Mean ionic activity coefficients

In general, the mean ionic activity coefficient is given by... 

The activity of any ion, a = 7m, where y is the activity coefficient and m is the molaHty (mol solute/kg solvent). Because it is not possible to measure individual ionic activities, a mean ionic activity coefficient, 7, is used to define the activities of all ions in a solution. The convention used in most of the Hterature to report the mean ionic activity coefficients for sulfuric acid is based on the assumption that the acid dissociates completely into hydrogen and sulfate ions. This assumption leads to the foUowing formula for the activity of sulfuric acid. 

Y- Mean ionic activity coefficient of solute Dimensionless Dimensionless... 

E6.12 The HC1 pressure in equilibrium with a 1.20 molal solution is 5.15 x 10 8 MPa and the mean ionic activity coefficient is known from emf measurements to be 0.842 at T = 298.15 K. Calculate the mean ionic activity coefficients of HC1 in the following solutions from the given HC1 pressures... 

Equations (7.35) and (7.36) can be used to calculate the activity coefficients of individual ions. However, as we discussed in Chapter 6, 7+ and 7- cannot be measured individually. Instead, 7 , the mean ionic activity coefficient for the electrolyte, M +AV-, given by... 

Figure 9.6 Mean ionic activity coefficients for HCl(aq) at T = 298.15 K obtained from the emf results of G. A. Linhart, J. Am. Chem. Soc.. 41, 1175-1180 (1919). The dashed line is the Debye-Huckel limiting law prediction.

As a result of these electrostatic effects aqueous solutions of electrolytes behave in a way that is non-ideal. This non-ideality has been accounted for successfully in dilute solutions by application of the Debye-Huckel theory, which introduces the concept of ionic activity. The Debye-Huckel Umiting law states that the mean ionic activity coefficient y+ can be related to the charges on the ions, and z, by the equation... 

Here/+ is the mean ionic activity coefficient defined, by analogy with the mean ionic activity, as... 

For symmetric electrolytes i=l for 1 2 electrolytes (e.g., Na2S04), 1 3 electrolytes (AICI3), and 2 3 electrolytes ([Al2(S04)3], the corresponding valnes of A, are 1.587, 2.280, and 2.551. Mean ionic activity coefficients of many salts, acids, and bases in binary aqneons solutions are reported for wide concentration ranges in special handbooks. 

As a result, we obtain for the mean ionic activity coefficient,... 

All quantities in Eq. (12.6) are measurable The concentrations can be determined by titration, and the combination of chemical potentials in the exponent is the standard Gibbs energy of transfer of the salt, which is measurable, just like the mean ionic activity coefficients, because they refer to an uncharged species. In contrast, the difference in the inner potential is not measurable, and neither are the individual ionic chemical potentials and activity coefficients that appear on the right-hand side of Eq. (12.3). 

Figure 7.9 The dependence of the mean ionic activity coefficient y on concentration for a few simple solutes...

We need a slightly different form of y when working with electrolyte solutions we call it the mean ionic activity coefficient y , as below. 

The mean ionic activity coefficient is obtained as a geometric mean via... 

We assume that the activity coefficient of the ion-pairs is unity and denote the mean ionic activity coefficient by y . The thermodynamic equilibrium constant for the dissociation is then given by the equation... 

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). 

DIPA) and methyldiethanolamine (MDEA) have also been employed. Earlier, Atwood et al. (J 5) proposed a thermodynamic model for the equilibria in I S+alkanol-amine+H20 systems. The central feature of this model is the use of mean ionic activity coefficient. The activity coefficients of all ionic species are assumed to be equal and to be a function only of ionic strengths. Klyamer and Kolesnikova ( 1j[) utilized this model for correlation of equilibria in C02+alkanol-amine+H O systems and Klyamer et al. (J 7) extended it to the H2S+C02+alkanolamine+H20 system. The model is restricted to low pressures as the fugacity coefficients are assumed unity and it has been found that the predictions are inaccurate in the four-component system since the activity coefficients are not equal when a number of different cations and anions are present. 

Recently, there have been a number of significant developments in the modeling of electrolyte systems. Bromley (1), Meissner and Tester (2), Meissner and Kusik (2), Pitzer and co-workers (4, ,j5), and" Cruz and Renon (7j, presented models for calculating the mean ionic activity coefficients of many types of aqueous electrolytes. In addition, Edwards, et al. (8) proposed a thermodynamic framework to calculate equilibrium vapor-liquid compositions for aqueous solutions of one or more volatile weak electrolytes which involved activity coefficients of ionic species. Most recently, Beutier and Renon (9) and Edwards, et al.(10) used simplified forms of the Pitzer equation to represent ionic activity coefficients. 

A wide variety of data for mean ionic activity coefficients, osmotic coefficients, vapor pressure depression, and vapor-liquid equilibrium of binary and ternary electrolyte systems have been correlated successfully by the local composition model. Some results are shown in Table 1 to Table 10 and Figure 3 to Figure 7. In each case, the chemical equilibrium between the species has been ignored. That is, complete dissociation of strong electrolytes has been assumed. This assumption is not required by the local composition model but has been made here in order to simplify the systems treated. 

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

Figure 3. Comparison of the calculated and experimental mean ionic activity coefficients of three uni-univalent electrolytes at 298.15 K (-) calculated (A, O,...

Activity Coefficients. The mean ionic activity coefficients of a cation-anion pair (here shortened to "activity coefficient" for convenience) are directly measureable in pure and occasionally in mixed solutions. The mean ionic activity coefficient is... 

Equation 1 is differentiated to calculate the ionic activity coefficients using the same technique as for non-electrolytes. Combining these ionic activity coefficients to form the mean ionic activity coefficients, y., we obtain the Bronsted-Guggenheim equations for two 1-1 electrolytes with a common ion ... 

Figure 1. Mean ionic activity coefficients for the HCl-NaCl-H20 system at 25°C...

For concentrated solutions, the activity coefficient of an electrolyte is conveniently defined as though it were a nonelectrolyte. This is a practical definition for the description of phase equilibria involving electrolytes. This new activity coefficient f. can be related to the mean ionic activity coefficient by equating expressions for the liquid-phase fugacity written in terms of each of the activity coefficients. For any 1-1 electrolyte, the relation is ... 

The standard state for the mean ionic activity coefficient is Henry s constant H., f is the standard-state fugacity for the activity coefficient f- and x. is the mole fraction of electrolyte i calculated as though thi electrolytes did not dissociate in solution. The activity coefficient f is normalized such that it becomes unity at some mole fraction xt. For NaCl, xi is conveniently taken as the saturation point. Thus r is unity at 25°C for the saturation molality of 6.05. The activity coefficient of HC1 is normalized to be unity at an HC1 molality of 10.0 for all temperatures. These standard states have been chosen to be close to conditions of interest in phase equilibria. 

---