Question

One of the various manganese oxides crystallizes with a cubic unit cell that contains manganese ions at the corners and in the center. Oxide ions are located at the center of each edge of the unit cell. What is the formula of the compound?

Short answer

The molecular formula of manganese oxide is \({\rm{M}}{{\rm{n}}_2}{{\rm{O}}_3}\).

Step-by-step solution

Definition of coordination number of sphere  

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 a tetrahedral structure is methane. Here, carbon atom is attached to four equidistance 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. A good example is sodium chloride. Here, the 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.

Finding the molecular formula

Manganese is present at the eight corners of the cube, and the number of atoms of manganese atoms in the cubic unit cell is \(\frac{1}{8} \times 8 = 1\)

Manganese is present at the center of the cube. So, the number of atoms of manganese atoms in the cubic unit cell is 1.

Therefore, the number of manganese atoms present in a cubic unit cell is 2.

Hence, the ratio of manganese and oxide ions is 2:3. Therefore, the molecular formula of rubidium iodide is \({\rm{M}}{{\rm{n}}_2}{{\rm{O}}_3}{\rm{. }}\)