The arrangement of ions in a crystal is greatly influenced by the ratio of radii of the ions. The limiting ratio for a cation to fit in an octahedral arrangement of anions is greater than 0.414 (i.e. r+/r– > 0.414).
Only in such a situation a cation will be able to keep the six anions from touching each other. Smaller cations will prefer to fit into tetrahedral holes in the lattice. For radius ratio (r+/r–) ranging between 0.225 to 0.414, tetrahedral sites are preferred. Above 0.414 octahedral coordination is favoured.
The application of radius ratio to predict coordination, such as, consider zinc sulphide in which rZn2+/rS– = 0.74/1.84 = 0.40. Zinc ions thus prefer the tetrahedral holes in the close-packed lattice of sulphide ions. In the same way, we can predict lattice of chloride ions (rNa+/rCl– = 0.95/1.81 = 0.52).
With larger cations, such as cesium, the radius ratio increases beyond the limit for a coordination number of 6(0.414 – 0.732). Cesium ions now occupy cubic sites, i.e., co-ordination number of cations increases to 8 in a lattice of chloride ion (rCs+/rCl– = 1.69/1.81 = 0.93). A list of limiting radius values is given in Table 2.8.
By using radius ratio rule, it is possible to predict the cation/anion coordination number in any compound. So radius ratio is a useful measure in establishing the structure of ionic solids. In Fig. 2.49 it is shown how the arrangement of ions in a single layer is affected by the differences in the sizes of the ions.
The sizes in picture (a) are “just right”. The X– ions are in contact with each other and also with M+. In contrast, M+ is too small in (b) and it is a poor fit. If X– ions move in closer to M+ then they would crowd each other. There would be repulsion amongst them and the energy of the system would be raised. If they hold each other apart then they are not very close to be oppositely charged M+. Finally (c) does not allow the X– ions to come close enough to feel any X– – X– repulsion.
Therefore, if M+ is too large as in (c) or too small (as in b), a different MX crystal packing might be obtained. If (a) represents the lattice of NaCl, then in situation (c) when M+ is quite large, we may have an arrangement in which M+ may have more X‑ as neighbours as in CsCl. On the other hand, if M+ is too small as in (b), only four negative ions pack around the positive ions. ZnS provides an example of this type.
Lattice Energy in Ionic Solids:
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The stability of an ionic solid is measured in terms of its lattice energy U0. It is defined as the energy released when one mole of an ionic crystal is formed from one mole of gaseous positive and one mole of gaseous negative ions, when these are separated from each other by infinite distance.
Greater the value of lattice energy more stable is the ionic crystal. The lattice energy, of sodium chloride is -185 kcal/mole. The negative sign indicates that the process is exothermic and that the energy of the system is lowered as the solid is formed. The high value suggests the stability of the solid.
Qualitatively, the force of attraction between oppositely charged ions is determined by – (i) their ionic radii (ii) the charges on the ions.
It is given by the relationship F = K q1 q2/r2
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Where, q1 and q2 are the charges on ions and r is the interionic distance. It may be observed that if the radii of the ions are large, the force of attraction will be less. The force of attraction between the large Cs+ ions and CF ions will be less than that between the smaller Na+ ions and CF ions. As a result CsCl is less stable than NaCl. It may be the stability of an ionic crystal increases with decreasing interionic distance for crystals with similar charge.
The role played by the charges on the ions (on the stability of solids) is demonstrated if we compare the lattice energies of BaO and NaCl, since both have very similar inter-nuclear distance (NaCl = 2.81 Å and BaO = 277 Å). Since BaO consists of doubly charged Ba2+ and O2- ions, the force of attraction is much large than that in NaCl which contains singly charged ions. Hence, BaO is more stable than NaCl (m. pts; NaCl = 800°C, BaO = 1921°C).
The quantitative evaluation of lattice energy of an ionic crystal is determined by coulombic interaction between all of its ion; attractive forces between oppositely charged ions and repulsive forces because of the interpenetration of electron clouds.
The potential energy of a pair of oppositely charged ions (representing force of attraction) varies inversely with distance, r, between the two ions (Fig. 2.50)
Where, Z1 and Z2 are the charges on the ions. As the ions come very close to each other, they repel one another because of interpenetration of their electron clouds. The repulsion energy is inversely proportional to the nth power of distance r between the ions.
