To determine the formula of the compound, we need to understand the relationships between the number of atoms (or ions) in the hexagonal close-packed (hcp) structure and the number of tetrahedral voids available, along with how many of these voids are occupied by the atoms of element
X.
In a hexagonal close-packed (hcp) structure, each atom of element
Y contributes to the closepacked structure. For convenience, we can assume that there are 6 atoms of
Y in the hcp arrangement. The reason for choosing 6 is because it's a multiple that simplifies calculations in the hcp lattice where each atom is surrounded by 12 others, but each atom itself is shared among multiple unit cells. Nonetheless, the exact number chosen for
Y is irrelevant to the ratio we are trying to calculate, as the ratio of tetrahedral voids to atoms in an hcp structure remains constant regardless of the number of atoms considered.
In a close-packed structure, the number of tetrahedral voids is twice the number of atoms present. Thus, if we have
6Y atoms in the hcp structure, we have
2×6=12 tetrahedral voids.
Given that atoms of element
X occupy one-third of the tetrahedral voids, the number of
X atoms occupying the tetrahedral voids is
‌×12=4.
Thus, we have
6Y atoms and
4X atoms. To find the simplest whole number ratio, we divide by the smallest number of atoms present among the elements, which gives us:
For
Y:‌=3For
X:‌=2This results in a formula of
X2Y3, indicating that the correct answer is Option A.