Crystallographic radii

Crystallographic radii, tabic, 266 Crystals, entropy of, 95, 180, 211 table, 267... 

R is the distance parameter, defining the upper limit of ion association. For spherical ions forming contact ion pairs it is simply the sum of the crystallographic radii of the ions a — a+ + a for solvent-shared and solvent-separated ion pairs it equals a + s or a + 2s respectively, where s is... 

The ionic radii are often difficult to ascertain. Mostly, crystallographic radii rc corrected by the additive term <5, with a constant common value for cations and a different constant value for all anions, are used ... 

Cations in aqueous solutions have an effective radius that is approximately 75 pm larger than the crystallographic radii. The value of 75 pm is approximately the radius of a water molecule. It can be shown that the heat of hydration of cations should be a linear function of Z /r where is the effective ionic radius and Z is the charge on the ion. Using the ionic radii shown in Table 7.4 and hydration enthalpies shown in Table 7.7, test the validity of this relationship. 

Fig. 3. Single ion conductivities in acetonitrile vs. reciprocal of estimated crystallographic radii...

Plots of limiting ion conductivities versus the reciprocal of the Pauling crystallographic radii for the alkali metal ions in various solvents are shown in Fig. 

Fig. 4. Limiting single ion mobilities vs. reciprocal of estimated crystallographic radii of alkali metal ions AN, acetonitrile NM, nitromethane DMF,dimethylformamide PY, pyridine NB, nitrobenzene DMSO, dimethylsulfoxide EC, ethylene carbonate...

To minimize the effects of viscosity for purposes of comparing data between solvents, plots areoften made using the product of the ion mobility and the viscosity (Walden product) in place of mobility alone. A plot of the Walden product against the reciprocal of the crystallographic radii for several solvents is shown in Fig. 6. Arbitrary curves have been drawn to indicate general trends. Values in solvents for which precise transference numbers and conductance data are available, such as acetonitrile and nitromethane, give smooth curves. 

Fig. 8. Limiting single ion mobility-viscosity product vs. reciprocal of estimated crystallographic radii for cations in methanol, ethanol, and acetonitrile (from Ref. 16 )...

Encounter radii, R, for reaction between hydrated electrons and various ions [17], the corrected encounter radius, i H, incorporating the hydrodynamic effect, and the sum of the hydrated electron and ion crystallographic radii, Rc... 

Figure 5. Influence of ion valence upon the sorption of Ar (a) Ar in K-L (6) Ar in BarL. The two ions have approximately the same crystallographic radii, o = adsorption points A = desorption points.

TABLE 4.1. Crystallographic Radii and Heats and Entropies of Ion Hydration at 25°C... 

For most of the salts the slope of the phoreogram is steeper (i.e., more negative or catabatic) than would be calculated by the Onsager limiting law. Most researchers have rejected the possibility of ionic association as a cause of this deviation - principally because the Bjerrum contact distance in NMA, for the temperature range of 30 to 60 °C, is less than 0.2 nm and thus less than the sum of the crystallographic radii of most possible cation-anion contact pairs. 

In the case of LiCl, NaCl, and KC1, Figure 6 enables experimental values to be compared with theoretical ones calculated by the Fuoss equation, substituting for the a parameter the lowest value with a physical meaning (a = 2.413, the sum of Li+ and Cl" crystallographic radii), in order to get larger theoretical values for KA. As seen from Figures 5 and 6, the observed trends in almost the whole range of solvent composition are linear but with different slopes from the theoretical one in the... 

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.

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). 

The larger monovalent alkali metals Rb and Cs (6-fold crystallographic radii of 1.47 and 1.67 A, respectively) are thought to be essentially unassociated with water molecules (1). However, the smaller monovalent and most divalent cations form strong aquo-complexes. Thus, the reaction... 

The contrast parameters entering into this comparison have been determined recently for the system under consideration here (see Fig. 1). The contrast A /jo, of the Br counterion was determined from their respective crystallographic radii. Hence, as already discussed in previous papers [14, 17, 18] the hydration shell of the ion is treated as bulk water. This may induce a small error if A /ion is small. The value calculated in this way for Br is A f ion =26 e/ion. For the contrast of the macroion we used the value Aprod = 25 e/nm3 [14]. 

An exactly analogous table could be constructed to assign the ry values that should be used in the above Wy terms. For the Keggin anions, this is simplified by the fact that all the species in Equation 1.69 possess the same crystallographic radii of 5.6 A. Therefore, all the ry values, approximated as the sum of radii, are 11.2 A or 1.12 x 10 7 cm. The individual wy terms are then evaluated as shown in Equation 1.43. 

The calculated crystallographic radii (95) for the LaF3-type actinide trifluorides (Ac — Bk) with coordination number six, and the YF3-type trifluorides (Bk, Cf) reproduced the two-third of the inclined W (Fig. 60(a)). From the crystallographic data for the isostructural tetravalent actinide dioxides (M = Th — Bk) Peterson and Cunningham (96) calculated the sixcoordinated radii for Th to Bk. These values are plotted (Fig. 60(b)) against the L-values of the tetravalent actinides. Like the case of the trivalent actinides here also a linearity within the two tetrads was observed. 

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