Activity coefficient, mean molal

Table 21.9 Stoichiometric mean molal activity coefficients (7,) for aqueous inorganic...

Of course, (4.9) cannot be used at higher ionic strengths. A more precise procedure is then required, such as the Robinson-Stokes equation for the mean molal activity coefficient [11] (see also [86a], chapter 1)... 

In equation 3 the terms of fNa+ and 7H + are the rational activity coefficients of exchanging cations in the zeolite phase and the terms yNa+ and XM + are the molal single ion activity coefficients in the solution phase. Equation 4 can be rewritten as equation 5 when the two salts, NaX and MX2 have a common anion. The mean molal activity coefficients usually can be estimated from literature data. The corrected selectivity coefficient includes a term that corrects for the non-ideality of the solution phase. Thus any variation in the corrected selectivity coefficient is due to non-ideality in the zeolite phase (see equation 3). 

In the concentration range regarding the ED processes, the effective diffusion coefficient (Z>B) can be predicted via the Gordon relationship (Reid et al, 1987), which accounts for the partial derivative of the natural logarithm of the mean molal activity coefficient (y+) with respect to molality (m) and solvent relative viscosity (rjr) ... 

FIG. 2 Effect of solute molality (m) on the mean molal activity coefficient (y ) for sodium chloride (O), sodium acetate (O), and sodium propionate ( ) at 25°C, as extracted from Robinson and Stokes (2002). The continuous lines were calculated using Eq. 9 and the empirical coefficients At extracted from Fidaleo and Moresi (2005a,b, 2006). 

Determine the mean molal activity coefficient at 25°C for A12(S04)3 present at 2 x 10-4 m concentration in a solution also containing a 10-4 m Na3P04. 

Two additional observations should be made. First, the methods used here treat each of the ionic types as a separate species that influences the thermodynamic properties of solutions very strongly by virtue of its associated charge. Second, it is instructive to examine the dependence of the mean molal activity coefficient for several different electrolytes as a function of the molality. Representative examples are shown in Fig. 4.3.1. One sees at first a very steep drop in as m is increased, and then either a gradual or a very sharp... 

FIGURE 4.3.1 Variation of mean molal activity coefficient as a function of molality for several salts in aqueous solutions. 

Determine the mean molal activity coefficient for LiCi at the concentrations c - 10-2, 3.3 x 10-2, 10 1 mol/liter. 

Determine the equilibrium constant for the dissociation of Hg2Ci2 into ionic constituents. Determine both the mean molal activity coefficient and molality of the ions in equilibrium with the undissolved salt at 25°C. 

The Mean Molal Activity Coefficient Calculation for the Macro-Component Salts. The mean molal activity coefficients for the macro-component chloride and sulfate salts are given in Figures 2a and 2b, respectively. Because of the unavailability of measured activity coefficient data in brines, validation of the results presented in Figures 2a and 2b was not possible. 

Figure 2. Mean molal activity coefficients of macrocomponent salts as a function of ionic strength for sea water concentrates. Normalized molar composition ratios Na = 1.00 = 0.021 Mg = 0.136 Ca Cd Fh Zn = 0.001 CU...

Considering the assumptions incorporated into the carbonate subroutine and in using the mole-fraction statistical model for the partitioning of the mean molal activity coefficients into single Ion activity coefficients, the agreement between predicted and measured trace metal concentrations in GSL brine was better than expected. 

In this equation is the experimentally determined selectivity coefficient, m is the molality of each ion, y is the activity coefficient of the ion in the gel phase and y- is the mean molal activity coefficient of the electrolyte in the external solution phase. 

Here it is a plot of P vs. [(v+y+(v-y-m P ] that is linear and that can be used to read off the desired mean molal activity coefficient. As earlier, P refers to the extrapolated pressure reached in Henry s Law plot for the component in pure form. 

