Spherical ion model

As discussed above, whereas for open structures the zero-pressure structures are calculated more accurately with the distorted ion model than with the spherical ion model, for close-packed systems the zero-pressure structures are calculated just as well with the spherical ion model. In the same way, the distorted ion model leads to improvements in the compression at high pressures for open structures but not for close-packed structures. Although the agreement with experiment is reasonable for calculated changes in volume with pressure, the calculated structures are generally less compressible than observed experimentally (i.e.,the bulk moduli are too high). 

The calculated enthalpies for silica in the quartz and stishovite phases are shown in Figure 3 as a funetion ofpressure. The stishovite structure beeomes more stable than the quartz strueture at 3.5 GPa with the distorted ion model, and at 21 GPa with the spherical ion model. In comparison, the experimental zero temperature transition pressure for the quartz to stishovite phase transition is estimated to be 5.5 GPa from thermodynamic data [53], and the transition pressure for the similar cristobalite to stishovite phase transition is caleulated to be 6 GPa by periodie Hartree-Fock methods [54]. The non-spherical distortions improve the modeling of this phase transition by stabilizing stishovite with respeet to quartz the greater stabilization ofstishovite occurs because the distortions strengthen three bonds per anion in stishovite, and only two bonds per anion in quartz (the bonds are significantly covalent in both structures, as shown above in the plots of the electron density distributions). 

Table 5.1. Gaseous monomeric alkalimetal halides, MX(g) experimental electric dipole moments, /Tgi electric dipole moments predicted by the spherical ion model, /rel(calc) equihbiium bond distances, Rg vibrational wavenumbers, ox, dissociation energies at zero K, Dq reduced masses, /xm force constants,/r dissociation energies calculated from the spherical ion model according to equation (5.16a), Do(calc).

How do these dipole moments compare with those calculated for the spherical ion model In the third column of Table 5.1 we list the electric dipole moments calculated from equation (5.6) with net ionic charges equal to e and the experimentally determined bond distances ... 

It is seen that all the calculated dipole moments are higher than the true, experimental values. The average deviation between estimate and experiment amounts to 32% of the experimental value, so it is clear that the charge distribution in alkali metal halide molecules deviates significantly from that predicted from the spherical ion model. 

We shall now return to the spherical ion model and use it to construct the potential energy curve of an ion pair. 

When the internuclear distance is reduced below 1000 pm, the energy continues to fall. The approach of the ions is, however, stopped at some point, presumably because the spherical ions repel each other at close range, much like two rubber balls when they are pressed together. The close range repulsion between ions is a topic of which we do not know very much today, even less was known when the spherical ion model was developed in the early 1920s. 

Problem 5.4 LiH is a colorless solid at room temperature. The melting point is about 700 and the melt conducts electricity. Both the solid and the hquid are therefore believed to consist of Li" " and H (hydride) ions. The electric dipole moment of the gaseous LiH is 5.88 D, the dissociation energy at zero K is 234 kJ moP, the bond distance is 160 pm, and the vibrational wavenumber 1406 cm. Calculate the electric dipole moment from the spherical ion model. Show that the dissociation energy calculated from the spherical ion model is in good agreement with the experimental if the exponent in the Born repulsion term is reduced from n = 10 to 5. Is there any reason... 

Problem 5.5 The gaseous HF molecule has a dipole moment of 1.83 Debye. The dissociation energy Do = 566 kJ mopi, the bond distance is 92 pm and the vibrational wavenumber 4138 cm . Use the spherical ion model to estimate the dipole moment and the dissociation energy (in kJ moP ) at 0 K. Would you describe HF as an ionic molecule ... 

Our calculation based on the spherical ion model thus indicates that M-X bond distances in all dimers should be 5% longer than in the monomers. Quantum chemical calculations, on the other hand, indicate an elongation by about 7%. 

We have seen that the electric dipole moments of the gaseous alkali metal halides estimated from the spherical ion model are significantly larger than the experimental values. The assumption that the ions remain spherical as they are brought close to each other is, in fact, an unreasonable one particularly for the anions which are easily polarized. 

Dipole moments calculated from the polarizable ion model, equation (5.32), using the polarizabilities listed in Table 3.4 and experimentally determined bond distances, reproduce the experimental dipole moments with an average deviation of only 5% as compared to 32% for the spherical ion model [5]. 

Problem 10 J Assume that a metal dihaUde molecule is ionic, i.e. that it consists of a cation and two X anions. Assume that the molecule is linear and denote the M-X bond distance by R. Use the spherical ion model to show that the potential energy curve may be written as... 

The spherical ion model and the mean bond energies of the Group 2 and 12 metal dichlorides... 

As long as we use the spherical ion model, the Coulomb interaction energy will obviously be at a minimum, and the molecule most stable, if the distance between the anions is as large as possible, i.e. if the molecule is linear. Would this stiU be so if we allow the ions to become polarized ... 

Throughout our discussion, we refer to ionic lattices, suggesting the presence of discrete ions. Although a spherical ion model is used to describe the structures, we shall see in Section 6.13 that this picture is unsatisfactory for some compounds in which covalent contributions to the bonding are significant. Useful as the hard sphere model is in describing common crystal structure types, it must be understood that it is at odds with modem quantum theory. As we saw in Chapter 1, the wavefunction of an electron does not suddenly drop to zero with increasing distance from the nucleus, and in a close-packed or any other crystal, there is a finite electron density everywhere. Thus all treatments of the solid state based upon the hard sphere model are approximations. 

We now return to the spherical ion model. According to the Bom-Lande model, the energy for one mole of a pair of spherical ions and X at a distance of R is given by... 

Fig. 2. Comparison of counter-ion concentrations about charged rod and spherical ion models.

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