Osmotic coefficient Pitzer equations

Using a constant error for the measurement of the osmotic coefficient, estimate Pitzer s parameters as well as the standard error of the parameter estimates by minimizing the objective function given by Equation 15.1 and compare the results with the reported parameters. 

Estimate Pitzer s electrolyte activity coefficient model by minimizing the objective function given by Equation 15.1 and using the following osmotic coefficient data from Rard (1992) given in Table 15.5. First, use the data for molalities less than 3 mol/kg and then all the data together. Compare your estimated values with those reported by Rard (1992). Use a constant value for in Equation 15.1. 

Pitzer et al (1972, 1973, 1974, 1975, 1976) have proposed a set of equations based on the general behavior of classes of electrolytes. Pitzer (1973) writes equations for the excess Gibbs energy, AGex, the osmotic coefficient activity coefficient Y+ for single unassociated electrolytes as... 

Generally, agreement has been found between our correlations and those of Pitzer, and others (1972, 1973, 1974, 1975, 1976) and Rard, and others (1976, 1977). Many of our correlations agree fairly well with Robinson and Stokes, (1965) and Harned and Owen, (1958) but in most cases a much larger data base and more recent measurements have been incorporated into the evaluations. It has been observed that agreement with Pitzer s equations is found below moderate concentrations (several molal), but often deviate at higher concentrations where the Pitzer equations do not contain enough parameters to account for the behavior of the activity (or osmotic) coefficient. 

The activity of water is obtained by inserting Eq. (6.12) into Eq. (6.11). It should be mentioned that in mixed electrolytes with several components at high concentrations, it is necessary to use Pitzer s equation to calculate the activity of water. On the other hand, uhjO is near constant (and = 1) in most experimental studies of equilibria in dilute aqueous solutions, where an ionic medium is used in large excess with respect to the reactants. The ionic medium electrolyte thus determines the osmotic coefficient of the solvent. 

In more complex solutions of high ionic strengths with more than one electrolyte at significant concentrations, e.g., (Na, Mg, Ca " ) (Cl, SOl ), Pitzer s equation may be used to estimate the osmotic coefficient the necessary interaction coefficients are known for most systems of geochemical interest. 

Equations for single ion activity coefficients [4], osmotic coefficients [17], and other thermodynamic quantities [28], as well as applications in different cases (e.g., H2SO4 and H3PO4 solutions) have been given by Pitzer and coworkers [4,20]. 

Figure 18.3 Comparison of osmotic coefficients at T= 298.15 K for three different electrolytes as calculated from Pitzer s equations (solid lines) with the experimental results (symbols).

Figure 18.5 gives the osmotic coefficient and the activity for NaCl(aq) as a function of temperature and molality as predicted by the equation of Silvester and Pitzer. Note that both and 7 increase as T changes from 273.15 K to 323.15 K, and then decrease in a regular manner with increasing temperature. At T= 523.15 K, 7 has become a small value at high m. 

Figure 18.5 Graph of (a) the osmotic coefficient and (b) the activity coefficient for NaCl(aqueous) at p = 0.1 MPa as a function of temperature. The curves were obtained by using temperature-dependent coefficients in Pitzer s equations. The dotted line is for r=273.15 K and the dashed line is for T = 298.15 K. The solid lines are for T= 323.15, 373.15, 423.15, 473.15, 523.15, and 573.15 K, with both and 7 decreasing with increasing temperature.

The log K values shown in Figure 18.10 are the values that best reproduce all of the heat of mixing curves.v The J1 values are obtained by estimating initial values using the activity coefficients for NaCl(aq).16 These initial values of Jy are then readjusted, as the value for Km is optimized, by adjusting the coefficients of Pitzer s equations, whose form is described in the previous section. Pitzer s equations are, of course, internally consistent so that adjustments to the activity or osmotic coefficient parameters result in adjustments to the thermal parameters (L, L2, 4>J, or J2), and hence, to the heat effects. 

Tables A7.2 to A7.6 summarize other coefficients needed to apply Pitzer s equations for the calculation of 7 and (f> for various 1 1, 2 1, 3 1, 4 1, 5 1, and 2 2 electrolytes. The equations for calculating the osmotic coefficient are...

To our knowledge, no one has ever worked out the mathematics for directly estimating the pressure dependence of the osmotic coefficient (or aw) using the Pitzer approach. However, Monnin (1990) developed an alternative model based on the Pitzer approach that allows calculation of the pressure dependence for the activity of water (aw). The density of an aqueous solution (p) can be calculated with the equation... 

Formalism According to Pitzer. The most common method for the evaluation of the activity and osmotic coefficients of an electrolyte in a binary mixture of strong electrolytes with a common ion is by Scatchard s Equations (23), the McKay-Perring treatment (24), Mayers Equations... 

Pitzer and co-workers (1973, 1974) have proposed a more detailed, but at the same time more complex, approach. Whitfield (1973, 1975) has applied these equations to seawater and has shown that this model gives good agreement with available experimental data for the osmotic coefficient and for the mean ion activity coefficient of the major electrolyte components. The results obtained yield numerical results similar to the predictions of the ion association model (see Table A6.2). 

The lysozyme solubilities in aqueous solutions of sodium acetate were calculated for pH =8.3 and the results are presented in Fig. 3. The experimental preferential binding parameters are listed in Table 2 (the values for pH=4. 68-4.7 were, however, used because those for pH=8.3 were not available). The concentration dependence of the water activity in solutions of sodium chloride was obtained from Eq. (18) using the Pitzer equation for the osmotic coefficient [38]. 

For the NBS report (4), the activity and osmotic coefficients of 2 2 charge type electrolyte (MgS04, CaS04, and MnS04) have been calculated from the Pitzer equations (7). 

Figures 1 and 2 show the results of fitting the osmotic coefficient of the aqueous electrolytes sodium perchlorate and potassium chloride, respectively. Analysis of the variance in fitting the osmotic coefficient indicates that the fits are about as good as those obtained using Pitzer s equations, despite the fact that our equations have one less fitting parameter. For sodium perchlorate, the standi d deviation in our fit is 0.0011, whereas Pitzer ( ) reports 0.001 using his equation. For potassium chloride, the standard deviation in our fit is 0.00036, that in Pitzer s,...

In Pitzer s model the Gibbs excess free energy of a mixed electrolyte solution and the derived properties, osmotic and mean activity coefficients, are represented by a virial expansion of terms in concentration. A number of summaries of the model are available (i,4, ). The equations for the osmotic coefficient (( )), and activity coefficients (y) of cation (M), anion (X) and neutral species (N) are given below ... 

Equation (17.38) can therefore be rewritten in terms of activity coefficients for specific ions and the osmotic coefficient of the solvent by taking derivatives with respect to the number of moles m, of each ionic constituent per Kg water (see Pitzer, 1987). These derivatives contain several functions B, C, and 5 ) defined below, and are as follows. 

We have presented the Pitzer equations describing activity and osmotic coefficients of multi-component salt solutions, but probably no brief summary will suffice to get you started if you actually want to use these equations to model real systems. The most useful summaries are probably Pitzer (in Pytkowicz (1979) and in Carmichael and Eugster (1987)) and Weare (in Carmichael and Eugster (1987)). 

Pitzer then used the same kind of expansion for the gij(a) term for the hard core effect of equation (4.53). Here the second term of the expansion dropped out, but the third term was kept. He then found the osmotic coefficient to be ... 

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