Electrolyte solutes mean ionic activity coefficients

Mean ionic activity coefficient in an electrolyte solution Mean ionic activity molality in an electrolyte solution Charge on cation and anion, respectively, in an electrolyte solution Numbers of cations and anions, respectively, from the ionization of an electrolyte... 

Activity coefficients of single ions cannot be measured directly. Instead, on the basis of experiments in which the free energy of an electrolyte is determined by various methods (such as by measuring freezing point depressions of solutions or by the measurement of the e.m.f. s of cells), a quantity called the mean ionic activity coefficient is obtained. For an electrolyte the mean ionic activity coefficient is defined as follows ... 

For electrolytes, the mean ionic activity coefficient f is defined as the geometrical mean of the rational activity coefficients of the imis in solution ... 

As a result of these electrostatic effects aqueous solutions of electrolytes behave in a way that is non-ideal. This non-ideality has been accounted for successfully in dilute solutions by application of the Debye-Huckel theory, which introduces the concept of ionic activity. The Debye-Huckel Umiting law states that the mean ionic activity coefficient y+ can be related to the charges on the ions, and z, by the equation... 

For symmetric electrolytes i=l for 1 2 electrolytes (e.g., Na2S04), 1 3 electrolytes (AICI3), and 2 3 electrolytes ([Al2(S04)3], the corresponding valnes of A, are 1.587, 2.280, and 2.551. Mean ionic activity coefficients of many salts, acids, and bases in binary aqneons solutions are reported for wide concentration ranges in special handbooks. 

We need a slightly different form of y when working with electrolyte solutions we call it the mean ionic activity coefficient y , as below. 

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). 

Recently, there have been a number of significant developments in the modeling of electrolyte systems. Bromley (1), Meissner and Tester (2), Meissner and Kusik (2), Pitzer and co-workers (4, ,j5), and" Cruz and Renon (7j, presented models for calculating the mean ionic activity coefficients of many types of aqueous electrolytes. In addition, Edwards, et al. (8) proposed a thermodynamic framework to calculate equilibrium vapor-liquid compositions for aqueous solutions of one or more volatile weak electrolytes which involved activity coefficients of ionic species. Most recently, Beutier and Renon (9) and Edwards, et al.(10) used simplified forms of the Pitzer equation to represent ionic activity coefficients. 

Table 1. Data and Results of Fit for Aqueous Solutions of uni-univalent electrolyte at 298.15°K - Mean Ionic Activity Coefficient Data... 

For concentrated solutions, the activity coefficient of an electrolyte is conveniently defined as though it were a nonelectrolyte. This is a practical definition for the description of phase equilibria involving electrolytes. This new activity coefficient f. can be related to the mean ionic activity coefficient by equating expressions for the liquid-phase fugacity written in terms of each of the activity coefficients. For any 1-1 electrolyte, the relation is ... 

The standard state for the mean ionic activity coefficient is Henry s constant H., f is the standard-state fugacity for the activity coefficient f- and x. is the mole fraction of electrolyte i calculated as though thi electrolytes did not dissociate in solution. The activity coefficient f is normalized such that it becomes unity at some mole fraction xt. For NaCl, xi is conveniently taken as the saturation point. Thus r is unity at 25°C for the saturation molality of 6.05. The activity coefficient of HC1 is normalized to be unity at an HC1 molality of 10.0 for all temperatures. These standard states have been chosen to be close to conditions of interest in phase equilibria. 

This expression is analogous to Eiq. (2.3), in that (1 — (p) expresses the contribution of the solvent and In y+ that of the electrolyte to the excess Gibbs energy of the solution. The calculation of the mean ionic activity coefficient of an electrolyte in solution is required for its activity and the effects of the latter in solvent extraction systems to be estimated. The osmotic coefficient or the activity of the water is also an important quantity related to the ability of the solution to dissolve other electrolytes and nonelectrolytes. 

As a reminder that the level of approximation in Equation (21) is the same as that of the Debye-Hiickel limiting law, the following example continues from this last result to the Debye-Hiickel expression for the mean ionic activity coefficient of an electrolyte solution. 

Figure 8.3 Schematic plot of mean ionic activity coefficient y versus molality (m) for typical strong electrolytes [1 1 (e.g., HC1), light dashed line 2 1 (e.g., H2S04), heavy dashed line], showing the extreme deviations from ideality (dotted line) even in dilute solutions.

With the help of stability constants valid for seawater, a model for distribution of the most important dissolved species can be calculated. Garrels and Thompson (1962) were the first to establish such a seawater model. Their calculations were based on stability constants (determined in simple electrolyte solutions and corrected or extrapolated to / = 0) and estimated activity coefficients of the individual ionic species in seawater. The mean ionic activity coefficients were assumed to be the same as those that would apply to a pure solution of the salt at the same ionic strength as seawater. This assumption is supported phenomenologically. 

