Activity coefficients of individual ions

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... 

In this development, attention is focused exclusively on activity coefficients of cation-anion pairs, with no use being made of activity coefficients of individual ions. 

As the values of the activity coefficients of individual ions are not known, the so called mean activity coefficient y i is used which is defined as follows ... 

The potential of such an electrode is, therefore, determined by the activity of its own ions in the electrolyte such an electrode is said to be reversible with respect to its own cations. It is well known that the activity + is expressed by the product of the molality (or molarity) and the activity coefficient of the respective ion. Since, however, the activity coefficients of individual ions are not known, they are being replaced by the mean activity coefficients y In sufficiently diluted solutions it is possible to substitute in the formula (VI-11) directly the concentrations for activities, since the activity coefficients in this instance would very nearly equal unity. 

Taghikhani, V. et al.. Application of the MSA to the modeling of the activity coefficients of individual ions. Fluid Phase Equilibria, 167, 161, 2000. 

The Activities and Activity Coefficients of Individual Ions tn Strong Electrolytes... 

Numerous models predict the activity coefficient of individual ions in solution. The one by Debye and Hiickel [8] considers only electrostatic (columbic) interactions between cations and anions in a dilute solution of a single, completely dissociated salt. It is assumed that ion-ion interactions (as opposed to other phenomena such as ion-solvent interactions, ion solvation effects, and variations in the solvent dielectric constant with salt concentration) cause the ion activity coefficients to deviate from 1.0. From a practical point, only the Debye-Hiickel activity coefficient relationship is needed, along with some knowledge of the theory s shortcomings, which restrict its application. For a dilute electrolytic solution containing a binary salt (i.e., a salt with one type each of cation and anion species), the ion activity coefficient from Debye-Hiickel theory is given by... 

Note In principle, the ion activity coefficient of a salt (y+) can be determined, but not those of individual ions (y and y,), because their concentrations cannot be varied independently. Nevertheless, the activity coefficients of individual ions are very useful, and as mentioned, one tries to calculate them from theory. Ion-selective electrodes measure chemical potentials (which depend on activities, not concentrations), but the standard potential is unknown. For the measurement of pH, which is the negative logarithm of the hydrogen ion activity, one has therefore arbitrarily chosen a reference potential for a certain buffer, which potential is of course as close to the real one as theory permits it to be calculated. 

The Debye-Huckel theory gives a calculation of the activity coefficients of individual ions. However, although the individual concentrations of the ions of an electrolyte solution can be measured, experiment cannot measme the individual activity coefficients. It does, however, furnish a sort of average value of the activity coefficient, called the mean activity coefficient, for the electrolyte as a whole. The term mean is not used in its common sense of an average quantity, but is used in a different sense which reflects the number of ions which result from each given formula. Such mean activity coefficients are related to the individual activity coefficients in a manner dictated by the stoichiometry of the electrolyte. 

As an example. Table II compares log values of the molality, activity, and activity coefficient of individual ions computed for seawater in equilibrium with aragonite at 25 °C and a CO2 partial pressure of lO atm. The results are presented for the problem computed on the Macinnes scale and without scaling. In the problem, pH was calculated from the various equilibria and is therefore internally consistent with the aqueous model and the respective scale. When all individual-ion activity coefficients are consistent with a single scale convention, the individual-ion molalities are independent of choice of scale, but the individual-ion activities and individual-ion activity coefficients are scale dependent. For example, the pH of seawater in equilibrium with aragonite at 25 °C and 10"3 atm Pco2 (using the data base of (9) ) is 7.871 on the Macinnes scale and 7.828 without scaling, as shown in Table II. 

We pointed out above that activity coefficients of individual cations or anions cannot be measured because of the requirement that solutions remain electrically neutral. Instead, the activity coefficient of the total solute (NaCl, CaCl2, etc.) is measured, and that is used to calculate mean activity coefficients, 7 . These are a kind of average of the activity coefficients for the individual cation and anion and are not the true values for the individual ions. This inability to estimate activity coefficients of individual ions is a problem because it is specific ion activities (rather than activities of total salts) that are most frequently required in thermodynamic calculations. 

