Page 29 - Materials Chemistry, Second Edition

16 2 Solid-State Chemistry

2

NMZ cation Z anion e n
ð1Þ U= 1
r o 4pe o r o
1
where: N ¼ Avagadro’s number (6.02 10 23 molecules mol )
Z cation, anion ¼ magnitude of ionic charges;
r 0 ¼ average ionic bond length
19
e ¼ electronic charge (1.602 10 C)
10 2 1 1
4pe 0 ¼ permittivity of a vacuum (1.11 10 C J m )
M ¼ the Madelung constant (see text)
n ¼ the Born exponent; related to the corresponding closed-shell electronic
configurations of the cations and anions (e.g., [He] ¼ 5; [Ne] ¼ 7; [Ar] or
10 10 10
[3d ][Ar] ¼ 9; [Kr] or [4d ][Kr] ¼ 10; [Xe] or [5d ][Xe] ¼ 12)
It is noteworthy that the calculated lattice energy is quite often smaller than
the empirical value. Whereas the ions in purely ionic compounds may be accurately
treated as hard spheres in the calculation, there is often a degree of covalency in
the bonding motif. In particular, Fajans’ rules describe the degree of covalency as
being related to the charge density of the cation and the polarizability of the anion.
In general, polarizability increases down a Periodic Group due to lower electrone-
gativities, and valence electrons being housed in more diffuse orbitals thus experien-
cing a much less effective nuclear charge. To wit, a compound such as LiI would
exhibit a significant degree of covalent bonding due to the strong polarizing potential
of the very small cation, and high polarizability of iodide. This is reflected in its lower
melting point (459 C) relative to a more purely ionic analogue, LiF (m.p. ¼ 848 C).

The Madelung constant appearing in Eq. 1 is related to the specific arrangement of
ions in the crystal lattice. The Madelung constant may be considered as a decreasing
series, which takes into account the repulsions among ions of similar charge, as well
as attractions among oppositely charged ions. For example, in the NaCl lattice
illustrated in Figure 2.1, each sodium or chloride ion is surrounded by six ions of
opposite charge, which corresponds to a large attractive force. However, farther
away there are 12 ions of the same charge that results in a weaker repulsive
interaction. As one considers all ions throughout the infinite crystal lattice, the
number of possible interactions will increase exponentially, but the magnitudes of
these forces diminishes to zero.
Ionic solids are only soluble in extremely polar solvents, due to dipole–dipole
interactions between component ions and the solvent. Since the lattice energy of the
crystal must be overcome in this process, the solvation of the ions (i.e., formation of
[(H 2 O) n Na] ) represents a significant exothermic process that is the driving force for
þ
this to occur.
2.2.2. Metallic Solids

Metallic solids are characterized by physical properties such as high thermal and
electrical conductivities, malleability, and ductility (i.e., able to be drawn into a thin
wire). Chemically, metals tend to have low ionization energies that often result in