With the reference states defined earlier for the resin and solution phases, the activity coefficient ratio in the preceding equations should approach a constant value with increasing dilution of the external solution, and ultimately become unity with increasing dilution of both the resin and external phases. Intuitively, the latter condition would not be expected to hold since the reference state of infinite dilution for the exchanger is not compatible with the physical existence of a crosslinked ion exchanger. The observed result, which has been confirmed by a large number of researchers, is that the mean ionic activity coefficient of the absorbed electrolyte in the resin phase decreases with increasing dilution of the external electrolyte. 

This simple relationship was derived before as equation 5.24, and was first used by Bauman and Eichorn in 1947 to predict selectivity sequences for simple monovalent cations from mean ionic activity coefficient data for pure aqueous electrolyte solutions containing a common anion. The inaccessibility of resin phase activity coefficients to direct measurement always remains a problem with thermodynamic equilibrium treatments. Therefore Glueckauf and others developed weight swelling and isopiestic water vapour sorption techniques to determine osmotic coefficients of pure salt forms of a resin, from which the mean ionic activity coefficients of mixed resinates could be computed using a modified form of Harned s Rule. Such studies predicted selectivity coefficient values which were in fair agreement with experiment and also demonstrated the fixed ion of the resin to be osmotically inactive. 

If we have in the same solution several electrolytes with common ions, the mean ionic activity coefficients corresponding to the various possible pairs of ions are not independent." Thus in an aqueous solution containing the ions Na+, K+, Cl ", NOg" we have... 

Relationships analogous to those given above may be derived in an exactb similar manner for the activities referred to mole fractions or molarities. As seen in 37c, the activities for the various standard states, based on the ideal dilute solution, can be related to one another by equation (37.7). The result is, however, applicable to a single molecular species the corresponding relationships between the mean ionic activity coefficients of a strong electrolyte, assumed to be completely ionized, are found to be... 

In the foregoing method the mean ionic activity coefficient of the solute has been calculated from actual vapor pressure data. If the osmotic coefficients for a reference substance are known over a range of concentrations, the activity coefficients of another electrolyte can be derived from isopiestic measurements, without actually determining the vapor pressures. If w, and p refer to an experimental electrolyte and wr, 0r and vr apply to a reference electrolyte which is isopiestic (isotonic) with the former, then by equation (39.46)... 

The actual e.m.f. of the cell in which the concentrations of the two solutions differ by an appreciable amount can now be obtained by the integration of equation (39.64), but as this is not needed for the determination of activities, the procedure will not be given here. For the present requirement the mean activity may be replaced by the product of the mean ionic molarity, which in this case, i.e., uniunivalent electrolyte, is equal to the molarity c of the solution, and the mean ionic activity coefficient y thus ... 

For solutions dilute enough for the Debye-Htickel equation to be applicable, the plot of log (t /7r) + A c against Qog 7r + log (t /7r)]Vc should be a straight line, the intercept for c equal zero giving the required value of -- log 7r, by equation (39.68). The values of log (y /yB,) are obtained from equation (39.67), and log yr, which is required for the purpose of the plot, is obtained by a short series of approximations. Once log yn has been determined, it is possible to derive log 7db for oy solution from the known value of log (y /yn)> The mean ionic activity coefficient of the given electrolyte can thus be evaluated from the e.m.f. s of concentration cells with transference, provided the required transference number information is available. ... 

These equations give the mean ionic activity coefficient in terms of the ionic strength of the solution. It is imj ortant to remember that 7 refers to a particular electrolyte whose ions have the (numerical) valences of 2+ and 2-, but the ionic strength contains terms for all the ions present in the solution. 

The various forms of equation (40.15), referred to as the Debye-HUckel limiting law, express the variation of the mean ionic activity coefficient of a solute with the ionic strength of the medium. It is called the limiting law because the approximations and assumptions made in its derivation are strictly applicable only at infinite dilution. The Debye-Hfickel equation thus represents the behavior to which a solution of an electrolyte should approach as its concentration is diminished. 

Compare the mean ionic activity coefficient of a 0.1 molal solution of (i) a uni-univalent, (ii) a bi-bivalent, electrolyte in water and methanol as solvents respectively, at 25 C. The mean ionic diameter d may be taken as 3A in each case. 

A 0.1 m solution of MgCl2 in water has a density of 1.0047 gmL at 25°C. The mean ionic activity coefficient on the molal scale is 0.528. Calculate the mean activity and electrolyte activity on this scale. Repeat the calculations for the molarity scale. 

The conclusion from the early work was that the equation was as successfid as the Debye-Hiickel theory relating to mean ionic activity coefficients was in coping with the effects of non-ideality in solutions of electrolytes. 

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