The Ion Size Parameter

One can now proceed rapidly to compare this theoretical expression for log4 with experiment but what value of the ion size parameter should be used The time has come to worry about the precise physical meaning of the parameter a that was introduced to allow for the finite size of ions. 

One can at first try to speculate on what value of the ion size parameter is appropriate. A lower limit is the sum of the crystallographic radii of the positive and negative ions present in solution ions cannot come closer than this distance [Fig. 3.31 (a)]. But in a solution the ions are generally solvated (Chapter 2). So perhaps the sum of the solvated radii should be used [Fig. 3.31(b)]. However when two solvated ions collide, is it not likely [Fig. 3.31 (c)] that their hydration shells are crushed to some extent This means that the ion size parameter a should be greater than the sum of the crystallographic radii and perhaps less than the sum of the solvated radii. It should best be called the mean distance of closest approach, but beneath the apparent wisdom of this term there lies a measure of ignorance. For example, an attempted calculation of just how cmshed together two solvated ions are would involve many difficulties. 

To circumvent the uncertainty in the quantitative definition of a, it is best to regard it as a parameter in Eq. (3.120), i.e., a quantity the numerical value of which is left to be calibrated or adjusted on the basis of experiment. The procedure (Fig. 3.32) is to assume that the expression for log [Eq. (3.120)] is correct at one concentration, then 

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Variable di in Equation 8.2 is the ion size parameter. In practice, this value is determined by fitting the Debye-Huckel equation to experimental data. Variables A and B are functions of temperature, and I is the solution ionic strength. At 25 °C, given I in molal units and taking a, in A, the value of A is 0.5092, and B is 0.3283. 

Just as in aqueous solutions, the activity of solute i (acl) in non-aqueous solutions is related to its (molar) concentration (sj by aCii = yCiiCi, where g is the activity coefficient that is defined unity at infinite dilution. For non-ionic solutes, the activity coefficient remains near unity up to relatively high concentrations ( 1 M). However, for ionic species, it deviates from unity except in very dilute solutions. The deviation can be estimated from the Debye-Hiickel equation, -log yci = Az2 /1/2/ (1+aoBf1 2). Here, I is the ionic strength and / (moll-1), a0 is the ion size parameter... 

The standard emf E° of the cell was determined by means of an extrapolation technique involving a function of the measured emf E (which was measured experimentally), taken to the limit of zero ionic strength /. A linear function of I was observed when the Debye-Hiickel equation (in its extended form) (12) was introduced for the activity coefficient of hydrobromic acid over the experimental range of molalities m. With this type of mathematical treatment, the adjustable parameter became a0, the ion-size parameter, and a slope factor / . This procedure is essentially the same as that used in our earlier determinations (7,10) although no corrections of E° for ion association were taken into account (e = 49.5 at 298.15°K). 

Values of the right-hand side of Equation 6 (that is, those denoting the term E0/) are expected to be linear in terms of m, when a suitable value for the ion-size parameter is chosen. Several different values for a0 (e.g., 0.2, 0.4, 0.6, and 0.8 nm) were tested, and the computer calculations showed that the deviations of E0/ vs. m from linearity were at a minimum when a0 is equal to 0.6 nm. These results are shown in Table VIII. By means of Equation 6, the intercepts of the extrpolation lines (E°) can be found and the slopes (—2k(3) computed. The values of the standard emf are summarized in Tables IV, V, and VI for x = 10,30, and 50 mass percent monoglyme, together with the standard deviations of the intercepts (E°). For a0 = 0.6 nm, the values of are entered in Table IX. 

Some comments should be made in regard to the arbitrary choice of the ion-size parameter, a0, in the determination of E°. The suitability of the value of the ion-size parameter was based on the standard deviation for regression of the experimental E0/ in Equation 6 as a function of m using a° equal to 0.2,0.4, 0.6, and 0.8 nm, successively. Table VIII indicates that a0 = 0.6 nm gives the best linear fit, and consequently a0 = 0.6 nm was used at all temperatures. It should be mentioned, however, that the standard deviation of regression is not a reliable guide to the best choice of the ion-size parameter. 

Table VIII. Values of the Standard Potentials (on the Molal Scale) and the Corresponding Variations of the Ion-Size Parameter, a0, with the Standard Deviations of a(EIJl)/mV for x = 10, 30, and 50 Mass Percent Monoglyme—Water Mixtures at 298.15°K...

