Osmotic coefficient, calculation

The osmotic coefficients from the HNC approximation were calculated from the virial and compressibility equations the discrepancy between ([ly and ((ij is a measure of the accuracy of the approximation. The osmotic coefficients calculated via the energy equation in the MS approximation are comparable in accuracy to the HNC approximation for low valence electrolytes. Figure A2.3.15 shows deviations from the Debye-Htickel limiting law for the energy and osmotic coefficient of a 2-2 RPM electrolyte according to several theories. 

The osmotic coefficients calculated from Eq. (9) can be brought into good agreement with solution data up to about 1M for aqueous solutions of alkali (26) and alkaline earth halides, (30) tetraalkyl ammonium halides, T3l) mixed electrolytes, where the Harned coefficients are measured, (32) and electrolyte-non electrolyte mixtures, where Setchenow coefficients are measured. 

Validation of the Osmotic Coefficient Calculation. The results for the osmotic coefficients calculated for a series of brines representing an evaporative concentration sequence of sea water brines are given in Figure 1 and Table I. In Figure 1, the agreement between the measured (10) and calculated osmotic coefficients are compared. Table I also includes the coefficient of variation between the calculated and measured values. The maximum coefficient of variation encountered was 0.90 percent and... 

Figure 1. Comparison of osmotic coefficients calculated in the model with measured osmotic coefficients for sea water concentrates. (------) NaCl, measured ...

Fig. 14 The osmotic coefficient calculated at different counterion concentrations using various models. The results for a finite rod calculated using the spherical cell model (empty circles), cylindrical cell with adjustable getnnetry (empty square, and bulk system (empty triangles) are found to be in very good agreemoit. The dotted line is a numerical fit to the bulk data drawn to guide the eye. The solid line is the prediction of the PB theory and the dashed line fits the results obtained from MMMID [135] simulations for an infinitely long rod (filled circles). Experimentally measured osmotic coefficient for iodide (open diamonds) and chloride (filled diamonds) counterions [131] are also showir Figure adapted liom [133]...

The water activity is calculated from the osmotic coefficient calculated by the following equation ... 

Fig. 17. The osmotic coefficient < ) calculated from the virial equation as a function of the electrolyte concentration c for...

Fig. 5.13 Osmotic coefficients calculated from experimental water activities at 25°C. - trisodium citrate and - tripotassium citrate...

Antypov D, Holm C (2007) Osmotic coefficient calculations for dilute solutions of short... 

Fig. 4. Osmotic coefficient calculated for the following values of the parameter A (1) 2.76, (2) 3.40, (3) 3.80, (4) 4.00. The circles represent the experimental values for solutions of HPSS. (Reference [15], Figure 2.)...

The themiodynamic properties calculated by different routes are different, since the MS solution is an approximation. The osmotic coefficient from the virial pressure, compressibility and energy equations are not the same. Of these, the energy equation is the most accurate by comparison with computer simulations of Card and Valleau [ ]. The osmotic coefficients from the virial and compressibility equations are... 

The solutions to this approximation are obtained numerically. Fast Fourier transfonn methods and a refomuilation of the FINC (and other integral equation approximations) in tenns of the screened Coulomb potential by Allnatt [M are especially useful in the numerical solution. Figure A2.3.12 compares the osmotic coefficient of a 1-1 RPM electrolyte at 25°C with each of the available Monte Carlo calculations of Card and Valleau [ ]. 

Figure A2.3.17 Theoretical (HNC) calculations of the osmotic coefficients for the square well model of an electrolyte compared with experimental data for aqueous solutions at 25°C. The parameters for this model are a = r (Pauling)+ r (Pauling), d = d = 0 and d as indicated in the figure.

The calculated osmotic coefficient is obtained by the next equation... 

Figure 15.1 Calculated and experimental osmotic coefficients for Na SiO The line represents the calculated values.

Rard (1992) reported the results of isopiestic vapor-pressure measurements for the aqueous solution of high-purity NiCl2 solution form 1.4382 to 5.7199 mol/kg at 298.1510.005 K. Based on these measurements he calculated the osmotic coefficient of aqueous NiCb solutions. He also evaluated other data from the literature and finally presented a set of smoothed osmotic coefficient and activity of water data (see Table IV in original reference). 

Thermodynamic methods also measure the activity coefficient of the solvent (it should be recalled that the activity coefficient of the solvent is directly related to the osmotic coefficient—Eq. 1.1.19). As the activities of the components of a solution are related by the Gibbs-Duhem equation, the measured activity coefficient of the solvent can readily be used to calculate the activity coefficient of a dissolved electrolyte. 

Helgeson, H. C., D. H. Kirkham and G. C. Flowers, 1981, Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high temperatures and pressures, IV. Calculation of activity coefficients, osmotic coefficients, and apparent molal and standard and relative partial molal properties to 600 °C and 5 kB. American Journal of Science 281, 1249-1516. 

As a means of verifying the model parameters of Table II, the osmotic coefficient was calculated from isopiestic vapor pressure measurement data (17) for the KCl-KBr-H20 system at 25°C (Table III). 

Using the model parameters of Table II the calculated osmotic coefficient is within 0.15% or better for all solutions investigated. Agreement with the experimental results (17) is within 0.02% or better if ( ci.Br.K = 0.0003 (Table III) instead of zero (Table II). We may conclude from this comparison that the thermodynamic model of Pitzer (Table II) is very realistic. An uncertainty of 0.0003 in i(ic Br K leads to uncertainties of less than 0.4% in log K(x). The largest uncertainty in equilibrium constants may thus be attributed to the original analytical data (j3). 

In addition, the critical evaluation of enthalpies of dilution and solution, as well as evaluations of heat capacities have been initiated. These evaluations will allow calculations and correlations of activity and osmotic coefficients as a function of temperature and composition. 

The solid curves below 7 A are calculated accurately for a model (Section 4) that fits the osmotic coefficient data. The curves above 7 A are merely schematic, showing in exaggerated form the oscillations that appear in gab at large r when the concentration is large, even for the models in Section 4. The dashed curve indicates the location and intensity of the peak in g+. (r) identified in aqueous NiCle in neutron diffraction and EXAFS studies, as reviewed in Section 5. 

The osmotic coefficient of water in NaCl solutions of varying concentration can be calculated from data in Ref. 15. From the resulting values of the osmotic coefficients, the effect of NaCl concentration on the equilibrium temperature for Equation (13.16) can be determined. The results of some calculations for a constant pressure of 1 atm are shown in Figure 20.5 (16). 

This expression is analogous to Eiq. (2.3), in that (1 — (p) expresses the contribution of the solvent and In y+ that of the electrolyte to the excess Gibbs energy of the solution. The calculation of the mean ionic activity coefficient of an electrolyte in solution is required for its activity and the effects of the latter in solvent extraction systems to be estimated. The osmotic coefficient or the activity of the water is also an important quantity related to the ability of the solution to dissolve other electrolytes and nonelectrolytes. 

Values of osmotic coefficients for single electrolytes have been compiled by various authors, e.g., Robinson and Stokes [22]. The activity of water can also be calculated from the known activity coefficients of the dissolved species. 

In the presence of an ionic medium NX of a concentration much larger than those of the reacting ions, the osmotic coefficient can be calculated according to Eq. (6.12) [23]. 

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. 

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