The principle of electroneutrality requires that the ionic species in an electrolyte solution remain charge balanced on a macroscopic scale. The requirement of electroneutrality arises from the large amount of energy required to separate oppositely charged particles by any significant distance against Coulombic forces (e.g., Denbigh, 1971). Because of this requirement, we cannot obtain a flask of sodium ions at the chemistry supply room, nor can we measure the activity coefficients of individual ions directly. 

The current density, J, of Eq. (2.1) must be summed up with contributions from each negative and positive ion species. Note that it may be difficult to find the activity coefficients of individual ion species because of electroneutrality—an electrolyte cannot consist of only anions or cations. 

Table 3.1. Values of activity coefficients of individual ions...

The determinaticMi of activity coefficients of individual ions is still a widely discussed application of UP models. From thermodynamic point of view it is impossible, so the problems of diffusion potential calcularimi and single ion activities determination form a sort of vicious circle. To calculate the former... 

The activity coefficients of individual ions (/n cit assumed to be unity) can be approximated in dilute solutions by the Debye-Hiickel expressions... 

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 ... 

Although there is no straightforward and convenient method for evaluating activity coefficients for individual ions, the Debye-Hiickel relationship permits an evaluation of the mean activity coefficient (y+), for ions at low concentrations (usually <0.01 moll-1) ... 

The degree of success of the ideal ion mixture model of ion exchange, which allows for nonideal behavior by introducing activity coefficients for individual ions in much the same way that the activities of ions in aqueous solutions are adjusted for nonideality. 

The treatment starts from a generalized virial expansion for Gxs, the total Gibbs energy of the solution minus the Gibbs energy of an ideal solution of the same composition. Although the virial coefficients are not individually measurable, measurable combinations of the virial coefficients have been identified - —/. Differentiation of Gxs with respect to the amount of water allows computation of the activity of the water and differentiation with respect to the amount of an ion yields the activity coefficient of that ion. 

The calculations of mean activity coefficients for various salts using the WATEQ model indicate that if Equation 2 is used for individual-ion activity coefficients of free ions, the results are reliable only for concentrations of 0.1 molal or less. If equations 3 or 5 are used (the amended WATEQ model), the calculated mean activity coefficients are accurate for the salts used to derive the individual-ion activity coefficients of the free ions, but are not accurate for other salts unless additional complexes are included. 

Because the ions Ag and Cl cannot exist separately in solution, the individual activity coefficients of these ions are not separately defined. Instead we define a mean activity y , which in this case is the geometric mean of the individual activity coefficients y = VyAg+ ycr For rn n electrolyte (a compound composed of cations of charge m+ and anions of charge rf, for example, MgCl2 is a 2 1 electrolyte), the mean activity coefficient is y = The mean activities for ionic compounds can be estimated... 

Express the mean activity coefficient of the ions in a solution of CaCl2 in terms of the activity coefficients of the individual ions. 

Table 8.1 Individual Activity Coefficients of Ions In Water at 25°C 8.3...

This is an equation for calculating the activity coefficient of an individual ion m (i.e., a parameter inaccessible to experimental determination). Let us, therefore, change to the values of mean ionic activity. By definition [see Eq. (3.27)],... 

Guggenheim used this assumption to employ Eq. (1.3.38) for the activity coefficient of the electrolyte, where the product aB was set equal to unity and the specific interaction between oppositely charged ions was accounted for in the term CL Consider a mixture of two uni-univalent electrolytes AlBl and AUBU with overall molality m and individual representations yl = milm and yn = mulm, where mx and mn are molalities of individual electrolytes. According to Guggenheim,... 

Two types of methods are used to measure activity coefficients. Potentiometric methods that measure the mean activity coefficient of the dissolved electrolyte directly will be described in Section 3.3.3. However, in galvanic cells with liquid junctions the electrodes respond to individual ion activities (Section 3.2). This is particularly true for pH measurement (Sections 3.3.2 and 6.3). In these cases, extrathermodynamical procedures defining individual ion activities must be employed. 

Kielland, J. "Individual Activity Coefficients of Ions in Aqueous Solutions," J. Amer. Chem. Soc., 1937, j>9, 1675-78. 

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