For a measurement of pH with cell (I) to be traceable to the SI, an uncertainty for the Bates-Guggenheim convention must be estimated. One possibility is to estimate a reasonable uncertainty contribution due to a variation of the ion size parameter. An uncertainty contribution of 0.01 in pH should cover the entire variation. When this contribution is included in the uncertainty budget, the uncertainty at the top of the traceability chain is too high to derive secondary standards as used to calibrate pH meter-electrode assemblies. 

It should be noted that, for solutions of ionic strength < 10 mol L , either Equation (3.4) or the Giintelberg approximation (3.5), which incorporates an average value of 3 for ai, can be used. The Guntelberg approximation is particularly useful in calculations where a number of ions are present in solution or when values of the ion size parameter are poorly defined. 

The term a0 is frequently called the ion-size parameter or apparent ionic diameter, B is an empirically fitted coefficient. In reality, however, B has important theoretical significance in interionic attraction theory, and it is a higher-order function of the ion-size parameter. 

Also a comparison of two alternative expressions for log y given by Equations 3 and 4 is of interest. A close examination of these equations will reveal certain inherent weaknesses of the linear extrapolation method for the evaluation of E°, even when using the extended terms. First, the linear extrapolation will require a precise value of D, the dielectric constant of the solvent, which in principle is experimentally measurable. Frequently, however, it is computed by using some empirical function or by graphical interpolation. Secondly, the uncertainty of the ion-size parameter is significant and no reliable method of computation exists, nor is it directly measurable experimentally. 

The coefficients of Equation 19 consist essentially of a number of constants among which are dielectric constant D of the medium and the ion-size parameter. It has been mentioned earlier that indeterminacy of the value of a makes prior computation of constants impossible. There are also doubts about the reliability of the values of D in many instances. 

A value of 5.2A for the ion-size parameter a yielded straight-line plots of E0 vs. m at each temperature and solvent composition. The intercepts were obtained by standard linear regression techniques. A graphical representation of the data for each of the H20/NMA solvent mixtures at 25°C is shown in Figure 1. The calculations were performed with the aid of a PDP-11 computer with a teletype output. The intercepts (E°) and the standard deviations of the intercepts are summarized in Table III. 

Equation 3 was obtained by combining the Nemst equation for the emf of Cell I with the equilibrium constant of the acidic dissociation of glycine. In Equations 2 and 3, E° is the standard emf of the cell in the respective solvent composition and these values were obtained from an earlier work (20). In Equation 4, /3 is the linear slope parameter for the plot of pK/ vs. I, a0 is the ion-size parameter, A and B are the Debye-Huckel constants on the molal scale (20) for the respective mixed solvent systems, and I is the ionic strength given by mi. 

In Equations 6-10 d0 is the density of the solvent which was measured at each solvent composition at each temperature, a is the ion-size parameter, z is an empirical constant, N is Avagadro s number, D is the dielectric constant of the solvent, k is the Boltzman constant, and e is the electronic charge. Other terms have been defined already. 

Table III contains the experimental quantities (except the potential, E) and the constants used to determine the standard potentials of the cell (Equation 3). The ion-size parameter a for water and terf-butanol-water solvents is 5.50 A, and for ethanol and ethanol-water it is 5.00 A.

As a first step, one can use for the lower limit of the integration a distance parameter that is greater than zero. Then one can go through the mathematics and later worry about the physical implications of the ion size parameter. Let this procedure be adopted and symbol a be used for the ion size parameter. 

Fig. 3.31. The ion size parameter cannot be (a) less than the sum of the crystallographic radii of the ions or (b) more than the sum of the radii of the solvated ions and is most probably (c) less than the sum of the radii of the solvated ions because the solvation shells may be crushed.

Fig- 3.32. Procedure for recovering the ion size parameter from experiment and then using it to produce a theoretical log 4 versus / curve that can be compared with an experimental curve. 

The values of the ion size parameter, or distance of closest approach, which are recovered from experiment are physically reasonable for many electrolytes. They lie around 0.3 to 0.5 nm, which is greater than the sum of the crystallographic radii of the positive and negative ions and pertains more to the solvated ion (Table 3.9). 

By choosing a reasonable value of the ion size parameter a, independent of concentration, it is found that in many cases Eq. (3.126) gives a very good fit with experiment, often for ionic strengths up to 0.1. For example, on the basis of a = 0.4 nm, Eq. (3.126) gives an almost exact agreement up to 0.02 Min the case of sodium chloride (Fig. 3.34 and Table 3.10). 

