Inter-ionic distance

In deriving theoretical values for inter-ionic distances in ionic crystals the sum of the univalent crystal radii for the two ions should be taken, and corrected by means of Equation 13, with z given a value dependent on the ratio of the Coulomb energy of the crystal to that of a univalent sodium chloride type crystal. Thus, for fluorite the sum of the univalent crystal radii of calcium ion and fluoride ion would be used, corrected by Equation 13 with z placed equal to y/2, for the Coulomb energy of the fluorite crystal (per ion) is just twice that of the univalent sodium chloride structure. This procedure leads to the result 1.34 A. (the experimental distance is 1.36 A.). However, usually it is permissible to use the sodium chloride crystal radius for each ion, that is, to put z = 2 for the calcium... 

The "new ionic radii (4, 5) split the interionic distance D into two ionic Structural Radii r at the point where the electron-density has its minimum value, hence by means of X-ray observations on the Electron-cloud Radius y. But electron clouds, as wave-mechanics would predict, have no sharp radius. One might therefore measure electron-cloud radii in some other way at a different electron-density, and split the inter-ionic distance in a different ratio this is, ls we shall see later, one of the two foundations of the Goldschmidt-Pauling system. 

When comparing ionic porosity of different minerals, for self-consistency, the same set of ionic radii should be used, and the same temperature and pressure should be adopted to calculate the molar volume of the mineral. Table 3-3 lists the ionic porosity of some minerals. It can be seen that among the commonly encountered minerals, garnet and zircon have the lowest ionic porosity, and feldspars and quartz have the highest ionic porosity. More accurate calculation of IP may use actual X-ray data of average inter-ionic distance and determine the ionic radius in each structure. 

Since ionization potentials of anionic donors and electron affinities of cationic acceptors are not readily available, Mulliken correlations for charge-transfer ion pairs are generally presented in a modified form using electrochemical oxidation or reduction potentials, respectively. A typical example of such a modified Mulliken plot with unit slope is shown in Figure 2 for the CT ion pairs of TpMo(CO)3 [Tp = hydrido-trM-(3,5-dimethylpyrazolyl)borate] as the donor and various pyr-idinium acceptors [127]. Similar (modified) Mulliken correlations with unit slopes have been found for numerous other ion pairs with pyridinium acceptors and Mn(CO)s [126], Co(CO)4 [118], or V(CO)e [118] as donors. It is important that the Coulombic work term (co) in Eq. 8 is explicitly included in all Mulliken evaluations of ion pairs with different structures since co reflects the electrostatic energy of the (ground-state) ion pair which strongly depends on the inter-ionic distance [125]. 

Bragg (1912) showed a sodium chloride crystal to consist, not of discrete molecules of NaCl, but of Na+ ions and Cl ions arranged in an indefinitely extended cubic lattice (Fig. 28, exterior view Fig. 84 on p. 138 shows co-ordination) X-ray analysis (p. 141) gave the internuclear distance as 2.81 A. Most alkali metal halides have the same type of crystal lattice but the inter-ionic distances differ. 

The value q == 0.345 A gives satisfactory results for all alkali halides. The inter-ionic distance r and the type of lattice are found from determinations of crystal structure (p. 118). 

As mentioned above, X-ray structures only give a precise knowledge of the distances between the atoms. While this is not so important for pure metals, as the interatomic distance is simply divided by two to obtain the metallic radius, this simple method will not work for ionic compounds. To begin, it is assumed that the individual ions are spherical and in contact. The strategy then used to derive ionic radii is to take the radius of one commonly occurring ion, such as the oxide ion, O2-, as a standard. Other consistent radii can then be derived by subtracting the standard radius from measured inter-ionic distances. 

Comment on the trends in the following values for inter-ionic distances (pm) ... 

The ionic bond results from the Coulomb attraction of oppositely charged ions. Its strength is characterised by the electrostatic energy in MX ionic crystals it is the crystal lattice energy C/(MX), which can be determined experimentally from the Born-Haber cycle or calculated theoretically from the known net charges of ions (Z, not to be confused with nuclear charges ) and inter-ionic distances d), as... 

A discussion of the statistical distribution of the population of the ion-pairs (9a) with respect to the inter ionic distance, based on the Bjerrum probability ratio and a continuum for the solvents reads ... 

Bjerrum probability ratios as function of the inter-ionic distance r for pairwise interaction between ions... 

In contrast to the relatively simple tendency of the local structure around Th" in monovalent cationic fluoride mixtures, the additional effect of a divalent cation is slightly complicated. The CaF2 concentration dependence of local structural parameters derived from EXAFS of the constant concentration of %i,p4 = 0.25 are shown in Figure 6.7.4. Although the inter-ionic distance is independent from the concentration of CaF2, the coordination number, Debye-Waller factor and C3 cumulant parameter increase until the concentration of XcaF2 = 0.17, but these values decrease with increases at XcaF2 > 0.17. The local structure of thorium cation is unstabilised up to a certain concentration of calcium fluoride and is then stabilised by further addition of calcium fluoride. 

Figure 6.7.4 (a-d) Structural parameters obtained by EXAFS of molten 0.25ThF -(0.75 - x)LiF-xCaF2 mixtures (0coordination number, (b) inter-ionic distance, (c) Debye-Waller factor, and (d) cumulant... 

Short-range repulsion A first improvement is related to the repulsive forces which become effective at short inter-ionic distances. In the original model, steric - or hard-core - repulsive forces prevent two ions i and J from coming closer than the sum of their ionic radii r,- and rj. The short-range repulsion energy is infinite if Rij < r, + ry, and zero otherwise, which may be written in the form ... 

The inter-ionic distances of the nearest neighbors in the molten salts (cation-anion distances) are obtained from the EXAFS method with the same accuracy as... 

Many authors have since then applied Monte Carlo (MC) and molecular dynamics (MD) simulations to molten salts. The simulations yielded the partial pair correlation functions, from which the inter-ionic distances and coordination numbers were deduced, as shown in Table 3.7. Generally the interionic distances were... 

The corresponding states theory was developed by Reiss, Mayer and Katz [138] to deal with molten salts. It employs the primitive model with a single distance parameter d (=2ri in Eq. 3.8) to which all the inter-ionic distances in the entire volume of the molten salt V are proportional = 4ijd for all ions I and J, whether 1 = J or 1 / J. Reduced thermodynamic quantities are then defined as ... 

The importance of ion association in a salt solution is largely dependent on the dielectric constant, e, of the solvent. As the salt concentration is increased, the inter-ionic distance decreases and ion-ion interactions become progressively more signiheant. The onset of ion-pair formation occurs at lower ion concentrations in solvents of low e. As e 5-10 for polyethers (cf. 78.5 for water), extensive ion-ion interactions are expected to be favourable. 

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