Single ionic activities

The Inoperability of Galvanl potential differences between dissimilar phases is closely related to that of single ionic activities. In a homogeneous phase the electrochemical potential of an ionic species i, is split into a chemical potential and an electrical work term (see sec. I.5.1c) according to... 

We note that, because monolayer and solution are electronetutral, the surface composition is completely determined by adsorption and depletion of electroneutral entites. Because of this, only two rd/i terms suffice. Some authors prefer to write the r.h.s. in terms of ionic components (r .d/i,., dyU., +, etc.) which gives one term more but also an auxiliary condition, viz. that of electroneutrality. It is to a certain extent a matter of taste which choice is preferred. From an academic point of view it is not elegant at the veiy outset to make the concession of introducing single ionic activities, l.e. thermodynamically inoperable quantities. On the other hand, in the later elaborations, working with single ionic activities is often unavoidable, particularly when the system contains many components. We discussed this matter in some detail in sec. II.3.4. Anyhow, we shall start with [4.6.6] and see how far we get. 

Thermodynamically, the activity of a single ionic species is an inexact quantity, and a conventional pH scale has been adopted that is defined by reference to specific solutions with assigned pH(5) values. These reference solutions, in conjunction with equation 3, define the pH( of the sample solution. 

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. 

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. 

Electrostatic and statistical mechanics theories were used by Debye and Hiickel to deduce an expression for the mean ionic activity (and osmotic) coefficient of a dilute electrolyte solution. Empirical extensions have subsequently been applied to the Debye-Huckel approximation so that the expression remains approximately valid up to molal concentrations of 0.5 m (actually, to ionic strengths of about 0.5 mol L ). The expression that is often used for a solution of a single aqueous 1 1, 2 1, or 1 2 electrolyte is... 

The LCM has proven to be useful in predicting data of molal ionic activity coefficients and vapor pressure depression of various single electrolyte, single solvent systems. The standard deviation of the natural logarithm of the mean activity coefficient was 0.01 for uni-univalent aqueous single electrolytes (17). Similar results... 

Solutions with a single ion cannot be prepared, and because ions influence each other, only the mean ionic activity coefficient can be measured. For a binary... 

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... 

Figure 2.1. Relationship between solution ionic strength and single-ion activity coefficients of ions with different valencies. Calculated utilizing the extended Debye-Huckle equation) (from Skoog and West, 1976, with permission).

The importance of single-ion activity in predicting the chemical behavior of a particular ionic species is demonstrated in Figure 2.1. For example, the y value of any divalent or monovalent ions at the highest possible ionic strength (7) causes 60% suppression in the activity of the divalent ion and 25% reduction in the activity of the monovalent ion. This implies that as Yj decreases, the apparent solubility of any given mineral increases, as demonstrated later in this chapter. 

Dividing the numerator and the denominator of Equation I by (Mg2+)2 Ca2+ produces Equation F. Therefore, a plot of [ExCa/CEC] versus [Ca2+]/[Ca2+ + Mg2+] (see Eq. E) produces a straight line with slope 1 (nonpreference isotherm). Furthermore, upon considering Equation F and introducing cation-solution activities, it appears that since yCa approximately equals VMg (where y. equals the single-ion activity coefficient for species i in solution), the relationship of ExCa/CEC versus [Ca2+]/[Ca2+ + Mg2+] is shown to be independent of ionic strength. 

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... 

At 7 = 0.064 mol the single-ion activity coefficient calculated by the DH model is 8% lower than that calculated by the extended DH model. Now consider a solution which has an ionic strength of only 0.001 mol L . ... 

Clearly, at lower ionic strength, the single-ion activity coefficient is much closer to unity. Also, the DH and the extended DH models give almost exactly the same value. This is because the denominator (l+0.33ai 7) of the extended DH equation approaches a value of unity, i.e. 0.33oi / approaches zero, as the ionic strength decreases towards zero. In other words, the two equations become identical at very low ionic strength. 

The Activity Coefficient of a Single Ionic Species Cannot Be Measured... 

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