Question

A metallic element crystallizes into a lattice containing a sequence of layers of ABABAB. Any packing of sphere leaves voids in the lattice. What percentage by volume of this lattice is occupied by voids?

Answer 1

Hint: Void: when a crystal lattice is formed by close packing of its constituting atoms, and hexagonal arrangement is formed, in its atoms creates some spaces between them which is known as void. They can be tetrahedral or octahedral void.

Complete step by step answer:
Now we will study about crystal lattice and packing of its atoms.
A crystal is defined as a solid material whose repeating units are arranged in a microscopic structure that gives rise to crystal lattice. The order which they follow is long range order which provides them definite shape and a sharp melting point.
Packing of atoms: The atoms of lattice are arranged in such a way that their maximum area is close to each other and shows the tight packing of the lattice they are occupying.
Packing of atom can occur in two dimension or three dimension:
First we will discuss about 2 dimension packing;
In 2-D packing, is obtained by combining rows of spheres together.
First layer can be represented as A, and another layer of the same atom is placed over it.This type of arrangement is known as square close packing.
Another arrangement is hexagonal close packing in which one layer of A atoms is laid over by another layer of B atom over them.
Now about 3-D packing;
It can be three dimensional square close packing when layer A is laid over by another layer of A atoms in the same orientation, exactly above it.
It can be three dimensional hexagonal close packing when layer A is laid over by another layer of B atoms creating empty spaces between them known as voids.
According to the question the sequence of lattice is ABABAB, and we learnt that this arrangement is shown by Hexagonal close packing.
Hexagonal lattices have packing fraction of = $0.74\% $
Converting it percentage = $0.74\% = \dfrac{{0.74}}{{100}} = 74$
Thus the void percentage of the lattice will be: $100 - 74 = 26$.



Note:
The packing efficiency of the cubic arrangements has already been calculated:
fccp = $74\% $
hcp = $74\% $
bcc = $68\% $
simple cubic = \[52.4\% \]