Lattice energies theoretical calculations

TABLE 4.21 The Lattice Energies, Theoretically Calculated Based On the Eq. (4.64) of Kaputinski Type and Compared with Those Experimentally Deduced from the Bom-Haber cycle, for Crystalline Substances ofNaCl Type ... 

An experimental study of barbituric acid found one new polymorph where molecules in the asymmetric unit adopted two different conformations [10]. The conformational aspect was investigated through the use of ab initio calculations, which permitted the deduction that the new form found would have a lower lattice energy than would the known form. It was also found that many hypothetical structures characterized by a variety of hydrogen-bonding structures were possible, and so the combined theoretical and experimental studies indicated that a search for additional polymorphs might yield new crystal structures. 

Several issues remain to be addressed. The effect of the mutual penetration of the electron distributions should be analyzed, while the use of theoretical densities on isolated molecules does not take into account the induced polarization of the molecular charge distribution in a crystal. In the calculations by Coombes et al. (1996), the effect of electron correlation on the isolated molecule density is approximately accounted for by a scaling of the electrostatic contributions by a factor of 0.9. Some of these effects are in opposite directions and may roughly cancel. As pointed out by Price and coworkers, lattice energy calculations based on the average static structure ignore the dynamical aspects of the molecular crystal. However, the necessity to include electrostatic interactions in lattice energy calculations of molecular crystals is evident and has been established unequivocally. 

Self-consistent energy band calculations have now been made through the LMTO method for all of the NaCl-type actinide pnictides and chalcogenides . The equation of state is derived quite naturally from these calculations through the pressure formula extended to the case of compounds . The theoretical lattice parameter is then given by the condition of zero pressure. 

At the right-hand side of Fig. 5, a comparison is made between two A-metal ions, Sr2+ and Caa+ (radii 1.13 and 0.99 A). Here the difference in radii is much greater than in any of the A/B pairs with which we have been concerned. However, the differences between one anion and another are now a good deal smaller moreover, these differences agree very well with values calculated on the basis of the differences in ionic size alone. The theoretical predictions given by ionic lattice-energy calculations are shown as circles to the left of the Sr/Ca points. 

It is possible to calculate a theoretical value of the lattice energy for a molecular crystal if data are available on the potential energy between atoms as a frmction of their separation. A commonly used form for the interatomic potential (see Fig. 1) is due to Lennard-Jones " ... 

Finally, use Eq. (8) to determine the experimental value of the lattice energy of argon at 0 K. X-ray diffraction data give 5.30 A for the cubic unit-cell parameter of solid argon at 4 K. Find the nearest-neighbor distance d, and use Eqs. (10) and (11) to calculate a theoreticaf value of o- Compare the theoretical and experimental values. 

It is important to note that the theoretical results obtained by Flory about the dependence of the critical concentrations v and vj on x are in good agreement with experimental data. It is sufficient to remember, as an example, the results obtained by Flory for PBLG solutions in dioxane (Fig. 4). The discrepancy between the experimental results (solid curves) and theoretical calculations (dashed curves) looks quite natural on the account of a number of assumptions made when deriving the equation for the free energy on the basis of the lattice model (see also ). 

Quite apart from its theoretical calculation, by the use of one of the expressions developed above, it is possible to relate the lattice energy of an ionic crystal to various measurable thermodynamic quantities by means of a simple Hess s law cycle. This cycle was first proposed and used by Bom 15) and represented in its familiar graphical form by Haber (45). It is now usually referred to as the Born-Haber cycle. The cycle is given below for a uni-univalent salt in terms of enthalpies. 

It must be remembered that the lattice energy given by the Bom-Haber cycle is an experimental lattice energy and is not,dependent upon the nature of the assumptions made about the bonding in the crystal. The classical theoretical calculations are of course dependent upon the assumption of the ionic nature of the bonding in the lattice. Because of this the Bom-Haber cycle has been used mainly for three purposes. 

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