Question

A compound forms a hexagonal close−packed structure. What is the total number of voids in $0.5mol$ of it? How many of these are tetrahedral voids?

Answer 1

First, we need to know what hexagonal close- packed structure is. A hexagonal close packing structure is made up of alternating layers of spheres or atoms stacked in a hexagon, with one extra atom in the middle. A triangular layer of atoms is sandwiched between these two hexagonal layers, and the atoms in this layer fill the tetrahedral gaps generated by the top and bottom layers.

Complete answer:
From the question it is given that,
Compound has a hexagonal close packed structure.
To find: Total number of voids = ? Number of tetrahedral voids = ?
We know that,
Number of atoms in closed packaging $= 0.5mol$
And 1 mole of compound has ${\text{6}}{\text{.022}} \times {\text{1}}{{\text{0}}^{23}} = 3.011 \times {10^{23}}$
Now, number of tetrahedral voids = ${\text{2}} \times$Number of atoms in close packaging
Putting the corresponding values, we now get,
Number of tetrahedral voids ${\text{ = 2}} \times {\text{3}}{\text{.011}} \times {\text{1}}{{\text{0}}^{23}} = 6.022 \times {10^{23}}$
The number of octahedral voids will be equal to the number of atoms in closed packaging.
Hence, the number of octahedral voids is given by,
Number of octahedral voids ${\text{ = 3}}{\text{.011}} \times {\text{1}}{{\text{0}}^{23}}{\text{ }}$
And,
Total number of voids ${\text{ = }}$ octahedral voids $+$ tetrahedral voids
${\text{ = 6}}{\text{.022}} \times {\text{1}}{{\text{0}}^{23}} + 3.011 \times {10^{23}}$
$= 9.03 \times {10^{23}}$
Thus, the total number of voids in $0.5mol$ of the compound is $9.03 \times {10^{23}}$ of which ${\text{6}}{\text{.022}} \times {\text{1}}{{\text{0}}^{23}}$ are tetrahedral voids.

Note:
We should note that there are two types of voids involved in hexagonal close packing (HCP), namely octahedral voids and tetrahedral voids. We know that in a lattice, there are twice as many tetrahedral voids as there are close-packed particles. While the number of octahedral voids produced is the same as the number of tightly packed particles. The arrangement of particles in these voids depends on other factors too.