What is the formula of the magnetic oxide of cobalt, used in recording tapes, that crystallizes with cobalt atoms occupying one-eighth of the tetrahedral holes and one-half of the octahedral holes in a closely packed aligned of oxide ions?

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

The 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, octahedron can be generated. The 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 as one octahedral hole per anion.

Assume \(x\) to be the number of cobalt ions present in the lattice. Hence, the number of tetrahedral holes is 2x.

One-eighth of the tetrahedral holes are occupied by cobalt ions.

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

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

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

Hence, the molecular formula of cobalt oxide is \({{\mathop{\rm Co}\nolimits} _{\frac{x}{4}}} + \frac{x}{2}{{\rm{O}}_x}\).

To get the whole number, multiply with \(\frac{4}{{3x}}\);then molecular formula of cobalt oxide is \({\rm{C}}{{\rm{o}}_3}{{\rm{O}}_4}\).

Thus the molecular formula of cobalt oxide is \({\rm{C}}{{\rm{o}}_3}{{\rm{O}}_4}\).