Question

Given,$Na$ and $Mg$ crystallize in BCC and FCC type crystals respectively, then the number of atoms of $Na$ and $Mg$ present in the unit cell of their respective crystal is:
a) $4$ and $2$
b) $9$ and $14$
c) $14$ and $9$
d) $2$ and $4$

Answer 1

Hint: Atoms in the body centered cubic lattice are present in the eight corners and in the body centre whereas atoms in the face centered lattice are present in the eight corners and in the centre of the six faces.

Complete step by step answer:
In a body centered cubic lattice the atoms are placed in the eight corners and one atom is present in the centre of the body.
The packing efficiency of body centered cubic lattice is $68\% $. The contributions of atoms in the body centered cubic lattice can also be determined. The contribution of each corner atom is one-eighth and the contribution of the atom present in the body centre is one.
$
   \Rightarrow no.of atoms \times contribution \\
   \Rightarrow 8 \times \dfrac{1}{8} + 1 \times 1 \\
   \Rightarrow 2 \\
$
Hence the total number of $Na$ will be $8corners + 1bodycentre = 9atoms$.
In a face centered cubic lattice the atoms are placed in the eight corners and six atoms are in the centre of the six faces of the cube.
The packing efficiency of body centered cubic lattice is $74\% $. The contributions of atoms in the body centered cubic lattice can also be determined. The contribution of each corner atom is one-eighth and the contribution of the atom present in face centre is half.
$
   \Rightarrow no.of atoms \times contribution \\
   \Rightarrow 8 \times \dfrac{1}{8} + 6 \times \dfrac{1}{2} \\
   \Rightarrow 4 \\
$
Hence the total number of $Mg$ will be $8corners + 6facecentre = 14atoms$.
So the correct option is (b).

Note:
As the packing efficiency of a crystal lattice increases, the stability and availability of such crystal lattices increases in nature. As compared to BCC more FCC lattices are available in nature. Also FCC lattices have tetrahedral and octahedral voids.