Hard sphere cations

If, on the other hand, the radius ratio rules are violated on the downside, that is to say the hard sphere cation and anion no longer touch, the structure is longer stable and another structure should be adopted. For instance, if r+/r < 0.224, then tetrahedral coordination is no longer possible but a threefold planar triangular coordination can be found by locating the cation in the trigonal hole centered in the plane of the close-packed layer. Such coordination should be stable for 0.15 < r+/r < 0.224. In fact, however, trigonal coordination is rare in chemical systems other than those of boron. 

As Table 6.3 shows, this classification into A- and B-type metal cations is governed by the number of electrons in the outer shell. A-type metal cations having the inert gas type d electron configuration correspond to those that were classified above as hard sphere cations. These ions may be visualized... 

When the hard-sphere cation-anion radius ratio exceeds 0.732, as it does for the cesium halides, a different crystal structure called the cesium chloride structure, is more stable. It may be viewed as two interpenetrating simple cubic lattices, one of anions and the other of cations, as shown in Figure 21.17. When the cation-anion radius ratio is less than 0.414, the zinc blende, or sphalerite, structure (named after the structure of ZnS) results. This crystal consists of an fee lattice of... 

Debye and Hiickel were concerned with treating a system of hard-sphere cations and anions of identical size and opposite charge in an isotropic environment without explicit boundaries. By linearizing the charge density with... 

Kapustinskii equation For an ionic crystal composed of cations and anions, of respective charge and z, which behave as hard spheres, the lattice energy (U) may be obtained from the expression... 

The simplest way to treat the solvent molecules of an electrolyte explicitly is to represent them as hard spheres, whereas the electrostatic contribution of the solvent is expressed implicitly by a uniform dielectric medium in which charged hard-sphere ions interact. A schematic representation is shown in Figure 2(a) for the case of an idealized situation in which the cations, anions, and solvent have the same diameters. This is the solvent primitive model (SPM), first named by Davis and coworkers [15,16] but appearing earlier in other studies [17]. As shown in Figure 2(b), the interaction potential of a pair of particles (ions or solvent molecule), i and j, in the SPM are ... 

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. 

An important reason for the exceptions to the radius ratio predictions is that ions are not hard spheres but somewhat compressible, hence do not have a truly constant radius. Another reason for the inadequacy of the radius ratio rules, particularly when the anions are much larger than the cations, is that some structures are determined by the close packing of the anions, leaving the cations in holes between the anions. In such a case more anions may be packed around a cation of a given fixed radius than are predicted by the radius ratio, so that although the anions are touching each other, they are not touching the cation. However, if... 

Molecular Probe Analysis. In an effort to understand how a molecule is seen by either another molecule or by a surface, molecular probes can be moved around a chemical to map out its surface. These probes include anions and cations (point charges) and hard spheres or can be constructed as a combination of these. The empirical potential energy is computed at a variety of points around the test molecule and an energy surface is thus generated. This can be examined graphically and compared as changes are made to the molecule. 

Based on the ionic radii, nine of the alkali halides should not have the sodium chloride structure. However, only three, CsCl, CsBr, and Csl, do not have the sodium chloride structure. This means that the hard sphere approach to ionic arrangement is inadequate. It should be mentioned that it does predict the correct arrangement of ions in the majority of cases. It is a guide, not an infallible rule. One of the factors that is not included is related to the fact that the electron clouds of ions have some ability to be deformed. This electronic polarizability leads to additional forces of the types that were discussed in the previous chapter. Distorting the electron cloud of an anion leads to part of its electron density being drawn toward the cations surrounding it. In essence, there is some sharing of electron density as a result. Thus the bond has become partially covalent. 

Equations (87)-(89) apply in aqueous solutions of two electrolytes in which the interaction potentials are conformal. For example, the assumptions utilized in the extensions of the Debye-Hiickel theory (e.g. water is considered as a continuous dielectric medium of dielectric constant D, that the cation-anion repulsive potential is that of hard spheres, and that all the... 

FIGURE 1.22. Solvent reorganization energies derived from the standard rate constants of the electrochemical reduction of aromatic hydrocarbons in DMF (with n-Bu4N+ as the cation of the supporting electrolyte) uncorrected from double-layer effects. Variation with the equivalent hard-sphere radii. Dotted line, Hush s prediction. Adapted from Figure 4 in reference 13, with permission from the American Chemical Society. 

The coordination number of a cation depends on the number of anions or ligand atoms that can be fit around it in three dimensions (Fig. 2). In the hard-sphere model the coordination number is determined by the ratio of the radius of the cation to that of the anion... 

The occurrence of different coordination numbers is traditionally explained by a model in which atoms are treated as hard spheres. Each cation is then assumed... 

One reason for the failure of the radius ratio rules is that ions do not behave like hard spheres. Even those that are hard in the Pearson (1973) sense can still be compressed. This is clearly seen in the way the bond length varies with the bond valence. If cation anion bonds can be compressed, so can the distance between the 0 ions in the first coordination sphere. The stronger the cation anion bonds, therefore, the closer the anions in the first coordination sphere can be pulled together (Shannon el al. 1975). 

Anion-anion repulsion places an upper limit on the coordination number that a cation can adopt but, since the 0 ions do not behave like hard spheres, the size of the limiting 0-0 distance depends on the effective valence of the bonds. Either Fig. 6.4 or eqn (6.3) can be used to decide whether or not a particular coordination number is physically possible. 

Lorentz1 advanced a theory of metals that accounts in a qualitative way for some of their characteristic properties and that has been extensively developed in recent years by the application of quantum mechanics. He thought of a metal as a crystalline arrangement of hard spheres (the metal cations), with free electrons moving in the interstices.. This free-electron theory provides a simple explanation of metallic luster and other optical properties, of high thermal and electric conductivity, of high values of heat capacity and entropy, and of certain other properties. 

It should be kept dearly in mind that the radius ratio rules apply strictly only to the packing of hard spheres of known size. As this is seldom the case, it is surprising that the rules work as well as they do. Anions are not hard like billiard balls, but polarizable under the influence of cations. To whatever extent such polarization or covalency occurs, errors are apt to result from application or the radius ratio rules. Covalent honds are directed in space unlike electrostatic attractions, and so certain orientations are preferred. 

The structure of ionic crystals usually corresponds to the maximum possible coordination number. If the ions are assumed to be hard spheres, there must be contact between ions of opposite signs and no contact between ions of like sign. The coordination depends on the ratio of anion-to-cation diameters. Figure 13.5 shows that the critical condition for threefold coordination is R/(R + r) = cos 30° = /3/2. Therefore, to prevent contact between like ions,... 

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