Where, e is the electronic charge, n is called the Born exponent and b is a repulsion coefficient. The Born equation gives the net potential energy for a pair of ions as the summation of the two terms, when these ions brought together from infinite distance, 
It is clear from Fig. 2.50 that repulsion term increases very rapidly as the separation between the ions decreases beyond the inter-nuclear separation. The potential energy is at a minimum at the equilibrium separation, r0. However, in ionic crystals no two separate ions to be considered, the entire crystal must be considered as a single entity.
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The force on any one ion will be determined not only by the oppositely charged ions which are directly surrounding it, but also somewhat due to other ions (both positive and negative) at greater distances. To consider the forces due to all the ions, we may take a specific example, say of sodium chloride. The potential energy of an ion will depend on the crystal structure of the solid.
In a crystal of sodium chloride (Fig. 2.44) the potential at a sodium ion result from:
(a) Six nearest Cl– ions at a distance r;
(b) Twelve next nearest Na+ ions at a distance √2r;
(c) Eight next nearest Cl– ions at a distance √3r;
(d) Six next nearest Na+ ions at a distance 2r;
(e) Twenty-four next nearest CF ions at a distance √5r.
These distances from a sodium ion are clearly shown in Fig. 2.51.
The potential energy expression for the energy released (attraction in bringing a sodium ion in a crystal lattice of NaCl from infinity is given by the summation of all such terms).
Equation (ii) is the summation of an infinite series which is called the Madelung constant. The summation value for sodium chloride is given as 1.747558 and is the Madelung constant for sodium chloride. Since the value of this constant depends only on the geometry of the crystal, the value will be the same for all other solids which exhibit the NaCl structure. The Madelung constant is represented by the symbol A. This quantity can be calculated in the same way for other crystal lattices. The value of constant for some common crystal structure is given in Table 2.9.
The second term in the potential energy expression involves the repulsive forces arising from the proximity of the electron clouds of Na+ and Cl– ions. The expression given by Born for the repulsion term will now be 6 be2/rn (as there are six CI ions immediately surrounding the Na+ ion). This is expressed in a more general form as Be2/rn, where B replaces the factor 6b.
The net potential energy of an ion considering all types of forces due to the neighbouring ions is given by-
Where, Z replaces Z1 and Z2. At the equilibrium inter-nuclear distance, r0, the attractive and repulsive forces are exactly balanced. The potential energy is minimum at r0. Differentiation of equation 3 for r = r0 and equating it to zero we obtain the values of r0 and B.
Equation (iii) may be written as-
By substituting the value of B in expression (iii), we get –
The lattice energy, U0, is the amount of energy released when one mole of NaCl crystal is formed from gaseous ions which are at infinite separation. So by definition –
U0 = -(PE)0 × N
When, N is the Avogadro’s number. So lattice energy of NaCl is given by the expression
Lattice energy is a useful property in predicting the behaviour of ionic solids, since the formation and destruction of a crystal is frequently an important step in reactions involving ionic solids. In these calculations, it is assumed that both cations and anions are spherical in shape: they are incompressible and are not destroyed by the neighbouring ions.
Role of Lattice Energy in Ionic Solids:
(i) Stability of Ionic Solids:
Greater the lattice energy, greater is the stability of an ionic solid. Because more energy will be needed to pull apart the positive and negative ions.
(ii) Melting Points:
Salts with high lattice energies require a greater input of thermal energy to breakdown the crystal lattice. Consequently, salts have high melting points.
(iii) Solubilities of Ionic Solids:
The magnitude of lattice energy of solid may give us an idea about the solubility of that substance in different solvents. Ionic solids in general are insoluble in non-polar (covalent) solvents such as carbon tetrachloride, whereas these are soluble in polar solvents, like water. For a solid to dissolve in a solvent, the strong forces of attraction between its ions (lattice energy) must be overcome. This energy can be overcome by the ion-solvent interactions.
The solvation of ions is referred in terms of solvation energy which is always negative, i.e., energy released in the process depends upon the nature of the solvent. In case of non-polar solvents, the solvation energy is small and it is not sufficient to overcome the lattice energy of the solid, thereby the substance does not dissolve. The solvation energy increases if the solvent has high dipole moment/or high polarizability. As a general rule, we may say that for a solid to dissolve in a particular solvent its solvation energy should be greater than the lattice energy of that solid.