Born-Lande

Madeluag constant For an ionic crystal composed of cations and anions of respective change z + and z, the la ttice energy Vq may be derived as the balance between the coulombic attractive and repulsive forces. This approach yields the Born-Lande equation,... 

The solvation energy described by the Born equation is essentially electrostatic in nature. Born equations 8.116 and 8.120 are in fact similar to the Born-Lande equation (1.67) used to define the electrostatic potential in a crystal (see section 1.12.1). In hght of this analogy, the effective electrostatic radius of an ion in solution r j assumes the same significance as the equilibrium distance in the Born-Lande equation. We may thus expect a close analogy between the crystal radius of an ion and the effective electrostatic radius of the same ion in solution. 

This is known as the Born-Lande equation the values of Fq and n can be obtained from X-ray crystallography and from compressibility measurements, respectively. The other terms in the equation are well-known constants, and when values for these are substituted, we get ... 

In the Born-Lande equation (and other similar equations) the main uncertainty lies with the n value, but that term only accounts for roughly 10% of the value of the lattice energy. Deviations from the experimental values for calculated enthalpies of formation greater than 10% usually mean that the compound is considerably covalent. 

The Born-Lande equation for the lattice energy of an ionic crystal was derived. 

The lattice energy of calcium oxide (Born-Lande) is 3554 kJ mol the enthalpy of lattice formation is -3554 - 6 = -3560 kJ mol1. The calculated standard enthalpy of formation of calcium oxide is -677 kJ mol about 7% more exothermic than the observed value, probably due to the inherent error in the calculation. 

The trends in Figure 3.3 are best interpreted by using the using the theoretical Born-Lande equation for the enthalpy change for lattice formation from the constituent gaseous ions ... 

This is the Born-Lande equation Tor 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, rc> both of which are readily available from X-ray diffraction studies... 

Performing the arithmetic, we obtain l/0 = — 733 kl mol-, which may be compared with the best experimental value (Table 4.3) of —770 kJ mol-1 We may feel confident using values predicted by the Born-Lande equation where we have no experimental values. 

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 ... 

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

Several methods have been proposed to check and correlate the data for the L F3 compounds. A semi-empirical method was proposed by Kim and Johnson (1981) who used the Born-Lande equation to estimate the lattice energy U ai ... 

A serious limitation of (but not an objection to) the hydridic model is its inability to rationalize the nonstoichiometry of metallic hydrides. The difficulty arises in part because of the use of the Madelung constant (or an approximate equivalent) and a Born-Lande or exponential repulsion term. It is not yet clear how these may be calculated for a defect structure where ions are randomly missing from their sites. It is reasonable that mutual repulsion of outer electrons will be less in such a structure, but no quantitative interpretation has been made. 

The variable n is knovra as the Born exponent, which depends upon the electronic configuration of the ions present. The Born-Lande equation calculates on the basis of the electrostatic attraction between... 

Using the Born-Lande etiuation and the data below, calculate the lattice enthalpies of MnO and MnO. ... 

Born exponent 46 Born-Haber cycle 44 Born-Lande equation 45... 

The enthalpy of formation of an ionic compound can be calculated with an accuracy of a few percent by means of the Born-Land equation (Eq. 4.13) and the Born-Haber cycle. Consider NaCI. for example. Wc 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 Born exponent, n, and various constants, a value of —755kJmor could be calculated for the lattice energy. The heat capacity correction is 2.1 kJ mol", which yields = —757 kJ moP. The Bom-Haber summation is then... 

The alkaline earth imides have a cubic, NaCl-type lattice and their lattice energies have been evaluated by A, P. Altshuller (4). Since the compressibilities were unknown he used the Born-Lande equation and adopted the value of n used by Sherman for the corresponding oxides. He did not incorporate a term to allow for the dispersion energy and found for the lattices energies of CaNH, SrNH and BaNH values of 787, 752 and 711 kcal/mole, respectively. Only the heat of formation of BaNH was available and from this a value of AH/ NH of 261 kcal/mole and hence a double electron affinity of —184 kcal/mole w as obtained. 

Sherman (114) and other workers have compared crystal energies obtained by the Born-Haber cycle from experimental thermochemical data with theoretical values calculated assuming strict ionic character. The differences obtained have been used to indicate deviations from strict heteropolarity. Sherman 114) in his review gives results for 50 crystals, the computations being made with the Born-Lande equation. 

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