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22 Materials
35 At 20 °C iron
adopts the bodycentered cubic
structure with
atoms of atomic
radius 0.124 nm.
Calculate the length
of the cube edge of
the iron unit cell.
The edge length of a unit cell can be determined by X-ray diffraction and the number of atoms
per unit cell can be deduced from knowledge of the type of lattice. Knowing the edge length, the
number of atoms per unit cell and molar mass of the element, the density of the unit cell can be
calculated as follows:
n Edge of unit cell = a pm = a × 10−10 cm
n Volume of unit cell = a3 × 10−30 cm3
n Mass of unit cell = number of atoms in the unit cell × mass of each atom = z × m
n Mass of each atom = molar mass/Avogadro constant = M/NA
n Density of the unit cell = mass of unit cell/volume of the unit cell
n Density of the unit cell = (z × M)/(a3 × NA × 10−30) g cm−3. If a is expressed in ppm, then the
density will be expressed in g cm−3. The density of the unit cell is the same as the density of
the substance.
Worked example
Niobium crystallizes in a body-centered cubic structure. The density of niobium is 8.55 g cm−3 and its molar
mass is 92.91 g mol−1. Calculate the atomic radius in ppm.
In a body-centred cubic structure, the number of atoms per unit cell is 2.
Density = (2 × 92.91)/(6.02 × 1023 × 10 −30 × 8.55) = 3.61 × 107 = 330.5 ppm
But in a body-centered cubic structure the diagonal is equal to four times the radius of the atom:
4r = √3a = √3 × 330.5; r = 143 ppm
■■Structure of simple ionic compounds
Structure of sodium chloride
Sodium chloride consists of sodium ions, Na+, and chloride ions, Cl−. The chloride ions are
larger than the sodium ions. The sodium ions are located at the corners and faces of a cube with
the chloride ions on the edge and in the middle of the cube. The lattice, known as the rock salt
structure, can be regarded as being composed of two face-centred cubic structures (one for each
type of ion) that overlap. There are six chloride ions around every sodium ion (and vice versa).
Hence, the coordination number of each ion is 6. The rock salt structure is described as showing
6 : 6 coordination. The ratio of 6 : 6 of sodium to chloride ions simplifies to 1:1, which agrees with
the formula, NaCl.
The unit cell of sodium chloride has 4 sodium ions and 4 chloride ions as calculated below:
1
n Number of sodium ions = 12 (at edge centres) × 4 + 1 (at body centre) × 1 = 4
1
1
n Number of chloride ions = 8 (at corners) × 8 + 6 (at face centres) × = 4
2
Hence the number of NaCl formula units per unit cell is 4. Most of the halides of alkali metals
(group 1) and oxides of alkaline earth metals (group 2) metals have the rock salt structure.
Caesium chloride structure
The caesium chloride unit cell consists of eight caesium ions in a simple cubic arrangement, with
a single chloride ion at the centre of the cube. The caesium chloride structure can be regarded as
being formed from two interlocking cubic arrangements of caesium and chloride ions.
This structure has an 8 : 8 coordination: each caesium ion is touching eight chloride ions and
each chloride ion is touching eight caesium ions.
The unit cell of caesium chloride has one caesium ion and one chloride ion as calculated below:
1
n Number of chloride ions = 8 (at corners) × 8 = 1
n Number of caesium ions = 1 (at the body centre) × 1 = 1
Hence, the number of CsCl formula units per unit cell is 1.
Chemistry for the IB Diploma, Second Edition © Christopher Talbot, Richard Harwood and Christopher Coates 2015