Electrolyte solutions, activity coefficient conventions

It has been emphasized repeatedly that the individual activity coefficients cannot be measured experimentally. However, these values are required for a number of purposes, e.g. for calibration of ion-selective electrodes. Thus, a conventional scale of ionic activities must be defined on the basis of suitably selected standards. In addition, this definition must be consistent with the definition of the conventional activity scale for the oxonium ion, i.e. the definition of the practical pH scale. Similarly, the individual scales for the various ions must be mutually consistent, i.e. they must satisfy the relationship between the experimentally measurable mean activity of the electrolyte and the defined activities of the cation and anion in view of Eq. (1.1.11). Thus, by using galvanic cells without transport, e.g. a sodium-ion-selective glass electrode and a Cl -selective electrode in a NaCl solution, a series of (NaCl) is obtained from which the individual ion activity aNa+ is determined on the basis of the Bates-Guggenheim convention for acr (page 37). Table 6.1 lists three such standard solutions, where pNa = -logflNa+, etc. 

A second example is provided by a semiempirical correlation for multi-component activity coefficients in aqueous electrolyte solutions shown in Fig. 2. This correlation, developed by Fritz Meissner at MIT [3], presents a method for scale-up activity-coefficient data for single-salt solutions, which are plentiful, are used to predict activity coefficients for multisalt solutions for which experimental data are rare. The scale-up is guided by an extended Debye-Hilckel theory, but essentially it is based on enlightened empiricism. Meissner s method provides useful estimates of thermodynamic properties needed for process design of multieffect evaporators to produce salts from multicomponent brines. It will be many years before sophisticated statistical mechanical techniques can perform a similar scale-up calculation. Until then, correlations such as Meissner s will be required in a conventional industry that produces vast amounts of inexpensive commodity chemicals. 

In the case of an - electrolyte dissociating in solution as Aj,+ Bj, < is+Az+ + z/ Bz where v+z+ = v z to ensure electroneutrality, and the total number of particles formed by each molecule is v = v+ + z/, then the only activity that can be measured is that of the complete species, and the individual ions cannot be assigned meaningful chemical potentials. Under these circumstances, a mean activity coefficient is defined through the equation yv = y++ yvs. Since individual ionic chemical potentials are not measurable, it has become conventional to assign to the chemical potential of the hydrogen ion under standard conditions the value of zero, allowing relative chemical potentials for all other ions to be formulated. 

The mean ion activity coefficient values can be obtained from experiments where the effect of electrolyte concentration on the A sp value for a salt is determined. The mean values are then compared with those for KCl under the same solution conditions. The single-ion activity coefficient for Ca " " can then be computed if an assumption is made about the individual values for and Cl. These ions have the same magnitude of charge and similar electronic configuration, ionic radii, and ionic mobilities. On the basis of these properties, the Macinnes convention (1919) states that... 

Frequently, solubility measurements may be used in mixed electrolytes to obtain mean molar activity coefficients. This method hinges on the use of an electrolyte solution which is saturated with respect to any particular salt, so that the equilibrium My Ay (s) = v+M + -F v A prevails. This situation may be characterized by (among others) use of the equilibrium constant Kf specified by Eq. (3.7.8b). It is conventional either to ignore the product term [afJ"(T, P)Y as being equal to (at unit pressure) or close to unity, or to absorb this constant factor into the equilibrium constant as well. This then gives rise to the expression... 

The Ionic Medium Scale This convention can be applied to solutions that contain a swamping concentration of inert electrolyte in order to maintiiin a constant ionic medium. The activity coefficient, f = A /[A], beconries unity as the solution approaches the pure ionic medium, that is, when all concentrations other than the medium ions approach zero ... 

In dilute solutions (/ < 10 M), that is, in fresh waters, our calculations are usually based on the infinite dilution activity convention and thermodynamic constants. In these dilute electrolyte mixtures, deviations from ideal behavior are primarily caused by long-range electrostatic interactions. The Debye-Huckel equation or one of its extended forms (see Table 3.3) is assumed to give an adequate description of these interactions and to define the properties of the ions. Correspondingly, individual ion activities are estimated by means of individual ion activity coefficients calculated with the help of the Guntelberg or Davies (equations 3 and 4 of Table 3.3) or it is often more convenient to calculate, with these activity coefficients, a concentration equilibrium constant valid at a given /,... 

When the standard and reference states for the exchanger and external solution phases are defined according to the conventional theory of electrolyte solutions, the thermodynamic exchange constant is by definition equal to unity. Therefore from equation 5.23 the observed selectivity in dilute solutions arises from the activity coefficient ratio in the exchanger phase thus ... 

If the conventional standard and reference states for dilute electrolyte solutions are adopted the pressure-volume term is usually neglected for simple non-hydrated ions, and further assuming the activity coefficient ratio in the external solution to equal unity, equation 5.29 becomes ... 

Corresponding to each chemical potential there is an activity coefficient defined in terms of equation (20.4). By convention, the activity coefficients of electrolytes are always expressed in terms of the ideal dilute solution as standard reference state, cf. chap. XXI, 3. Thus in the case of an aqueous NaCl solution we may write... 

