Question

There is a collection of crystalline substances in a hexagonal closed packing, if the density of matter is $2.6{\text{ g/c}}{{\text{m}}^3}$ , what would be the average density of matter in collection? What fraction of shape is actually unoccupied?

Answer 1

First use the formula of density of matter which is given by-
Density of matter= Packing fraction × Total density. Now, we know that the packing efficiency of the HCP structure is$74\%$. So put this value and the given value in the formula to find the density of matter. Then to find the fraction of unoccupied space we subtract occupied space from the total space.

Complete step by step answer:
Given,
The collection of crystalline substances is in Hexagonal close packing whose total density=$2.6{\text{ g/c}}{{\text{m}}^3}$
We have to find the average density of matter in the collection and the fraction of space which is unoccupied.
We know that density of matter is given by-
Density of matter= Packing fraction × Total density-- (i)
Here we have to know the packing fraction.
We know that HCP is arranged in ABA type of arrangement.
The packing fraction/efficiency is the fraction of total space occupied. $74\%$ . And We know that the packing efficiency of the HCP structure is $74\%$. So on using this value and the given value in eq. (i) we get,
Density of matter= $\dfrac{{74}}{{100}} \times 2.6$
On solving the equation we can write
Density of matter=$\dfrac{{1924}}{{100}}$
On division we get,
Density of matter=$1.924{\text{ g/c}}{{\text{m}}^3}$
Now we see that packing efficiency of the HCP structure is $74\%$. Which mean out of $100$, $74$ fraction of the space is occupied.
So to find the fraction of unoccupied space we subtract occupied space from the total space.
$\Rightarrow$ Empty space= $100 - 74$
On solving we get,
$\Rightarrow$ Empty space=$26$
Hence we can say that $26\%$ of the space is unoccupied.
Answer- Density of matter=$1.924{\text{ g/c}}{{\text{m}}^3}$ and $26\%$ of the space is unoccupied.

Note:
The properties of HCP are-
-In this structure, a sphere present in one layer is in contact with six spheres of its own layers, three spheres of the layer above and three spheres of the layer below it.
-Coordination number of each sphere in this type of arrangement is twelve.
-In this packing, the top layer cannot have coordination number twelve as the top layer has no sphere above it.