of dissolved species are used to calculate the ion activity coefficients, then the expressions are quite unwieldy. However, for the applications discussed in this paper many of the dissolved species are of low concentration and only the major dissolved species need be considered in the calculation of ion activity coefficients. For lime or limestone applications with a high chloride coal and a tight water balance, calcium chloride is the dominant dissolved specie. For this situation Kerr (5) has presented these expressions for the calculation of ion activity coefficients. 

Beutier and Renon(6) have used a similar model but adjusted their ionic activity coefficient parameters to fit selected ternary data. Our own approach, initiated before we became aware of the Prausnitz work, was to analyze the ternary data by modification of the method of van Krevelen, Hoftijzer, and Huntjens(7 and to extend its range by use of ionization constants, Henry s law correlations, and correlations of activity coefficients as needed. Thus, in many areas we needed the same basic data as in the Prausnitz or Renon approach. 

The effect of concentration of free (molecular) ammonia on the activity of the electrolyte was derived mainly from two 80 C data points of Miles and Wilson having 16 to 17 molal free ammonia concentration. Data points below 0.2 ionic strength were fitted by application of Kielland s estimation of ionic activity coefficients(6 2). Details are presented elsewhere(45), together with graphs giving partial pressures of ammonia and hydrogen sulfide for temperatures from 80 to 260 F over a range of liquid concentration. 

Equations 11 and 12 were fit to the experimental activity coefficients of HC1 and NaCl as described by Harned s rule. 

This model has been applied by Vera and Vega (1 7) to the NaCl-HCl-H20 system. Table 2 presents their fit to the vapor pressures of water and the activity coefficients in the NaCl-H20 system. As can be seen in Table 2, the agreement between the model and the experimental data is very good down to 0.2 molality. In a similar way, it was also found possible to obtain an excellent fit to the experimental data for the HC1-H20 system. 

With the experimental methods described, as well as with several others, the activity coefficients of numerous strong electrolytes of various valence types have been calculated. Many of these data have been assembled and examined critically by Harned and Owen [2], More recent evaluations for uni-univalent electrolytes have been made by Hamer and Wu [6], and for uni-bivalent electrolytes by Goldberg [7], Data used by the authors in Refs. 6 and 7 are fitted to an expression of the form... 

Raji Heyrovska [18] has developed a model based on incomplete dissociation, Bjermm s theory of ion-pair formation, and hydration numbers that she has found fits the data for NaCl solutions from infinite dilution to saturation, as well as several other strong electrolytes. She describes the use of activity coefficients and extensions of the Debye-Hiickel theory as best-fitting parameters rather than as explaining the significance of the observed results. ... 

The activity coefficients can be calculated using any of the existing models if the binary parameters for all combinations of binary pairs are known. These parameters are obtained by fitting to experimental data. For ternary systems, one can either simultaneously fit all six parameters or first determine the parameters using binary data for those binary systems that have a phase separation and the rest of the parameters from ternary data. 

Recently, Rubingh ll) and Scamehorn et al. (9) have shown that the activity coefficients obtained by fitting the mixture CMC data can be correlated by assuming the mixed micelle to be a regular solution. This model proposed by Rubingh for binary mixtures has been extended to include multicomponent surfactant mixtures by Holland and Rubingh (10). Based on this concept Kamrath and Frances (11) have made extensive calculations for mixed micelle systems. 

Results for the various binary mixed surfactant systems are shown in figures 1-7. Here, experimental results for the surface tension at the cmc (points) for the mixtures are compared with calculated results from the nonideal mixed monolayer model (solid line) and results for the ideal model (dashed line). Calculations of the surface tension are based on equation 17 with unit activity coefficients for the ideal case and activity coefficients determined using the net interaction 3 (from the mixed micelle model) and (equations 12 and 13) in the nonideal case. In these calculations the area per mole at the surface for each pure component, tOj, is obtained directly from the slope of the linear region in experimental surface tension data below the cmc (via equation 5) and the maximum surface pressure, from the linear best fit of... 

The basis of the UNIFAC approach is the definition of submolecular groups (e.g., CH2, CH3, CH3O, CH2CI, OH) and the fitting of a given molecular property or activity coefficient to a sum of contributions based on the subgroup molecular volume and interaction terms between the groups. 

Turning now to the solvent-solvent binary, the effect of the value of a 2 on the quality of the obtained fit is well established (12,16). Since this binary had the largest number of experimental activity coefficients—for the solvent-salt binaries only the y of the solvent is used—it was decided to let a vary between +1.0 and —3.0 with the best fit of the ternary data as criterion for its optimum value. The possibilities of varying the other two as (ai3 and 23) to obtain the best ternary fit was rejected although it would probably lead to better correlation of the ternary results, it could not lead to any predictive scheme. The number of available systems is simply too limited for the establishment of optimum a values for all three binaries. 

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