Ion activity coefficient

The Orientation of Water Molecules Adjacent to an Ion. Order and Disorder in the Vicinity of Solute Particles. Coulomb Attraction and Repulsion between Ions. Activity Coefficients. The Distance of Closest Approach. Activity Coefficients of Various Solutes. Forces Superimposed on the Coulomb Forces. 

Comparison of the concentrations of either the cation or the anion in the two phases thus has potential for evaluating the polyanion valence provided that estimates of the mean ion activity coefficient (y ) are available. Furthermore, as realized by Svensson [165], expression of the Donnan distribution of small ions in this manner has two advantages in that (i) Eq. 31 applies to each type of small ion in situations where the supporting electrolyte is not restricted to single cationic and anionic species and (ii) multivalence of a small ion is also accommodated. 

Fig. 2.3 was constructed using a K2-3 value at 250°C extrapolated from high-temperature data by Orville (1963), liyama (1965) and Hemley (1967). Ion activity coefficients were computed using the extended Debye-Hiickel equation of Helgeson (1969). The values of effective ionic radius were taken from Garrels and Christ (1965). In the calculation of ion activity coefficients, ionic strength is regarded as 0.5 im i ++mci-) (= mc -)- The activity ratio, an-f/aAb, is assumed to be unity. 

The formal Galvani potential, described by Eq. (22), practically does not depend on the concentration of ions of the electrolyte MX. Since the term containing the activity coefficients of ions in both solutions is, as experimentally shown, equal to zero it may be neglected. This results predominantly from the cross-symmetry of this term and is even more evident when the ion activity coefficients are replaced by their mean values. A decrease of the difference in the activity coefficients in both phase is, in addition, favored by partial hydration of the ions in the organic phase [31 33]. Thus, a liquid interface is practically characterized by the standard Galvani potential, usually known as the distribution potential. 

In contrast with the individual ion activity coefficients fit the mean activity coefficient ft can be measured, calculation of which can be achieved through eqn. 2.46 as follows ... 

In the foregoing derivations we have assumed that the true pH value would be invariant with temperature, which in fact is incorrect (cf., eqn. 2.58 of the Debye-Hiickel theory of the ion activity coefficient). Therefore, this contribution of the solution to the temperature dependence has still to be taken into account. Doing so by differentiating ET with respect to T at a variable pH we obtain in AE/dT the additional term (2.3026RT/F) dpH/dT, which if P (cf., eqn. 2.98) is neglected and when AE/dT = 0 for the whole system yields... 

An expression for the ion activity coefficients y,- follows from differentiating Equation 8.10 with respect to m,. The result in general form is,... 

Here m>u and m>UCd++ are molal concentrations of the unoccupied and occupied sites, respectively, and aCd++ is the activity of the free ion. Activity coefficients for the surface sites are not carried in the equation they are assumed to cancel. Equilibrium constants reported in the literature are in many cases tabulated in terms of the concentrations of free species, rather than their activities, as assumed here, and hence may require adjustment. 

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). 

For applications where the ionic strength is as high as 6 M, the ion activity coefficients can be calculated using expressions developed by Bromley (4 ). These expressions retain the first term of equation 9 and additional terms are added, to improve the fit. The expressions are much more complex than equation 9 and require the molalities of the dissolved species to calculate the ion activity coefficients. If all of the molalities of dissolved species are used to calculate the ion activity coefficients, then the expressions are quite unwieldy. However, for the applications discussed in this paper many of the dissolved species are of low concentration and only the major dissolved species need be considered in the calculation of ion activity coefficients. For lime or limestone applications with a high chloride coal and a tight water balance, calcium chloride is the dominant dissolved specie. For this situation Kerr (5) has presented these expressions for the calculation of ion activity coefficients. 

The typical system for which the equilibrium composition is desired however does not contain a single salt in solution but more usually the equivalent of several salts in solution. In addition, the activities required in equilibrium expressions arising from the law of mass action are single ion activities or in general, single ion activity coefficients. And, we are interested in the ionic activity coefficeint of each species in a multicomponent system. 

