Ion-size parameter

Figure A2.3.16. Theoretical HNC osmotic coefTicients for a range of ion size parameters in the primitive model compared with experimental data for the osmotic coefficients of several 1-1 electrolytes at 25°C. The curves are labelled according to the assumed value of a+- = r+ + r-...

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. 

A considerable volume of literature has accumulated on conductance measurements in mixtures of solvents. Ion mobilities and association constants have been measured over a range of bulk dielectric constants with the aim of correlating bulk solvent properties with mobilities, ion association, and ion size parameters. An example of a widely used solvent mixture is water and 1,4-dioxane, which are miscible over all concentrations, providing a dielectric constant range of 2 to 78. The data obtained in systems containing two or more solvents must be treated with circumspection, as one solvent may interact more strongly with a given species present in solution than the other, and the re-... 

Note A is the activity correction term, z is the charge of the ion, I is the ionic strength of the background electrolyte I = 0.5 xY Ci x zj = 0.05), a is the hydrated analyte ion size parameter, set to 5A, and and OTb are the maximum mobility (constants for a given ion). For simplicity, the activity term A was not taken into account in some model equations. 

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...

To obtain the pH, it is necessary to evaluate the activity coefficient of the chloride ion. So the acidity function is determined for at least three different molalities mci of added alkali chloride. In a subsequent step, the value of the acidity function at zero chloride molality, lg(flHyci)°, is determined by linear extrapolation. The activity of chloride is immeasurable. The activity coefficient of the chloride ion at zero chloride molality, yci, is calculated using the Bates-Guggenheim convention (Eq. 5) which is based on the Debye-Hiickel theory. The convention assumes that the product of constant B and ion size parameter a are equal to 1.5 (kg mol1)1/2 in a temperature range 5 to 50 °C and in all selected buffers at low ionic strength (I < 0.1 mol kg-1). 

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. 

Table 1.2. Ion size parameters for the Debye-Huckel equation.

Dielectric constant Viscosity (millipoise) Ao (Q 1-cm2-equiv 1) Dissociation constant, Kd Ion size parameter Bjerrum (A)... 

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. 

Originally the extended DH term was similar to that in Equation (3.5), i.e. with an average ion size parameter, a = 3. Scatchard showed that a better fit with experimental data was obtained using an average value of fli = 4.6, i.e. 0.33ai = 1.5, and so the extended DH expression used in the SIT model is often... 

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. 

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. 

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. 

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