Formula of a Compound and Number of Voids Filled

When particles are close-packed resulting in either ccp or hcp structure, two types of voids are generated. While the number of octahedral voids present in a lattice is equal to the number of close packed particles, the number of tetrahedral voids generated is twice this number. In ionic solids, the bigger ions (usually anions) form the close packed structure and the smaller ions (usually cations) occupy the voids. If the latter ion is small enough then tetrahedral voids are occupied, if bigger, then octahedral voids. All octahedral or tetrahedral voids are not occupied. In a given compound, the fraction of octahedral or tetrahedral voids that are occupied, depends upon the chemical formula of the compound.

Atoms of element B form hcp lattice and those of the element A occupy \(\frac{2}{3}\)^rd of tetrahedral voids. What is the formula of the compound formed by the elements A and B?

A₄B₃

Atoms of element B form hcp lattice

FCC is the unit cell in hcp lattice

Total no. of atoms of element B = \(\frac{1}{8}\) x 8 + \(\frac{1}{2}\) x 6 = 1 + 3 = 4

Element A occupy \(\frac{2}{3}\)^rd of tetrahedral voids

Total number of T-voids = 8

No. of elements of A = \(\frac{2}{3}\) x 8 = \(\frac{16}{3}\)

A : B = \(\frac{16}{3}\) : 4

or 

A : B = 16 : 12

or

A : B = 4 : 3

Hence, the chemical formula of the compound is A₄B₃.