Ionic Madelung constant

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... 

Fumi F. G. and Tosi M. P. (1957). Naor relations between Madelung constants for cubic ionic lattices. Phil Mag, 2 284-285. 

For ionic crystals > = 1, and the A are known (Madelung constants). For van der Waals crystals m — 6 (though small terms in and exist) but in view of the difficulties of calculation we obtain B from the observed heat of vapourisation (from A. 3). The repulsion exponent n varies from about 6 for LiF to 12 for Csl, for gases (Lennard-Jones) a value of about 12 seems the best. We assume a constant value of 11 throughout. 

The geometry of the crystal introduces a factor multiplying the pairwise ionic interaction, which is the Madelung constant /t. It is a dimensionless constant, dependent on the geometry of the crystal under consideration. For an ionic binary crystal, consisting of N each positive and negative ions, fi is defined by... 

When we have an ordered assembly of atoms called a lattice, there is more than one bond per atom, and we must take into account interactions with adjacent atoms that result in an increased interionic spacing compared to an isolated atom. We do this with the Madelmg constant, ckm. This parameter depends on the structure of the ionic crystal, the charge on the ions, and the relative size of the ions. The Madelung constant fits directly into the energy expression (Eq. 1.25) ... 

The term inside the brackets is known as the Madelung constant, A, in this case for the NaCl structure. The series is slow to converge, but values of the Madelung constant have been computed, not only for NaCl, but also for most of the simple ionic structures. For one mole of NaCl, we can write ... 

TABLE 1.14 Madelung constants for some common ionic lattices... 

The electrostatic (Madelung) part of the lattice energy (MAPLE) has been employed to define Madelung potentials of ions in crystals (Hoppe, 1975). MAPLE of an ionic solid is regarded as a sum of contributions of cations and anions the Madelung constant. A, of a crystal would then be the sum of partial Madelung constants of cation and anion subarrays. Thus,... 

A compound A+B has a larger Madelung constant per equivalent than a compound Therefore, while, for equal ionic radii, a... 

The covalent curve of Figure 3-5 has been drawn in the same way as for the HX curves. The ionic curve represents a Coulomb attractive potential and a repulsive potential b/R9f in which the constant b is given values which lead (with use of the Madelung constant and corresponding constant in the repulsive potential) to the correct interatomic distance in the corresponding crystal (Chap. 13). Polarization is neglected. 

It was noted in Chapter 4 that the Madelung constant of a structure may be expressed in various ways. The way that is conceptually simplest in terms of the Bom-Lande equation is the simple geometric factor. A, such that when combined with the true ionic charges, Z and Z, the correct electrostatic energy is formulated. It was noted that some workers have favored using another constant, A. combined with the highest common factor of Z and Z. Z -... 

The enthalpy of formation of an ionic compound can be calculated with an accuracy of a few percent by means of the Born-Land6 equation (Eq. 4.13) and the Bcrn-Haber cycle. Consider NaCI. for example. We have seen that by using the predicted internuclear distance of 283 pm (or the experimental value of 281.4 pm), the Madelung constant of 1.748, the Rorn exponent, n, and various constants, a value of — 755 kJ mol-1 could be calculated for the lattice energy. The heat capacity correction is 2.1 kJ mol-1, which yields U 9i = —757 kJ mol-1. The Bom-Haber summation is then... 

The ionic solids NaCl and KC1 form the same type of crystal structure, so they have the same values of the Madelung constant A. We can decide in which compound the ions bind together more strongly by taking the ratio of U for the two compounds. All the constants cancel when the ratio is taken, including the ionic charges, and we are left with... 

The crystal lattice energy can be estimated from a simple electrostatic model When this model is applied to an ionic crystal only the electrostatic charges and the shortest anion-cation intermiclear distance need be considered. The summation of all the geometrical interactions be/Kveeti the ions is called the Madelung constant. From this model an equatitWjor the crystal lattice energy is derived ... 

Kapustinskii noted that if the Madelung constant A is divided by the number of ions per formula unit for a number of crystal structures, nearly the same value is obtained. Furthermore, as both A/n and re increase with the coordination number, their ratio A/nre is expected to be approximately the same from one structure to another. Therefore, Kapustinskii proposed that the structure of any ionic solid is energetically equivalent to a hypothetical rock-salt structure and its lattice energy can be calculated using the Madelung constant of NaCl and the appropriate ionic radii for (6,6) coordination. 

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