Sodium Madelung constant

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... 

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 series 6 - 12/21/2 + 8/31/2 - 6/41/2 + 24/51/2 -. .. eventually becomes convergent and gives the value for the Madelung constant for the sodium chloride lattice (the standard description of lattices which have the same form as that adopted by sodium fluoride). The values of Madelung constants for some common crystal lattices are given in Table 7.5. 

Alternatively, we might examine the radius ratio of Oj BF and get a crude estimate of = 0.8. The accuracy of our values does not permit us to choose between coordination number 6 and 8, but since the value of the Madelung constant does not differ appreciably between the sodium chloride and cesium chloride structures, a value of 1.75 may be taken which will suffice for our present rough calculations. We may then use the Bom-Lande equation (Eq. 4.13), which provides an estimate of —616 kJ mor1 for the attractive energy, which will be decreased by about 10% (if... 

The factor A is a positive numerical constant called the Madelung constant its value depends on how the ions are arranged about one another. For ions arranged in the same way as in sodium chloride, A = 1.748. 

He assumed that for salts made up of combinations of spherical or tetrahedral ions the lattice energy could be well approximated by assigning to the crystal structure (usually unknown) a Madelung constant equal to that of sodium chloride and estimating the repulsive contribution to the lattice energy by a Bom-Mayer expression similar to that which holds approximately for the alkali halides. These assumptions lead to the following expressions for the lattice energy U ... 

Using Fig- 4.7 generate the first five terms of the senes for the Madelung constant for NnCL How close is the summation of these terms to the limiting value given in Table 4.1 The enthalpy of formation of sodium fluoride is —571 kj mol. Estimate Ihe electron affinity of fluorine Compare your value with that ven in Table 23. 

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. 

Madelung constant A constant arising in calcuiations of the cohesion of Ionic crystals. The electrostatic interaction per ion pair, U, is given by U[r) = -a lr, where a is the Madelung constant and e /ris the Coulomb interaction between the ions, with rbeing the lattice constant. The value of a depends on the type of lattice. For the sodium chloride lattice, a has a value of about 1.75. A more realistic calculation of cohesion is obtained if short-range repulsions with an inverse power law are included, i.e. 

The geometric series in the parentheses is a constant that depends on the crystal structure. In other words, if the sodium cations and chloride anions assumed the CsCl, zinc blende, Avurtzite, or some other crystal structure, the series in the parentheses would be different from that given for NaCl. These unique series for each crystal structure are known as Madelung constants M) and are listed in Table 8.1. Using the symbol A/njCi for the Madelung constant unique to the sodium chloride... 

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