A compound containing zinc, aluminium, and sulfur crystallizes with a closest-packed aligned of sulfide ions. Zinc ions are found in one-eighth of the tetrahedral holes and aluminium ions in one-half of the octahedral holes. What is the empirical formula of the compound?

The coordination number of a sphere occupying tetrahedral hole is 4. They are smaller than the cubic or octahedral holes. The face centred cubic arrangement has both tetrahedral (8) and octahedral holes (4). A good example of tetrahedral structure is methane. Here, a carbon atom is attached to four equidistance hydrogen atoms placed at the corners of tetrahedrons.

Octahedron is defined as a polyhedron with 8 regular triangles. It is made by one sphere surrounded by 6 equal spheres. By connecting 6 centres of the surrounding spheres, an octahedron can be generated. A good example is sodium chloride. Here, octahedral holes are formed by chloride ions which are occupied by smaller sodium ions. In a face centred cube, there are four octahedral holes, with one octahedral hole per anion.

Assume \(x\)to be the number of oxide ions present in the lattice.

One-eighthof the tetrahedral holes are occupied by zinc ions.

The mathematical representation is as follows: \(\frac{1}{8} \times 2x = \frac{x}{4}\)

One half of the octahedral holes are occupied by aluminium ions.

The mathematical representation is as follows: \(\frac{1}{2} \times x = \frac{x}{2}\)

Hence, the molecular formula of the compound having zinc, aluminium, and sulfur \({\rm{Z}}{{\rm{n}}_{\frac{x}{4}}}{\rm{A}}{{\rm{l}}_{\frac{x}{2}}}\;{{\rm{S}}_x}\)

To get the whole number, multiply with ; then the molecular formula of is \({\rm{ZnA}}{{\rm{l}}_2}\;{{\rm{S}}_4}\).

Thus, the molecular formula of the compound comprising zinc, aluminium, and sulfur is \({\rm{ZnA}}{{\rm{l}}_2}\;{{\rm{S}}_4}\).