Chemistry - A Molecular Science

8.4

COORDINATION NUMBER AND GEOMETRY

(a)

(b)

(c)

(d)

Figure 8.6 Coordination numbers (CN) and geometries a) CN = 4 is tetrahedral b) CN = 6 is octahedral c) CN =8 d) CN = 12 (closest packed)

f

f

f

f

(a) (b)

Example 8.3 Unit cell of rutile (a) Ti atoms (blue) are situated

on the eight corners and in the

body center. The O atoms (red) labele

d with ‘f’ are in faces, while

those that are not labeled are tota

lly within the unit cell. (b) The

octahedral coordination of the central Ti atom. Note it is a distorted octahedron because the bonds to the two O atoms in the cell are longer than those to the

four O atoms in the faces.

An understanding of the unit cell of a crysta

lline solid also provides an understanding of

the local environment of each atom or ion. The

coordination number

(

CN

) and the

coordination geometry

of an ion or atom indicate the number and geometry of atoms or

ions that surround it in the crystal lattice, wh

ich indicates the nature of the bonding in a

crystal. Figure 8.6 shows the most comm

on coordination numbers adopted by atoms

(represented by the red spheres). Four particl

es coordinated to a central particle with

CN

=

4 generally exhibit tetrahedral geometry (Figure 8.6a). The atoms in a simple cubic unit cell have octahedral c

oordination geometry and

CN

= 6 (Figure 8.6b). Atoms in a body-

centered cubic unit cell have

CN

= 8 (Figure 8.6c). Each atom in a face-centered cubic

unit cell has

CN

= 12 (Figure 8.6d), which can be viewed as three planes of particles: the

top (green spheres) and bottom (blue spheres) planes with three particles each and one plane in the middle that contains the central atom and six particles (yellow spheres) that form an equatorial belt around it. The packing of equal sized particles in this geometry represents the tightest possible packing arrangement for spheres and is frequently described as

closest packed

.

Example 8.3

Rutile is a titanium oxide mineral that crystallizes in the tetragonal (a=b>c, α=

β=

γ=90

o) unit cell shown in the margin. What

is the formula of the titanium oxide

in rutile, and what are the coordination number and geometry of Ti in the mineral? Determine the number of each type of atom in the uc.

NTi

= eight atoms on the corners + one atom in the cell = (8)(

1 /^8

) + 1 = 2 Ti atoms

NO

= four atoms on faces + two atoms in the cell = (4)(

1 /^2

) + 2 = 4 O atoms.

Thus, the unit cell stoichiometry is Ti

O 2

. The formula of the oxide is the simplest whole 4

number ratio, which is TiO

, so there are two TiO 2

molecules in the unit cell. 2

All of the O atoms in the unit cell are nearest neighbors of the Ti atom

in the center, so its

coordination number is 6, and it

s coordination geometry is oct

ahedral (Figure b). Note that

the Ti atoms are arrayed in a body centered fa

shion, and the Ti atoms on the corners and

in the center are equivalent, so the coordi

nation number of all Ti atoms is six.

Chapter 8 Solid Materials

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