Solids Madelung constants

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... 

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

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,... 

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... 

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. 

It is convenient also to generalize the Madelung constant to structures with more than one magnitude of charge (as in CaF2). This can be done for the case of a solid with formula in terms of a reduced Madelung constant (discussed, for... 

A few AB solids have body-centered cubic structures, with a CN of 8. An example is CsCl, which is calculated to be much more stable in the ionic model. Again ionic bonding seems to favor a high CN, since the Madelung constant increases, being 1.763 for bcc. This theory proves to be short-lived, however, when we consider bonding in the metals. By definition, the bonding here must be covalent, since identical atoms are bonded. 

The largest change is in the cohesive energy, which increases m all cases. For ionic solids, the increase is due to the larger Madelung constant. For metals, the increase is due to the better delocalization energy with a higher coordination number. For covalent solids, the increase occurs because all of the valence-shell... 

The latter is less by a factor of M, the Madelung constant. Then (M — 1) must be the effect of the rest of the lattice. This suggests the corrected value 7 = nMjZ for ionic solids. 

Here, e is the electronic chaige unit (e = 4.8 X 10"esu), Nq is Avogadro s number, A is the Madelung constant, and n is a parameter arising from repulsive forces that build up as atoms come into contact and atomic orbitals begin to overlap. The distance between ion pairs, r, can be determined for an ionic solid from the known radii of the constituent ions. For solid KQ, as an example, the radii of and Cl" are 1.33 A (Angstrom) and 1.81 A, respectively. The value of is then 1.33 + 1.81 = 3.14 A, and n is known to be 9 for this particular lattice. Equation 1.3 can then be used to calculate the overall lattice energy of KCl and other ionic solids. 

Calculate the lattice energy (in kcal/mole) for NaO if the Madelung constant for this solid is 1.75 and the value of n is 9, The radii of Na" and CF are 1.02 and 1.81 A, respectively. If KCl has about the same Madelung constant as NaCl, how does its lattice energy compare with that of NaCl ... 

The attractive forces within the crystalline ionic structure are of the form M (q X q )/(r+ + r ), where the value of M, the Madelung constant, depends on the pattern of packing of the ions. We can expect that the product (q X q ) should give an indication of the cohesive energy of the solid. (See Table 2.)... 

Because the Madelung constant has been computed by a summation over all lattice sites, it adopts characteristic values for all structure types [5,8]. To give a few examples, M arrives at (dimensionless) values of 1.6381 (zinc-blende-type), 1.7476 (sodium chloride-type), 1.7627 (caesium chloride-type), 5.0388 (fluorite-type), and 25.0312 (corundum-type) and does not scale with (= is independent of) the interionic distances. For the case of NaCl, the Madelung constant shows that the three-dimensional lattice surpasses the ionic pair in energy by almost 75%. This is what has made the formation of solid NaCl possible, a collective stabilization. 

If the lattice type and Madelung constant for an ionic solid are not yet known, the lattice energy can still be approximated using an empirical equation developed by Kapustinskii, shown in Equation (12.7). Only the charges on each ion, the stoichiometric total number of ions (v), and the interionic separation need to be known in order to make the calculation. If Tq is entered into Equation (12.7) in units of pm, the calculated lattice energy will be in units of kj/mol. Using the values in Example 12-1 for NaCI and v = 2,Uq = -746 kj/mol, a difference of only 5% from the experimental value is obtained. 

Hoppe, R., 1970b, Madelung constants as a new guide to the structural chemistry of solids, in Advances in Fluorine Chemistry, Mrl. 6, eds M. Stacey, J.C. Tatlow and A.G. Sharpe (Butterwoiths, London) p. 387. 

Madelung Constant. An important constant for calculating the lattice energy of an ionic crystalline solid. If the distance between nearest adjacent ions is R, and the distance of the jth ion from a (negative) reference ion is rj (always taken as positive), then the Madelung constant... 

Glasser, L., Solid-state energetics and electrostatics Madelung constants and Madelung energies, Inorg. Chem. 51 (4), 2420-2424 (2012). 

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