Repulsion, interionic

Takes attractive and repulsive interionic interactions into account Is given by the Born-Meyer equation below... 

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. 

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 ... 

The above calculation applies to independent sodium and fluoride ions, and does not take into account the electrostatic attraction between the oppositely charged ions, nor the repulsive force which operates at small interionic distances. In the crystal of NaF the distance of nearest approach of the sodium and fluoride ions is 231 pm, and Coulomb s law may be used to calculate the energy of stabilization due to electrostatic attraction between individual ion pairs ... 

The allowance for polarization in the DH model obviates the need for separation of long-range and short-range attractive forces and for inclusion of additional repulsive interactions. Belief in the necessity to include some kind of covolume term stems from the confused analysis of Onsager (13), and is compounded by a misunderstanding of the standard state concept. Reference to a solvated standard state in which there are no interionic effects can in principle be made at any arbitrary concentration, and the only repulsive or exclusion term required is that described by the DH theory which puts limits on the ionic atmosphere size and hence on the lowering of electrical free energy. The present work therefore supports the view of Stokes (34) that the covolume term should not be included in the comparison of statistical-mechanical results with experimental ones. 

If the Coulomb forces were to correspond to 2 = 1 (univalent ions), with the characteristic repulsive coefficient B unchanged, the equilibrium interionic distance would be... 

Anion Contact and Double Repulsion.2 —The explanations of the deviations from additivity are indicated by Figure 13-6, in which the circles have radii corresponding to the crystal radii of the ions and are drawn with the observed interionic distances. It is seen that for LiCl, LiBr, and Lil the anions are in mutual contact, as suggested in 1920 by Land6.14 A simple calculation shows that if the ratio p = r+/r of the radii of cation and anion falls below /2 — 1 = 0.414 anion-anion contact will occur rather than cation-anion contact (the ions being considered as rigid spheres). A comparison of apparent anion radii in these crystals and crystal radii from Table 13-8 is given in Table 13-7. 

A Detailed Discussion of the Effect of Relative Ionic Sizes on the Properties of the Alkali Halogenides.—A simple detailed representation of interionic forces in terms of ionic radii has been formulated that leads to complete agreement with the observed values of interionic distances for alkali halogenide crystals and provides a quantitative theory of the anion-contact and double-repulsion effects. 0... 

It might be thought that this treatment would provide a poor approximation because of the neglect of polarization of each of the two ions in the electric field of the other.36 However, there is reason to think that the neglect of polarization does not introduce great error. First, the effect of multipole polarization as well as of the partial covalent character of the bonds is taken into account in the treatment of the crystals by the evaluation of the Bom exponent n from the observed compressibility and of the repulsion factor from the observed interionic distance. Second, in the gas molecule, in which there is dipole polarization mainly of the anion, its effect in causing increased attraction of the ions may be largely neutralized by the increased repulsion caused... 

Crystals with the Rutile and the.Fluorite Structures Interionic Distances for Substances of Unsymmetrical Valence Type.—In a crystal of a substance of unsymmetrical valence type, such as fluorite, CaFs (Fig. 13-10), the equilibrium cation-anion interionic distance cannot be expected necessarily to be given by the sum of the crystal radii of the bivalent calcium ion and the univalent fluoride ion. The sum of the univalent radii of calcium and fluoride, 2.54 A, would give the equilibrium interionic distance in a hypothetical crystal with attractive and repulsive forces corresponding to the sodium chloride arrangement. 

It is to be emphasised that equilibrium interionic distances are less well defined than covalent bond lengths their values depend not only on ligancy, but also on radius ratio (anion contact, double repulsion), amount of covalent bond character, and other factors, and a simple discussion of all the corrections that have been suggested and applied cannot be given. On the other hand, we have a reliable picture of the forces operating between ions, and it is usually possible to make a reliable prediction about interionic distances for particular structures. 

Ionic radii are discussed thoroughly in Chapters 4 and 7. For the present discussion it is only necessary to point out that the principal difference between ionic and van der Waals radii lies in the difference in the attractive force, not the difference in repulsion. The interionic distance in UF, for example, represents the distance at which the repulsion of a He core (Li+) and a Ne core (F ) counterbalances the strong electrostatic or Madelung force. The attractive energy for Lt F"is considerably over 500 kJ mol"1 anti the London energy of He-Ne is of the order of 4 kJ mol-1. The forces in the LiF crystal are therefore considerably greater and the interioric distance (201 pm) is less than expected for the addition of He and Ne van der Waals radii (340 pm). 

Ionic strength a measure of interionic interactions that are primarily derived from electrical attractions and repulsions. 

In this the Madelung constant (A) and repulsive parameter (n) are put equal to average values and the interionic distance is assumed to be equal to (r+ + r). This equation then becomes... 

For compounds in which the radius ratio is small, another term may be added to (5) to include also the appreciable effect of anion-anion repulsion. Pauling has indeed proposed such a treatment, analogous to Born s, but refined to include radius-ratio effects in doing so, he has been able to predict just how much the interionic distances in each of the alkali halides should depart from strict additivity. In using his modified treatment further for calculation of lattice energies, he has been able to show that the anomalies in melting points and boiling points, mentioned earlier in this chapter, may be correlated, at least semiquantitatively, with the radius ratios. 

As the coordination number of an ion is thus increased from 6 to 8, it appears that the ionic radius also suffers a slight increase, presumably because the repulsion forces, exerted by the electron clouds of neighboring ions, increase. For the rubidium halides and the ammonium halides which can assume either structure (depending upon pressure and temperature), the interionic distances in the structures having coordination number 8 are about 3 percent greater than the distances in structures with coordination number 6. This increase is close to the value that would be... 

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