The Macinnes convention leads to = Tci = 7 kci, We can now compute individual ion activity coefficients from their mean values measured in solutions of strong electrolytes using y Kci values as our starting point. (In the ideal strong electrolyte, cations and anions are unassociated with each other and thus do not form complexes [see Chap. 3].) It is important to remember that all such calculations must be done with y values for KCl and other salts measured at the same ionic strength, which is not the same molality except for monovalent-monovalent salts. 

In this chapter we discuss some of the properties of electrolyte solutions. In Sec. 12-1, the chemical potential and activity coefficient of an electrolyte are expressed in terms of the chemical potentials and activity coefficients of its constituent ions. In addition, the zeroth-order approximation to the form of the chemical potential is discussed and the solubility product rule is derived. In Sec. 12-2, deviations from ideality in strong-electrolyte solutions are discussed and the results of the Debye-Hiickel theory are presented. In Sec. 12-3, the thermodynamic treatment of weak-electrolyte solutions is given and use of strong-electrolyte and nonelectrolyte conventions is discussed. 

In principle, the conventions used for nonelectrolyte solutions developed in Chap. 11 could be employed for electrolyte solutions which are subject to the condition of electroneutrality. Agreement with experimental data could be obtained by choosing the molecular weight to be some fraction of the formula weight. However, these conventions generally lead to activity coefficients which are rapidly varying functions of composition. In order to avoid this, we formally define chemical potentials and activity coefficients for ionic components. The definition of chemical potentials for ionic components does not have operational significance since their concentrations cannot be varied independently. 

Thus it is useful to associate a with an activity coefficient at low concentrations. At low concentrations, it is useful to represent Pha by Eq. (12-59). Equation (12-59) is similar to Eq. (12-30) which expresses //ha in terms of strong-electrolyte conventions. This completes our discussion of weak-electrolyte solutions. [More details concerning computations involving Eq. (12-56) can be found in F. H. MacDougall, Thermodynamics and Chemistry, chap. XV, 3d ed., John Wiley, Sons, Inc., New York, 1939.]... 

The practical matter of choosing an arbitrary conventional means of evaluating the activity of an individual ionic species that cannot be exactly defined may be discussed from the standpoint of dilute solutions, concentrated solutions and mixtures of electrolytes (1,28). In dUbite solutions, multiple pathways to the activity coefficients, /, of single unassociated ions exist and as can be seen from Table 2.2, the values of pM and pX obtained by various simple conventions do not differ greatly at an ionic strength of 0.1. 

The standard acidity scale, although well defined theoretically, has the limitation in practice that only the mean activity coefficient, but not the single-ion activity coefficient, is thermodynamically assessible. The single-ion coefficient depends on the composition of the solution as well. One way to circumvent this problem would be to have a defined value of the activity coefficient for one selected ion. Given that, all other activity coefficients could be obtained firom the activity coefficients of the particular electrolytes and that special single-ion coefficient. The value of this selected coefficient could be used, then, as the base of the conventional acidity scale. This single-ion activity coefficient is derived for chloride by the Debye-Hiickel theory. This choice is made by convention, initially proposed for aqueous solutions it is accepted also for other amphiprotic, polar solvents. Note that the measurements of the proton activity are carried out in cells without liquid junction. 

Bromley felt that treating a multicomponent solution as a single complex salt solution would be the simplest approach towards calculating the activity coefficients of electrolytes in solution. The Fj terms would then be based on the ionic interactions of this "complex salt". Using the convention that odd number subscripts denote cations and even number subscripts indicate anions, he proposed for a cation ... 

Nesbitt (44,45) has pointed out that ratios of the activity coefficients of ions of the same charge in mixtures can be obtained without ambiguity from mean activity coefficients of electrolytes with a common anion or cation. If HCl is one of the electrolytes, a pH measurement might provide a reference point for calculating the activity coefficient of a second cation as well as that of the anion involved. Equilibrium theory suggests that pH measurements of saturated solutions of a metal hydroxide or carbonate might also lead to the activity coefficient of the metal ion concerned (46). In these cases, a convention is necessary to provide numerical values of the pH. 

The Macinnes, Debye-HUckel, and pH conventions describe the ionic activity as a function only of ionic strength. It is, however, not reasonable to expect the chloride ion, for example, always to have the same activity coefficient at a fixed temperature and ionic strength, regardless of the nature of the counter cation. Bjerrum (58) showed in 1920 that the behavior of electrolytes, including the minima observed in plots of In y+ vs. I, provides evidence for ion-solvent interactions. Hydration of the ions must be considered, and "single-parameter conventions are inadequate from the standpoint of solution theory. 

Pitzer and co-workers have developed an ion interaction model and published a series of papers (Pitzer, 1973a-b, 1974a-b, 1975, 1977, 1995, 2000 Pabalan Pitzer, 1987) which gave a set of expressions for osmotic coefficients of the solution and mean activity coefficient of electrolytes in the solution. Expressions of the chemical equilibrium model for conventional single ion activity coefficients derived are more convenient to use in solubility calculations (Harvie Weare, 1980 Harvie et al.l984 Felmy Weare, 1986 Donad Kean, 1985). 

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