Energy interionic repulsion

It has been found that the contribution of the interionic repulsive forces to the lattice energy can be well represented by a term of the... 

There is yet a third component of the lattice energy which must be considered, namely, that due to the van der Waals forces of attraction between the ions. These forces, like those of interionic repulsion, cannot be described in simple physical terms, but they, too, have been the subject of much recent work to which we shall have occasion to refer in chapter 6. The effect of these forces cannot be treated in isolation, for it can be shown that its inclusion involves a corresponding modification in the estimate of the effect of the repulsive forces. In consequence it is found that the necessary correction to the lattice energy so far computed is very small, and for the alkali halides never exceeds 3 kcal/mole. 

Theoretical values for lattice energy may be calculated. The crystal is assumed to be made up of perfectly spherical ions. From the geometry of the crystal, the interionic distance is known. The energy of attraction between all the oppositely charged ions and the energy of repulsion between ions of same charge are calculated. The final equation obtained is Bom-Lande equation -... 

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. 

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). 

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. 

The distance between the ions in a crystal is determined by the equilibrium between the forces of attraction and repulsion. Values of the interionic distances may be obtained from x-ray data. On the basis of the Born theory of lattice energies we have,... 

One of the disadvantages of the fully theoretical approach is that it is necessary to know the crystal structure and the interionic distances to estimate the lattice energy. The Kapustinskii equations overcome this limitation by making some assumptions. The Madelung constant and the repulsive parameter n are put equal to average values, and it is also assumed that the interionic distance can be estimated as the sum of anion and cation radii r+ and r (see Topic D4Y The simpler... 

Figure 2.2 The total potential energy, I7l, between monovalent ions as a function of the ionic separation, r. The total energy is the sum of the attractive and repulsive potential energy terms. The lattice energy, U corresponds to the minimum in the total energy curve, reached at an interionic separation of ro...

Ions are not simply point charges and as they are brought together their closed electron shells begin to overlap and, for quantum mechanical reasons, repulsion sets in. This increases sharply as the interionic distance, r, decreases until, neglecting other forces, a balance is obtained with the electrostatic attractive forces (Figure 2.2). The repulsive potential energy, E, can be formulated in a number of ways. One of the first to be used was an empirical expression of the type... 

Consider a hypothetical —Y ion pair for which O the equilibrium interionic spacing and bonding energy values are 0.38 nm and —5.37 eV, respectively. If it is known that n in Equation 2.17 has a value of 8, using the results of Problem 2.18, determine explicit expressions for attractive and repulsive energies E and E of Equations 2.9 and 2.11. 

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