Although one wishes activity coefficients for neutral combinations of ions, it is convenient to use equations for single-ion activity coefficients which can then be combined appropriately. 

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. 

These individual-ion activity coefficients have the desired property of approaching 1 at infinite dilution, because each ratio a,/(m,/m°) approaches 1. However, individual-ion activity coefficients, like individual-ion activities, cannot be determined experimentally. Therefore, it is customary to deal with the mean activity coefficient 7+ and the mean activity a which for a uni-univalent electrolyte can be related to measurable quantities as follows ... 

According to the Debye-Htickel theory, in the limit of the infinitely dilute solution, individual-ion activity coefficients are given by the equation... 

Various empirical relations are available for calculating individual ion activity coefficients [discussed by Stumm and Morgan (1996) for natural waters and Sposito (1984a, b), for soil solutions]. In the calculations in this book I used the Davies equation ... 

Table 4.1. Single ion activity coefficients (molal scale) for uni-univalent chlorides at 25° C derived from hydration theory [11].

Note that in all ion interaction approaches, the equation for mean activity coefficients can be split up to give equations for conventional single ion activity coefficients in mixtures, e.g., Eq. (6.1). The latter are strictly valid only when used in combinations that yield electroneutrality. Thus, while estimating medium effects on standard potentials, a combination of redox equilibria with H " + e 5112(g) is necessary (see Example 3). 

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]. 

Single-ion activity coefficients, DEBYE-HUCKEL TREATMENT... 

The activity coefficient for the counterions was taken to be a constant value of 0.862. This value was derived from the experimental value of 0.745 for NaCJl, which was reported by Moore (20). The sodium ion activity coefficient was obtained as the square root of the sodium chloride activity coefficient. 

The stability constants are defined here in terms of concentrations and hence have dimensions. True thermodynamic stability constants K° and (3° would be expressed in terms of activities (Section 2.2), and these constants can be obtained experimentally by extrapolation of the (real) measurements to (hypothetical) infinite dilution. Such data are of limited value, however, as we cannot restrict our work to extremely dilute solutions. At practical concentrations, the activities and concentrations of ions in solution differ significantly, that is, the activity coefficients are not close to unity worse still, there is no thermodynamically rigorous means of separating anion and cation properties for solutions of electrolytes. Thus, single-ion activity coefficients are not experimentally accessible, and hence, strictly speaking, one cannot convert equations such as 13.6 or 13.8 to thermodynamically exact versions. 

From the association AGextra = kT In y with the ion activity coefficient y, we can therefore write... 

Both associated and nonassociated electrolytes exist in sea water, the latter (typified by the alkali metal ions U+, Na-, K+, Rb+, and Cs-) predominantly as solvated free cations. The major anions. Cl and Br, exist as free anions, whereas as much as 20% of the F in sea water may be associated as the ion-pair MgF+. and 103 may be a more important species of I than I-. Based on dissociation constants and individual ion activity coefficients the distribution of the major cations in sea water as sulfate, bicarbonate, or carbonate ion-pairs has been evaluated at specified conditions by Garrels and Thompson (19621. 

We have chosen the formulation of Equation 3.28 because it seems to be more consistent with our discussion in Section 3.1 about the nature of Bronsted acid—base reactions. Since the quantity h0 is empirically determined and cannot be broken down experimentally into its component parts, it makes little difference in practice which derivation is used. For direct measurements of hydrogen ion activity coefficients in these solvents, see T. A. Modro, K. Yates, and J. Janata, J. Amer. Chem. Soc., 97, 1492 (1975). 

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). 

The concentration dependencies of both the equivalent conductivity (A) and the chloride ion activity coefficient (fa) of the monomer DADMAC are not different... 

The X values for K+ and CL at the infinite dilution are 73.5 and 76.4 mho-cm2/equivalent, respectively (Table 1). Therefore, the conductivity of 0.01 M (or 0.01 N) KC1 solution is estimated to be 735 + 764 or 1499 pmhos/cm, which is a distinct deviation from the measured value of 1412 pmhos/cm. This problem may be overcome by using ion activity coefficient as shown below in the following equations and examples. 

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