Pitzers Water Activity

Pitzer (P14) presented two equations for calculating osmotic coefficients. One is for single electrolyte solutions, the other is for multicomponent solutions. For pure solutions of electrolyte ca  

The and i parameters for some Interactions are tabulated in Appendix 5.1. The 0 term can be neglected for solutions of electrolytes of similar or not too different charges. 

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The ideality of the solvent in aqueous electrolyte solutions is commonly tabulated in terms of the osmotic coefficient 0 (e.g., Pitzer and Brewer, 1961, p. 321 Denbigh, 1971, p. 288), which assumes a value of unity in an ideal dilute solution under standard conditions. By analogy to a solution of a single salt, the water activity can be determined from the osmotic coefficient and the stoichiometric ionic strength Is according to,... 

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]. 

Alternatively, water activities can be taken from Table B-1. These have been calculated for the most common ionic media at various concentrations applying Pitzer s ion interaction approach and the interaction parameters given in [91 PIT]. Data in italics have been calculated for concentrations beyond the validity of the parameter set applied. These data are therefore extrapolations and should be used with care. 

Water activities for the most common ionic media at various concentrations applying Pitzer s ion interaction approach and the interaction parameters given in [91 PIT].362... 

If determination of osmotic and activity coefficients is limited only to NajCit and KjCit salts then the Pitzer formalism [167] can be applied. In this procedure trisodium citrate and tripotassium citrate are treated as fully dissociated electrolytes and the water activities a (T m)=p(T m)lp%T), at constant temperature are calculated using Eqs. (5.27) and (5.31). These equations represent the best fit of vapour pressures as a functions of concentration. 

Typical behaviour of osmotic and activity coefficients as calculated using Eqs. (5.36) and (5.37), is illustrated for trisodium citrate and tripotassium citrate in Fig. 5.15. It can be observed, that values of the (/w) and y+(/w) coefficients after a strong fall in very dilute solutions depend rather weakly on the citrate concentration. Since a T-,m) values are nearly temperature independent, the same is observed in the case osmotic and activity coefficients. It is worthwhile to mention that the Pitzer model was also used by Schunk and Maurer [163] when they determined water activities at 25 °C in ternary systems (citric acid + inorganic salt). The interactions parameters between ions, which were applied to represent activities in ternary systems, were calculated by taking into account the dissociation steps of citric acid and the formation of bisulfate ions for solutions with sodium sulfate. 

In applying this equation to multi-solute systems, the ionic concentrations are of sufficient magnitude that molecule-ion and ion-ion interactions must be considered. Edwards et al. (6) used a method proposed by Bromley (J7) for the estimation of the B parameters. The model was found to be useful for the calculation of multi-solute equilibria in the NH3+H5S+H2O and NH3+CO2+H2O systems. However, because of the assumptions regarding the activity of the water and the use of only two-body interaction parameters, the model is suitable only up to molecular concentrations of about 2 molal. As well the temperature was restricted to the range 0° to 100 oc because of the equations used for the Henry1s constants and the dissociation constants. In a later study, Edwards et al. (8) extended the correlation to higher concentrations (up to 10 - 20 molal) and higher temperatures (0° to 170 °C). In this work the activity coefficients of the electrolytes were calculated from an expression due to Pitzer (9) ... 

About the same time Beutier and Renon (11) also proposed a similar model for the representation of the equilibria in aqueous solutions of weak electrolytes. The vapor was assumed to be an ideal gas and < >a was set equal to unity. Pitzer s method was used for the estimation of the activity coefficients, but, in contrast to Edwards et al. (j)), two ternary parameters in the activity coefficient expression were employed. These were obtained from data on the two-solute systems It was found that the equilibria in the systems NH3+ H2S+H20, NH3+C02+H20 and NH3+S02+H20 could be represented very well up to high concentrations of the ionic species. However, the model was unreliable at high concentrations of undissociated ammonia. Edwards et al. (1 2) have recently proposed a new expression for the representation of the activity coefficients in the NH3+H20 system, over the complete concentration range from pure water to pure NH3. it appears that this area will assume increasing importance and that one must be able to represent activity coefficients in the region of high concentrations of molecular species as well as in dilute solutions. Cruz and Renon (13) have proposed an expression which combines the equations for electrolytes with the non-random two-liquid (NRTL) model for non-electrolytes in order to represent the complete composition range. In a later publication, Cruz and Renon (J4J, this model was applied to the acetic acid-water system. 

An important application of Pitzer s work is that of Whitfield (30) who developed a model for sea water. Single ion activity coefficients for many trace metals in sea water are tabulated over the ionic strength range of 0.2m to 3.0m. 

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. 

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... 

Note that the equations for estimating the pressure dependencies of 7 and aw (Eqs. 2.87 and 2.90) depend on the Pitzer equations (Eqs. 2.76, 2.80, and 2.81) but this is not the case for the pressure dependence of the equilibrium constants (Eq. 2.29) the latter equation is based entirely on partial molar volumes at infinite dilution, which are independent of concentration. Also, compared to the pressure-dependent equation for the equilibrium constant (Eq. 2.29), the pressure equations for activity coefficients (Eq. 2.87) and the activity of water (Eq. 2.90) do not contain compressibilities (K) because the database for these terms and the associated Pitzer parameters are lacking at present (Krumgalz et al. 1999). The consequences of truncating Eqs. 2.80 and 2.81 for ternary terms and Eqs. 2.87 and 2.90 for compressibilities will be discussed in Sect. 3.6 under limitations. 

Equations 2.87 (activity coefficient), 2.88 (density), and 2.90 (activity of water) are all indirectly dependent on the temperature and pressure dependence of B v, B v, BC2), and Cv (Eqs. 2.76, 2.80, and 2.81). Table B.10 (Appendix B) lists the temperature dependence of these volumetric Pitzer parameters. The pressure dependence of these parameters were evaluated with the density equation (Eq. 2.88). All three terms in the denominator of Eq. 2.88 are temperature and pressure dependent. The density of pure water (p°) as a function of temperature and pressure is evaluated with Eqs. 3.14-3.16 and 3.20. Similarly, the molar volume of ions as a function of temperature and pressure is calculated by... 

Fig. 4 to Fig. 8 show the severe divergence for activity coefficients such as given here for calcium, chloride, sulfate, sodium and water ions, calculated with different equations. The activity coefficients were calculated applying Eq. 13 to Eq. 17 for the corresponding ion dissociation theories, whereas the values for the PITZER equations were gained using the program PHRQPITZ. The limit of validity of each theory is clearly shown. 

Pitzer (1973) re-examined the statistical mechanics of aqueous electrolytes in water and derived a different but semi-empirical method for activity coefficients, commonly termed the Pitzer specific-ion-interaction model. He fitted a slightly different function for behavior at low concentrations and used a virial coefficient formulation for high concentrations. The results have proved extremely fruitful for modeling activity coefficients over a very large range of molality. The general equation is... 

Clegg S. L. and Whitfield M. (1991) Activity coefficients in natural waters. In Activity Coefficients in Electrolyte Solutions (ed. K. S. Pitzer). CRC, Boca Raton, FL, pp. 279-434. 

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