Ionic radii radius ratio rule

The dominant features which control the stoichiometry of transition-metal complexes relate to the relative sizes of the metal ions and the ligands, rather than the niceties of electronic configuration. You will recall that the structures of simple ionic solids may be predicted with reasonable accuracy on the basis of radius-ratio rules in which the relative ionic sizes of the cations and anions in the lattice determine the structure adopted. Similar effects are important in determining coordination numbers in transition-metal compounds. In short, it is possible to pack more small ligands than large ligands about a metal ion of a given size. 

Even if there are exceptions to the radius ratio rule, or if exact data are hard to come by, it is still a valid guiding principle. Cite three independent examples of pairs of compounds illustrating structural differences resulting from differences in ionic radii. 

The alkali metals react with many other elements directly to make binary solids. The alkali halides are often regarded as the most typical ionic solids. Their lattice energies agree closely with calculations although their structures do not all conform to the simple radius ratio rules, as all have the rock salt (NaCl) structure at normal temperature and pressure, except CsCl, CsBr and Csl, which have the eight-coordinate CsCl structure. The alkali halides are all moderately soluble in water, LiF being the least so. 

Pauling subsequently introduced three mles governing ionic sfructures (Pauling, 1928, 1929). The first is known as the radius ratio rule. The idea is that the relative sizes of the ions determine the sfructure adopted by an ionic compound. Pauling proposed specific values for the ratios of the cation radius to the anion radius as lower limits for different coordination types. These values are given in Table 3.5. 

Unlike ionic structures where co-ordination numbers are determined by the radius ratio rule and the bonds are not directed, homopolar structures have directed bonds. An element in the group of the Periodic Table can form 8—N bonds per atom and co-ordination numbers are small, usually four or less. 

Determining the ionic structure in the cubic system from ionic radii using the radius-ratio rule Determining the density of the ionic solid from the density of the unit cell Note Usually within 10% of the experimental value... 

The radius ratio rule is only applicable to ionic compounds. In silicate minerals, however, it is the bonds between oxygen and silicon and between oxygen and aluminium (Al) which are structurally important. These bonds are almost equally ionic and covalent in character and the radius ratio rule predicts the coordination of these ions adequately. 

The radius ratio rule (Section 4.2.1) predicts that divalent Ca2+, Mg2+ and Fe2+ will have six-fold coordination because of their similar ionic radii (0.106nm Ca2+, 0.078nm Mg2+, 0.082 nm Fe2+). They are therefore interchangeable without upsetting either the physical packing or the electrical stability of an ionic compound. 

An ionic solid should achieve maximum electrostatic stability when (i) each ion is surrounded by as many as possible ions of opposite charge, and (ii) the anioncation distance is as short as possible. There is, however, a play-off between these two factors. Consider an octahedral hole in a close-packed array of anions (see Topic D3) The minimum radius of the hole, obtained when the anions are in contact, is 0.414 times the anion radius. A cation smaller than this will not be able achieve the minimum possible anion-cation distance in octahedral coordination, and a structure with lower coordination (e.g. tetrahedral) may be preferred. These considerations lead to the radius ratio rules, which predict the likely CN for the smaller ion (usually the cation) in terms of the ratio r where... 

We should mention that in the few cases in which the variation in electron density in a crystal has been accurately determined (e.g. NaCl), the minimum electron density does not in fact occur at distances from the nuclei indicated by the ionic radii in general use e.g. in LiF and NaCl, the minima are found at 92 and 118 pm from the nucleus of the cation, whereas tabulated values of / l + and rj4a+ are 76 and 102 pm, respectively. Such data make it clear that discussing lattice structures in terms of the ratio of the ionic radii is, at best, only a rough guide. For this reason, we restrict our discussion of radius ratio rules to that in Box 5.4. 

Before we can estimate the lattice enthalpy of CaCI, we select a lattice with the aid of the radius-ratio rule. The ionic radius for Cl- is 181 pm use the ionic radius of K+ (1381) for Ca+... 

Our review of simple ionic compounds shows that the radius-ratio rules are a useful guide to the stereochemistry of halides and oxides, but that the quantitative predictions are not to be taken too seriously. This is no doubt in part attributable to the crudeness of the model from which the rules are derived, and in part to uncertainties in the values of the ionic radii. There are many examples in which the coordination number increases with increasing radius of the metal ion in closely related compounds for example, Mg++ in MgF2 is 6-coordinated but CaF2 has an 8-coordinated structure. It is in accounting, at least semiquantitatively, for such relations that the radius-ratio rules are most important. 

For simple ionic compounds of symmetrical valence type, the radius ratio rules are ... 

These rules enable us to predict the structure of the compound from the relative sizes of the two ions. Applied to many different ionic crystals of different valence types, the rules are quite good. There are at least two reasons for the exceptions to the radius ratio rules (1) the ions are not rigid spheres (2) the ions of opposite charge are not in contact. 

Rule I. You can draw a polyhedron of anions around every cation in a crystalline lattice, such that (i) the interionic separation can be determined as the sum of the ionic radii, according to Equation (12.10), and (ii) the coordination number of the cation can be determined using the radius ratio rule (vide infra). The radius ratio rule sets the minimum r+/r ratio that can exist for a cation with a given ccxjrdina-tion number. This ratio can be determined from a geometrical consideration of the minimum cationic radius necessary to keep the anions in a particular coordination geometry from just touching each other. 

Example 12-7. Employ the radius ratio rule using the Pauling ionic radii listed in Table 12.5 to predict the lattice types of (a) CsCI, (b) SrF2, and (c) KBr. 

In the crystalline lattice of BaTiOj, the ionic radii of the Ba " ", TP, and ions are 149, 75, and 121 pm. Use the radius ratio rule to predict the coordination number of Ba " and Ti in BaTiOj. Then calculate the electrostatic bond strengths of each cation and use this information to determine how many Ba ions and how many Ti " " ions each 0 will be coordinated to in the crystalline lattice. 

In conclusion, many of the bonding models that we examined in previous chapters rely on circular arguments, to a given extent, because we define the radius of an atom depending on its molecular context. Thus, the radius ratio rule for ionic solids was self-consistent only in the sense that it relied on the ionic radii... 

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