Average lattice energy

Thus, on a typical lattice energy of 150 kJ mol , a change of just 0.02 A in C—H distance results in an average lattice energy difference of about 1 kJmol while stabilizations or destabilizations of up to 2-3 kJ moU for single cases are frequent. 

It is also useful to define an average lattice energy per point centre as... 

Born-Haber cycle A thermodynamic cycle derived by application of Hess s law. Commonly used to calculate lattice energies of ionic solids and average bond energies of covalent compounds. E.g. NaCl ... 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

For an atom in a solid, vibratory motion involves potential energy as well as kinetic ener, and both modes will contribute a term l/2kT, resulting in an average total energy of 3kT. Thus, it is the entropy of mixing that forces the creation of a certain number of vacant lattice positions above 0.0 °K. Hence, vacancies are the natural resultof thermod5mamic equilibrium md not the result of accidental growth or sample preparation. 

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. 

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. 

Referred to the averaged eight-coordination of a B2 lattice, the equivalent separation, d8 =fd6 — (v/3/2)a. The ionic lattice energies of Table 5.3 are... 

Fig. XXIX-5.—Potential energy of an electron in a metallic crystal, for two different distances of separation, illustrating the decrease of average potential energy with decreasing lattice spacing.

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