Question

What is the formula of a compound in which the element Y forms HCP lattice and of X occupies $1/{3^{rd}}$ of octahedral voids.

Answer 1

Hint: We know that the arrangement of a solid sphere with close packing which remains some space is called voids. Even in the closest packing of a sphere, there is some space left in between the spheres called interstitial sites.

Complete step by step answer: We have to remember that the packing of atoms or molecules in crystals requires several possible arrangements but the one in which maximum available space is occupied.
Close packing arrangements are classified into square packing and hexagonal packing.
In hexagonal arrangement occupied space is \[60.4\% \] and remaining \[39.6\% \] called voids.
In square packing, the arrangement occupied space is \[52.4\% \] and the remaining \[47.6\% \] called voids.
Let the number of atoms Y in hcp lattice is 6.
As the number of octahedral voids is double the number of atoms in close packing and the number of octahedral voids is eight.
As atoms X occupy $1/{3^{rd}}$ of octahedral voids, the number of atoms in the lattice is,
The number of atoms contributed to one unit cell is $\dfrac{1}{4} \times 8 = 2$
$\dfrac{1}{3} \times 2 = \dfrac{2}{3}$
The ratio of X and Y is $\dfrac{2}{3}:6 = 2:18$
The formula of the compound is${X_2}{Y_{18}}$.

Note:
In three dimensions it shows interstitial sites in close packing of the sphere. Consider a triangular or trigonal site. Also of the above sites shows two different sites.
Tetrahedral sites: if one sphere is placed on the three other spheres which are touching one other. Even four spheres touch one another some gaps available that know as the tetrahedral sites. In these sites, one sphere is surrounded by eight tetrahedral sites, and one site surrounded by four spheres. Coordination number is four and the ratio is \[2:1\]
Octahedral sites: The close packing of a sphere in hcp, as well as ccp arrangement, is known as octahedral sites. This shows six spheres attached in two forms. In these sites, one sphere is surrounded by six octahedral sites, and one site surrounded by six spheres. Hence the coordination number is six. Ratio is\[1:1\].