Interionic distances

Silver bromide crystals, formed from stoichiometric amounts of silver nitrate and potassium bromide, are characterized by a cubic stmcture having interionic distances of 0.29 nm. If, however, an excess of either ion is present, octahedral crystals tend to form. The yellow color of silver bromide has been attributed to ionic deformation, an indication of its partially covalent character. Silver bromide melts at 434°C and dissociates when heated above 500°C. 

Of the three principal classes of crystals, ionic crystals, crystals containing electron-pair bonds (covalent crystals), and metallic crystals, we feel that a good understanding of the first class has resulted from the work done in the last few years. Interionic distances can be reliably predicted with the aid of the tables of ionic radii obtained by Goldschmidt1) by the analysis of the empirical data and by Pauling2) by a treatment based on modem theories of atomic structure. The stability,... 

The comparison of observed intensities of reflection in eight orders from (110) and intensities calculated for a range of parameter values suggested by assumed minimum interionic distances led to the values w=0.175, x = 0.135, y = 0.440, = 0.150. These values were verified by a large number of other reflections. The resultant structure is shown in Fig. 1 and 3. 

Although the rule the reactivity of tight ion-pairs increases with increasing interionic distance still holds, the above results show that the ionic radius is not the only factor that determines the reactivity of tight ion-pairs. 

A typical special feature of the melts of ionic crystals (ionic liquids) are their high concentrations of free ions, of about 25 M. Because of the short interionic distances, considerable electrostatic forces act between the ions, so that melts exhibit pronounced tendencies for the formation of different ionic aggregates ion pairs, triplets, complex ions, and so on. 

Experience shows that the potentials of metal electrodes in melts of their own salts (i.e., the activities of the cations) depend on the natnre of the anions. However, the variation in the valnes of activity in melts is not very pronounced. This is dne to the relatively small spread of interionic distances fonnd in different melts (their entire volume is filled up with ions of similar size) compared to the spread found in aqueous solutions. For this reason the electrostatic forces between the ions (which are very significant) do not differ greatly between different melts. 

The stability of a certain structure type depends essentially on the relative sizes of cations and anions. Even with a larger Madelung constant a structure type can be less stable than another structure type in which cations and anions can approach each other more closely this is so because the lattice energy also depends on the interionic distances [cf. equation (5.4), p. 44], The relative size of the ions is quantified by the radius ratio rm/rx rM being the cation radius and rx the anion radius. In the following the ions are taken to be hard spheres having specific radii. 

A straightforward estimate of the maximum hardness increment can be made in terms of the strain associated with mixing Br and Cl ions. The fractional difference in the interionic distances in KC1 vs. KBr is about five percent (Pauling, 1960). The elastic constants of the pure crystals are similar, and average values are Cu = 37.5 GPa, C12 = 6 GPa, and C44 = 5.6 GPa. On the glide plane (110) the appropriate shear constant is C = (Cu - C12)/2 = 15.8 GPa. The increment in hardness shown in Figure 9.5 is 14 GPa. This corresponds to a shear flow stress of about 2.3 GPa. which is about 17 percent of the shear modulus, or about C l2n. 

In liquefied rare gases (LRG) the ejected electron has a long thermalization distance, because the subexcitation electrons can only be thermalized by elastic collisions, a very inefficient process predicated by the small mass ratio of the electron to that of the rare gas atom. Thus, even at a minimum of LET (for a -1-MeV electron), the thermalization distance exceeds the interionization distance on the track, determined by the LET and the W value, by an order of magnitude or more (Mozumder, 1995). Therefore, isolated spurs are never seen in LRG, and even at the minimum LET the track model is better described with a cylindrical symmetry. This matter is of great consequence to the theoretical understanding of free-ion yields in LRG (see Sect. 9.6). 

The compounds [Fe(C5Me4SCMe3)2][M(mnt)2] (M = Ni, Pt) are the only cases of salts based on metallocenium derivatives and on [M(mnt)2 complexes where the crystal structure is based in parallel arrangements of the type I chain motif [49]. In the chains the [Pt(mnt)2]- units are considerably tilted in relation to the chain direction, and short interatomic D+A- intrachain contacts were observed, involving one C from the Cp and a S atom from the anion, with a C-S distance of the order of dtf/. Relatively short interchain interionic distances were observed, where the closest corresponds to a S... C, involving one S atom from the anion and a C atom from a Me group of the cation, exceeding dw by 10%. 

If the ions are large, it is to be expected that the ratio of free ions to ion-pairs will be relatively great. For instance, it follows from the Fuoss equation [72] that if the interionic distance is 10 A, then in ethyl chloride at -78° (eT = 3.29 x 103) [73], the dissociation constant of ion-pairs is 2.5 x 10"3 mole/1. At a total concentration of electrolyte of 5 x 10 2 mole/1, the degree of dissociation is 0.2, and the ratio [cations]/... 

The terms j32 and I3 have the values given above. Approximately, D3 = D5 = 80 kcal/ mole, and A2 = 88 kcal/mole. The value of D4 is unknown, but 70 kcal/mole is probably a realistic estimate. The Coulombic energy of the ion-pair is q, and assuming an interionic distance of 5 AU, this amounts to 70 kcal/mole. A value of AHSS can be estimated as approximately three times the latent heat of vaporization of methylene dichloride (6.7 kcal/ mole), giving AiTSJ = 20 kcal/mole. Thus... 

In order for a solvated ion to migrate under an electric field, it must be prevented from forming close ion pairs with its counterions by the solvating solvent. The effectiveness of the solvent molecule in shielding the interionic Coulombic attraction is closely related with its dielectric constant. The critical distance for the ion pair formation q is given by eq 4 according to Bjerrum s treatment, with the hypothesis that ion-pair formation occurs if the interionic distance is smaller than... 

The additivity of cationic and anionic radii in reproducing the interionic distance (eq. 1.24) is valid, provided that coordination number, electron spin, degree of covalence, repulsive forces, and polyhedral distortion (eq. 1.49) are all taken into account. 

We now introduce two new parameters that describe the changes in interionic distances and volume with pressure isothermal linear compression coefficient ft, and isothermal volumetric compression coefficient jiy ... 

For simple compounds such as NaCl, where a single interionic distance (r) is sufficient to describe the structure, if we adopt the simplified form of equation 1.97 and combine the partial derivatives of U and V on r, we obtain... 

For cubic structures with more than one interionic distance—as, for instance, spinels (multiple oxides of type AB2O4 with A and B cations in tetrahedral and octahedral coordination with oxygen, respectively)—it is stiU possible to use equation 1.109, but the partial derivatives must be operated on the cell edge, which is, in turn, a function of the various interionic distances (OttoneUo, 1986). 

The vibrational motion of atoms in diatomic molecules and, by extension, in crystals cannot be fully assimilated to harmonic oscillators, because the potential well is asymmetric with respect to Xq. This asymmetry is due to the fact that the short-range repulsive potential increases exponentially with the decrease of interionic distances, while coulombic terms vary with 1/Z (see, for instance, figures 1.13 and 3.2). To simulate adequately the asymmetry of the potential well, empirical asymmetry terms such as the Morse potential are introduced ... 

---