The ion size parameter a has done part of the job of extending the range of concentration in which the Debye-Htickel theory of ionic clouds agrees with experiment. Has it done the whole job One must start looking for discrepancies between theory and fact and for the less satisfactory features of the model. 

The best illustration of the fact that a has to be adjusted is its concentration dependence. As the concentration changes, the ion size parameter has to be modified (Fig. 3.35). Further, for some electrolytes at higher concentrations, a has to assume quite impossible (i.e., large negative, irregular) values to fit the theory to experiment (Table 3.11). 

Evidently there are factors at work in an electrolytic solution that have not yet been reckoned with, and the ion size parameter is being asked to include the effects of all these factors simultaneously, even though these other factors probably have little to do with the size of the ions and may vary with concentration. If this were so, the ion size parameter a, calculated back from experiment, would indeed have to vary with concentration. The problem therefore is What factors, forces, and interactions were neglected in the Debye-Hiickel theory of ionic clouds ... 

However, the value of the ion size parameter a could not be theoretically evaluated. Hence, an experimentally calibrated value was used. With this calibrated value fora, the values of. 4 at other concentrations [caicuiated fromEq. (3.119)] were compared with experiment. 

The finite-ion-size model yielded agreement with experiment at concentrations up to 0.1 N. It also introduced through the value of a, the ion size parameter, a specificity to the eiectrolyte (making NaCi different from KCl), whereas the point-charge model yielded activity coefficients that depended only upon the valence type of electrolyte. Thus, while the limiting law sees only the charges on the ions, it is biind... 

It is because it is implicitly attempting to represent all these various aspects of the real situation inside an ionic solution that the experimentally calibrated ion size parameter varies with concentration. Of course, a certain amount of concentration variation of the ion size parameter is understandable because the parameter depends upon the radius of soivated ions and this time-averaged radius might be expected to... 

In order to test Eq. (3.130), which is a quantitative statement of the influence of ionic hydration on activity coefficients, it is necessary to know the quantity and the activity of water, it being assumed that an experimentally calibrated value of the ion size parameter is available. The activity of water can be obtained from independent experiments (Table 3.12). The quantity can be used as a parameter. If Eq. (3.130) is tested as a two-parameter equation (n and a being the two parameters Table 3.13), it is found that theory is in excellent accord with experiment. For instance, in the case of NaCl, the calculated activity coefficient agrees with the experimental value for solutions as concentrated as 5 mol dm" (Fig. 3.39). 

Bjerrum concluded therefore that ion-pair formation occurs when an ion of one type of charge (e.g., a negative ion) enters a sphere of radius q drawn around a reference ion of the opposite charge (e.g., a positive ion). However, it is the ion size parameter that defines the distance of closest approach of a pair of ions. The Bjerrum hypothesis can therefore be stated as follows If a <, then ion-pair formation can occur if a> q, the ions remain free (Fig. 3.45). 

Fig. 3.47. Variation of 0, the fraction of associated ions, with a, the distance of closest approach (i.e., the ion size parameter).

What about the difficult problem of modeling electrolyte solutions at higher concentrations In the MSA approach, attempts were made to extend the treatment by allowing the ion size parameter to be a function of ionic strength. Unfortunately, such an approach became unrealistic because of the sharp reduction of effective cation size with increasing electrolyte concentration. 

Although a correction for ion size is necessary, the correction to be used in terms of (2-19) can be predicted only roughly. For many ions the ion-size parameter a is of the order of 3 x 10 cm, and hence... 

Unfortunately, the calculated values of yt cannot be confirmed by direct experiment, because in principle all experimental methods yield the mean activity coefficient y rather than the individual ionic values. By use of the definition given in (2-16), the experimentally determined value can be apportioned to give nd y. This procedure is theoretically justified only at high dilution, where the DHLL is valid because the limiting slope of log y plotted against /n is found experimentally to be O.SZ Zb, as required by (2-17). At higher values of n the ion-size parameter a must be introduced. 

With the present feasibility of evaluating experimental data with the aid of a computer, more refined or optimum values of the ion-size parameter can be obtained. For example, Paabo and Bates, in a study of the dissociation of deuteriophosphoric acid or deuteriocarbonate ion, selected the ion-size parameter as that which gave the smallest standard deviation for the least-squares intercept for the ionization constant. 

In (4-6), i is 8.3145 x 10 ergs/deg-mole. The radius r is usually estimated to be equal to the crystal radii and should not be confused with the ion-size parameter used in Debye-Hiickel calculations. 

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