Question

The neon atom has a radius of 160pm. What is the edge of the unit cell of a face centered structure of neon?

Answer 1

Hint: In the FCC arrangement, there are eight atoms at corners of the unit cell and one atom centred in each of the faces. The atom in the face is shared with the adjacent cell. FCC unit cells consist of four atoms, eight eighths at the corners and six halves in the faces.

Complete step by step solution:
In the given question the neon atom has a radius of 160pm.
\[r = 160pm\]
In FCC there are 8 atoms on the eight corners of edges, there are six faces.
So total number of atom = $8 \times \dfrac{1}{8} + 6 \times \dfrac{1}{2} = 4$ atom

In the figure length of each edge of is ‘a’ then length of AB is written by

Pythagoras theorem: $c = \sqrt {{a^2} + {b^2}} $
$ \Rightarrow $AB = $\sqrt {{a^2} + {a^2}} $
$ \Rightarrow $AB= $\sqrt 2 $a
Let that radius of each atom is ‘r’ then distance of AB in terms of radius = 4r
$ \Rightarrow $AB = 4r
$ \Rightarrow $4r=$\sqrt 2 $a
$ \Rightarrow $$a = \dfrac{{4r}}{{\sqrt 2 }}$
Putting the value of ‘r’ which is given in the question
$
\Rightarrow a = \dfrac{{4 \times 160(pm)}}{{\sqrt 2 }} \\
\Rightarrow a = \dfrac{{640}}{{\sqrt 2 }} \\
\Rightarrow a = 453pm \\
 $

Hence, the edge of the unit cell of a face centred structure of neon is 453pm.

Note: In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.
There are three main varieties of these crystals:
1. Primitive cubic (abbreviated cP and alternatively called simple cubic)
2. Body-centred cubic (abbreviated cI or bcc)
3. Face-centred cubic (abbreviated cF or fcc, and alternatively called cubic close-packed or ccp)
In this given question FCC which means face centred cubic formula is used.