The Mean Ionic Activity Coefficient

Consider a uni-univalent electrolyte MA (e.g., NaCl). The chemical potential of the ions is CExj. (3.57)] 

What has been obtained here is the change in the free energy of the system due to the addition of 2 moles of ions—1 mole of ions and 1 mole of A ions—which are contained in 1 mole of electroneutral salt MA. 

suppose that one is only interested in the average contribution to the free energy of the system from 1 mole of both M and A ions. One has to divide Eq. (3.67) by 2 

What is the significance of these quantities fi, x, and It is obvious they are all average quantities—the mean chemical potential the mean standard chemical potential the mean ionic mole fraction x, and the mean ionic-activity coefficient f. In the case of and fi% the arithmetic mean (half the sum) is taken because free energies are additive, but in the case of x and f , the geometric mean (the square root of the product) is taken because the effects of mole fraction and activity coefficient on free energy are multiplicative. 

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In general, the mean ionic activity coefficient is given by... 

The activity of any ion, a = 7m, where y is the activity coefficient and m is the molaHty (mol solute/kg solvent). Because it is not possible to measure individual ionic activities, a mean ionic activity coefficient, 7, is used to define the activities of all ions in a solution. The convention used in most of the Hterature to report the mean ionic activity coefficients for sulfuric acid is based on the assumption that the acid dissociates completely into hydrogen and sulfate ions. This assumption leads to the foUowing formula for the activity of sulfuric acid. 

E6.12 The HC1 pressure in equilibrium with a 1.20 molal solution is 5.15 x 10 8 MPa and the mean ionic activity coefficient is known from emf measurements to be 0.842 at T = 298.15 K. Calculate the mean ionic activity coefficients of HC1 in the following solutions from the given HC1 pressures... 

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... 

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... 

Here/+ is the mean ionic activity coefficient defined, by analogy with the mean ionic activity, as... 

As a result, we obtain for the mean ionic activity coefficient,... 

All quantities in Eq. (12.6) are measurable The concentrations can be determined by titration, and the combination of chemical potentials in the exponent is the standard Gibbs energy of transfer of the salt, which is measurable, just like the mean ionic activity coefficients, because they refer to an uncharged species. In contrast, the difference in the inner potential is not measurable, and neither are the individual ionic chemical potentials and activity coefficients that appear on the right-hand side of Eq. (12.3). 

Figure 7.9 The dependence of the mean ionic activity coefficient y on concentration for a few simple solutes...

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

The mean ionic activity coefficient is obtained as a geometric mean via... 

We assume that the activity coefficient of the ion-pairs is unity and denote the mean ionic activity coefficient by y . The thermodynamic equilibrium constant for the dissociation is then given by the equation... 

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. 

Activity Coefficients. The mean ionic activity coefficients of a cation-anion pair (here shortened to "activity coefficient" for convenience) are directly measureable in pure and occasionally in mixed solutions. The mean ionic activity coefficient is... 

Equation 1 is differentiated to calculate the ionic activity coefficients using the same technique as for non-electrolytes. Combining these ionic activity coefficients to form the mean ionic activity coefficients, y., we obtain the Bronsted-Guggenheim equations for two 1-1 electrolytes with a common ion ... 

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. 

To leam that activity a and concentration c are related by the expression a = c X (equation (3.12)) where y is the mean ionic activity coefficient, itself a function of the ionic strength I. 

With the above considerations in mind, Debye and HUckel devised a means of calculating the ionic strength I and the mean ionic activity coefficient y . Before we can progress, however, we must define the extent to which a solute promotes association, and thus screening. The parameter of choice is the ionic strength /, which is defined formally as follows ... 

Knowing the ionic strength, we are now in a position to determine the mean ionic activity coefficient y by using the Debye-HCickel laws. There are two such laws, namely the limiting law and the extended law. 

Sulfuric acid is a 2 1 electrolyte, and so (by using the data in Table 3.1) the ionic strengthlis three times the concentration, i.e.l = 0.03 moldm f Next, from the Debye-Huckel extended law equation (3.15), we can obtain the mean ionic activity coefficient y as follows ... 

Worked Example 3.11. We know the concentration of copper sulfate to be 0.01 mol dm from other experiments, and so we also know (from suitable tables) that the mean ionic activity coefficient of the copper sulfate solution is 0.404. The measured electrode potential was Ec j+ — 0.269 V and = 0.340 V. We will calculate the... 

Therefore, by including the mean ionic activity coefficient into the calculation, the real concentration is seen to be the same as the apparent concentration without the mean ionic activity coefficient, the apparent (computed) concentration is about 2.4 times too small. 

From the above, it is clear that knowledge of y is vital for an accurate answer. Unfortunately, calculations of this second type are often quite impossible to carry out since the mean ionic activity coefficient y is normally unknown. 

The mean ionic activity coefficient, of copper sulfate decreases to 0.158 when the concentration was raised from 0.01 to 0.1 mol dm" . By using the Nemst equation, calculate the electrode potential, for the following situations ... 

When will the effect of the mean ionic activity coefficients be most likely to affect the accuracy of the concentrations calculated by the Nemst equation (Hint - when is the value of activity most like the concentration )... 

A third (and usually more profound) cause of error lies in the way that the Nemst equation is formulated in terms of activities rather than concentration. Even if the emf and E are correct, the proportionality constant between concentration and activity (the mean ionic activity coefficient y ) is usually wholly unknown. Errors borne of ignoring activity coefficients (i.e. caused by ionic interactions) are discussed in Sections 3.4 and 3.6.3. 

Determination of Concentration when the Mean Ionic Activity Coefficient is Unknown... 

While this relationship is simple, it introduces more errors because the activity coefficient (or more normally, the mean ionic activity coefficient y ) is wholly unknown. While y can sometimes be calculated (e.g. via the Debye-Huckel relationships described in Section 3.4), such calculated values often differ quite significantly from experimental values, particularly when working at higher ionic strengths. In addition, ionic strength adjusters and TISABs are recommended in conjunction with calibration curves. 

The activity a and concentration c are related by a = (c/c ) x y (equation (3.12)), where y is the mean ionic activity coefficient, itself a function of the ionic strength /. Approximate values of y can be calculated for solution-phase analytes by using the Debye-Huckel relationships (equations (3.14) and (3.15)). The change of y with ionic strength can be a major cause of error in electroanalytical measurements, so it is advisable to buffer the ionic strength (preferably at a high value), e.g. with a total ionic strength adjustment buffer (TISAB). 

Note that we can safely assume that all of the mean ionic activity coefficients are essentially the same because / is so high. In fact, we do not even need to know this common value of the y , since all of the coefficients cancel out in equation (4.3). For this reason, we can safely write K in terms of concentrations rather than activities (a). 

During a redox reaction, a potentiometric titration can be employed to determine a concentration of analyte rather than an activity, since we are only using the emf as a reaction variable in the accurate determination of an end point volume. For this reason, an absolute value of reference electrode need not be known, as we are only concerned with changes in emf. It is, however, advisable to titrate at high ionic strength levels in order to minimize fluctuations in the mean ionic activity coefficients. 

It is also desirable that the mean ionic activity coefficient y + approach unity in the limit of infinite dilution. We can achieve this result if, as in Equation (19.9), we define... 

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. 

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