Single-ion activities

Moreover, for the constants j2m and Mm-, the critical potential coefficient with respect to the single-ion activity a of the cation, the minority ion, is written using Eq. (51)... 

Rabinovich et al. have shown that it is possible to propose an extrather-modynamic definition of single-ion activity, a, as a function of the real potentials of those particles. "" By carrying out the measurements of voltaic cells containing electrodes reversible to the same ionic species in solutions of different concentrations in the same solvent. 

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). 

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. 

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 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. 

However, because the determination of o(H+), a single ion activity, is thermodynamically impossible, an operational pH definition is given as outlined in (1) and (2) below ... 

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). 

Equation (1) involves the single ion activity of the hydrogen ion, and it might be said that thus the problems commenced. Activities of individual ions can never be measured without non-thermodynamic assumptions being made. 

In order to improve the primary method for pH, investigations into solution theory and into the concept of single ion activity are necessary. 

Future improvements of the concept of single ion activity, e.g. the Pitzer treatment will open up the possibility for pH values to be traceable to the SI with acceptable uncertainties for calibration purposes. 

Even with the definition of the Reference State, chemical thermodynamics alone cannot provide a unique methodology for the measurement of single-ion activity coefficients. An infinitude of possibilities exists, each of that calls upon its own extra thermodynamic set of conventions according to criteria of experimental convenience and intended application. However, chemical thermodynamics does provide general constraints that limit any set of arbitrary conventions defining single-ion activities. 

By analogy with Eq. 1.13, one can define single-ion activity coefficients 9... 

The mean ionic and single-ion activity coefficients are conceptually different parameters, but both must conform to the Debye-Hiickel infinite-dilution limit. This theoretical constraint on activity coefficients takes on a particular mathematical form, depending upon the way in which an electrolyte solution is characterized. In a strictly thermodynamic picture of aqueous solutions, the Debye-Hiickel limit can be expressed as follows 9... 

This equation represents a general theoretical constraint on single-ion activity coefficients. 

For a discussion of single-ion activity coefficients and their calculation, see G. Sposito, The future of an illusion Ion activities in soil solutions, Soil Sci. Soc. Am. J. 48 531 (1984). 

These concepts are developed in greater detail in Chap. 7 of W. Stumm and J. J. Morgan, Aquatic Chemistry, Wiley, New York, 1981. As with any single-ion activity, it is always possible to relate the electron activity to a chemical potential formally (see, e.g., Eq. s2.22) and thus define an equivalent electrochemical emf (see Eq. s2.23). Accordingly, the (electro)chemical potential of an electron in aqueous solution is related to the pE value by the equation (see Special Topic 2) ... 

Given the generic relationship for a single-ion activity coefficient,... 

Only mean activity coefficients can be experimentally determined for salts, not activity coefficients for single ions. The Maclnnes Convention is one method for obtaining single ion activity coefficients and states that because of the similar size and mobility of the potassium and chloride ions ... 

---