Bom-Lande equation

There is another use of the Kapustinskii equation that is perhaps even more important. For many crystals, it is possible to determine a value for the lattice energy from other thermodynamic data or the Bom-Lande equation. When that is done, it is possible to solve the Kapustinskii equation for the sum of the ionic radii, ra + rc. When the radius of one ion is known, carrying out the calculations for a series of compounds that contain that ion enables the radii of the counterions to be determined. In other words, if we know the radius of Na+ from other measurements or calculations, it is possible to determine the radii of F, Cl, and Br if the lattice energies of NaF, NaCl, and NaBr are known. In fact, a radius could be determined for the N( )3 ion if the lattice energy of NaNOa were known. Using this approach, which is based on thermochemical data, to determine ionic radii yields values that are known as thermochemical radii. For a planar ion such as N03 or C032, it is a sort of average or effective radius, but it is still a very useful quantity. For many of the ions shown in Table 7.4, the radii were obtained by precisely this approach. 

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 strength ofa purely ionic bond between two ions can be obtained quite accurately by means of the Bom-Lande equation (Chapter 4).8 Neglecting repulsive forces, van der Waals forces, and other small contributions, we can estimate the energy of an ion pair simply as... 

As long as we do not neglect to understand each or the factors in the Born-Lande equation (4.13), we can simplify the calculations. It should be realized that the only variables in the Bom-Lande equation are the charges on the ions, the inlernuclear distance, the Madelung constant, and the value of n. Equation 4.13 may thus be simplified with no loss of accuracy by grouping the constants to give ... 

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... 

Fig. 21. The enthalpy of formation of the lanthanide trilluorides as a function of the atomic number, o and indicate the experimental results from fluorine combustion studies at ANL and Kyoto University, respectively the broken curve shows the estimated values using the Bom-Lande equation (Kim and lohnson, 1981) the solid curve shows...

Performing the arithmetic, we obtain (7q = — 755 kl mo( which may be compared with the best experimental value (TaUe 4.3) of —770 kJ mol We may feel confident using values predicted the Bom-Lande equation where we have no experimental values. 

TheMadelung constant., A, is also added to the equation. This is a factor which takes into account all the interactions between the ions, since an ion in a lattice is influenced by more than just one other single ion of opposite charge, as considered above. Fortunately, these attractions and repulsions form a mathematical series which depends on the lattice type. The values of the Madelung constant have been calculated by computer and some are shown in Table 2.3. The final expression of the Bom-Lande equation then becomes ... 

For ionic solids, in which both attractive and repulsive electrostatic forces as well as short-range repulsive forces complicate the description of the overall energy, the Bom-Lande equation has been shown to provide an adequate estimate of lattice energy, /. ... 

This is the Bom-Lande equation for the lattice energy of an ionic compound. As we shall see, it is quite successful in predicting accurate values, although it omits certain energy factors to be discussed below. It requires only a knowledge of the crystal structure (in order to choose the correct value for A) and the interionic distance, ro. both of which are readily available from X-ray diffraction studies. 

The enthalpy of formation of an ionic compound can be calculated with an accuracy of a few percent by means of the Bom-Lande equation (Eq. 4.13) and the Bom-Haber cycle. Consider NaCI, for example. We have seen that by using the predicted intemuclear distance of 283 pm (or the experimental value of 281.4 pm), the Madelung constant of 1.748, the Born 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 f/ 9a = —757 kJ mol-1. The Born-Haber summation is then... 

In this case the reason tor the correlation is fairly obvious. The parameter r n is equal to the ionic radius plus a constant, 85 pm, the radius of the oxygen atom in water. Therefore, rM is effectively the interatomic distance in the hydrate, and the Bom-Lande equation (Eq. 4.13) can be applied. 

The Bom-Lande equation gives an expression for A(7(0K) assuming an electrostatic model and this is appropriate for the group 1 metal halides ... 

Compound Born-Haber cycle Bom-Lande equation... 

This expression has its origins in the Bom Lande equation, with a value of 9 for the Bom exponent (the value for NaCl) and half the value of the Madelung constant for NaCl the inclusion of the factor v shows why half of A is included. Although the Kapustinskii equation is useful, it is a gross approximation and values obtained in this way must be treated with caution. 

Theoretical values for lattice energy may be calculated. The crystal is assumed to be made up of perfectly spherical ions. From the geometry of the crystal, the interionic distance is known. The energy of attraction between all the oppositely charged ions and the energy of repulsion between ions of same charge are calculated. The final equation obtained is Bom-Lande equation -... 

Write the Bom-Lande equation and explain the terms involved. 

The change in internal energy (the lattice energy) is given by the Bom-Lande equation ... 

Lattice energies obtained from the Bom-Lande equation are approximate, and for more accurate evaluations, several improvements to the equation can be made. The most... 

Equation 8.9, page 201, Bom-Lande equation for lattice energies ... 

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