If one specifies the activities of the solids as described in Section 3.7 (ordinarily very close to unity) then a measurement of the cell emf yields the mean molal activity coefficient of ZnCla in solution. 

The product 7 + 7- is experimentally measurable. The quantity (7 referred to as the mean molal activity coefficient The mean ionic molality is defined as im+mJ) and is simply m for a univalent-univalent electrolyte. Summarizing these definitions for a nonideal, univalent-univalent solution, where the solute is component 2. 

A similar expression can be obtained for the symmetric and the unsymmetric mole fraction activity coefficient. In a completely dissociated solution of n mol Na2S04, the mean molal activity coefficient is... 

Values of electrolyte activities, as measured by osmotic pressures, freezing point depression, and other experimental methods are in the literature (References 5 and 6, for example) or one can calculate activity coefficients based on models of molecular-level interactions between ions in electrolyte solutions. For illustrative purposes, mean molal activity coefficients for various salts at different aqueous molal (mj concentrations at 25°C are listed in Table 26.3 [7]. 

Mean Molal Activity Coefficients (y ) of Aqueous Salt Solutions at 25°C... 

Equation (26.41) predicts to within approximately 10% mean molal activity coefficients for salt concentrations up to 0.1 molal. The more accurate form of the activity coefficient equation [Equation (26.40)] allows the model to be extended to salt concentrations up to 0.5 molal. To expand the applicability of the Debye-Hiickel theory to higher concentrations, additional terms are added to Equation (26.40), such as [4]... 

Mean molal activity coefficient 5 Actual mass transfer boundary-layer thickness (cm)... 

There was justification as well for utilization of the osmotic coefficient data for the linear polyelectrolyte analogue in the Gibbs Duhem equation to compute resin-phase activity coefficients for the exchanging ions. However, since the trend, with dilution, of osmotic coefficient data for fully dissociated polyelectrolytes cannot be deduced beyond the lowest measurable concentj ation as it can with simple electrolytes, where the Debye-Hilckel limiting law applies, the computation with this equation of mean molal activity coefficients meaningfully related to a value of unity for the polyelectrolyte at infinite dilution was impossible. It was necessary to use the equation as shown below to compute mean molal activity coefficient values, y, as a function of counterion concentration, m, relative to 5n indeterminate mean molal activity coefficient,, at the low... 

Figure 3. The mean molal activity coefficient of sodium perchlorate (left) and potassium chloride (right). The curves show the predictions of our model, fit to data for the osmotic coefficient. The squares represent the data tabulated by Robinson and Stokes (11).

Equation 9.10-18 is frequently satisfactory for the correlation of the mean molal activity coefficient data for solutions of low molality (see Figs. 9.10-1 and 9.10-2). In general, the accuracy of this equation is best for a z+ = 1, z = —1 electrolyte (termed a 1 1 electrolyte) and becomes progressively less satisfactory for electrolytes of 1 2, 2 2, and so on, which are increasingly more nonideal. This should be kept in mind in using the equations here. 

Here the standard state for the ionic species is a 1-molal ideal solution the enthalpies and Gibbs energies of formation for some ions in this standard state at 25 C are given in Table 13.1-4. In Eq. 13.1-27 the standard state for the undissodated molecule has also been chosen to be the ideal 1-molal solution (see Eq. 9.7-20), although the pure component state could have been used as well (with appropriate changes in AfG, b aA B ). Finally, we have used the mean molal activity coefficient, y , of Eq. 9.10-11. Also remember that for the 1-molal standard state, y ° 1 as the solution becomes veiy dilute in the component. 

Table VII. Computation of the Mean Molal Activity Coefficient Ratio of Simple Electrolytes in Mixed Media...

The increasing deviation between the mean molal activity coefficient of KI and Nal with methanol content is to be expected. As the dielectric of the medium decreases with larger methanol content, differences in the ion pair formation capability of Na+ and K+ are believed to be enhanced with the more highly polarizable iodide ion. 

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