Constant Madelung

The total energy of the system may be found by taking the sum of these two potential functions over all of the ions in the system. 

It is convenient to write r/y as rpij, where r is the nearest neighbor distance and p,y is a dimensionless parameter relating the distance between the I th and /th charges in units of r. Now instead of summing over both i and j, we arbitrarily chose some i as the 0th charge, sum over j and multiply by N, the number of ion pairs. 

A structure with a coordination number of 8 is known as the CsCl structure. It can be described as two interpenetrating simple cubic lattices with the anions on the comers of one lattice and the cations located on the comers of the second interpenetrating lattice arranged such that they sit in the centers of the cubes of the anion lattice so that each ion is surroimded by 8 coimterions. The Madelung constant for this structure is equal to 1.763. 

The NaCl or rock salt stmcture has a coordination number of 6. It consists of two interpenetrating face-centered cubic (fee) lattices with the anions on the lattice sites of one lattice and the cations on the lattice sites of the interpenetrating lattice arranged such that the cations sit on the octahedral interstitial sites (points on the edges of the face-centered cubes half-way between the comers. Despite the terminology, the coordination 

The zinc blende structure has a coordination number of 4 and is a diamond-like structure with opposite charges on every other occupied lattice point. The A for this structure is equal to 1.638. (These and other lattices will be discussed in more detail in Chapter 5.) 

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Many of the spinel-type compounds mentioned above do not have the normal structure in which A are in tetrahedral sites (t) and B are in octahedral sites (o) instead they adopt the inverse spinel structure in which half the B cations occupy the tetrahedral sites whilst the other half of the B cations and all the A cations are distributed on the octahedral sites, i.e. (B)t[AB]o04. The occupancy of the octahedral sites may be random or ordered. Several factors influence whether a given spinel will adopt the normal or inverse structure, including (a) the relative sizes of A and B, (b) the Madelung constants for the normal and inverse structures, (c) ligand-field stabilization energies (p. 1131) of cations on tetrahedral and octahedral sites, and (d) polarization or covalency effects. ... 

The overall lattice energies of ionic solids, as treated by the Born-Eande or Kaputin-sldi equations, thus depends on (i) the product of the net ion charges, (ii) ion-ion separation, and (iii) pacldng efficiency of the ions (reflected in the Madelung constant, M, in the Coulombic energy term). Thus, low-melting salts should be most... 

Madelung constant (A) A number that appears in the expression for the lattice energy and depends on the type of crystal lattice. Example A = 1.748 for the rock-salt structure. 

Table 5.1 MADELUNG constants for some structure types...

Derive the first four terms of the series to calculate the MADELUNG constant for CsCl (Fig. 7.1). 

The stability of a certain structure type depends essentially on the relative sizes of cations and anions. Even with a larger Madelung constant a structure type can be less stable than another structure type in which cations and anions can approach each other more closely this is so because the lattice energy also depends on the interionic distances [cf. equation (5.4), p. 44], The relative size of the ions is quantified by the radius ratio rm/rx rM being the cation radius and rx the anion radius. In the following the ions are taken to be hard spheres having specific radii. 

For compounds of the composition MX (M = cation, X = anion) the CsCl type has the largest Madelung constant. In this structure type a Cs+ ion is in contact with eight Cl-ions in a cubic arrangement (Fig. 7.1). The Cl- ions have no contact with one another. With cations smaller than Cs+ the Cl- ions come closer together and when the radius ratio has the value of rM/rx = 0.732, the Cl- ions are in contact with each other. When rM/rx < 0.732, the Cl- ions remain in contact, but there is no more contact between anions and cations. Now another structure type is favored its Madelung constant is indeed smaller, but it again allows contact of cations with anions. This is achieved by the smaller coordination number 6 of the ions that is fulfilled in the NaCl type (Fig. 7.1). When the radius ratio becomes even smaller, the zinc blende (sphalerite) or the wurtzite type should occur, in which the ions only have the coordination number 4 (Fig. 7.1 zinc blende and wurtzite are two modifications of ZnS). 

R. Hoppe, Madelung constants as a new guide to the structural chemistry of solids. Adv. Fluorine Chem. 6 (1970) 387. 

We have already mentioned that for sodium chloride approximately 1.78 times as much energy is released when the crystal lattice forms as when ion pairs form. This value, the Madelung constant (A) for the sodium chloride lattice, could be incorporated to predict the total energy released when 1 mole of NaCl crystal is formed from the gaseous Na+ and Cl- ions. The result would be... 

We have already defined the Madelung constant as the ratio of the energy released when a mole of crystal forms from the gaseous ions to that released when ion pairs form. In order to understand what this means, we will consider the following example. Suppose that a mole of Na+ and a mole of Cl... 

The series inside the parentheses converges to a sum that is 2 ln2 or 1.38629. This value is the Madelung constant for a hypothetical chain consisting of Na+ and Cl- ions. Thus, the total interaction energy for the chain of ions is —1.38629N0e2/r, and the chain is more stable than ion pairs by a factor of 1.38629, the Madelung constant. Of course NaCl does not exist in a chain, so there must be an even more stable way of arranging the ions. 

Table 7.3 Madelung Constants for Some Common Crystal Lattices.

In this series, the terms neither lead to a recognizable series nor converge very rapidly. In fact, it is a rather formidable process to determine the sum, but the value obtained is 1.74756. Note that this is approximately equal to the value given earlier for the ratio of the energy released when a crystal forms to that when only ion pairs form. As stated earlier, the Madelung constant is precisely that ratio. 

Details of the calculation of Madelung constants for all of the common types of crystals are beyond the scope of this book. When the arrangement of ions differs from that present in NaCl, the number of ions surrounding the ion chosen as a starting point and the distances between them may be difficult to determine. They will most certainly be much more difficult to represent as a simple factor of the basic distance between a cation and an anion. Therefore, each arrangement of ions (crystal type) will have a different value for the Madelung constant. The values for several common types of crystals are shown in Table 7.3. 

Each of the compounds shown in Eq. (9.101) has the same crystal structure, the sodium chloride structure, so the Madelung constant is the same for all of them. The term containing 1/n is considered to be a constant for the two pairs of compounds (reactants and products). Actually, an average value of n... 

The principle that embodies this relationship can be stated as the products will be those in which the smaller ions will combine with oppositely charged ions of higher charge. In these cases, the Madelung constants may be different, so other factors may be involved. However, the principle correctly predicts reactions such as the following ... 

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