Question

\[CsBr\] crystallizes in a body – centered cubic lattice. The unit cell length is \[436.6pm\]. Given that atomic mass of \[Cs=133\] and that of \[Br=80\text{ amu}\] and Avogadro number being \[6.02\times {{10}^{23}}mo{{l}^{-1}}\]the density of \[CsBr\] is:
A. $42.5\,g/c{{m}^{3}}$
B. $0.425\,g/c{{m}^{3}}$
C. $8.5\,g/c{{m}^{3}}$
D. $4.25\,g/c{{m}^{3}}$

Answer 1

Hint: In a body centered cubic cell atom belongs entirely to one-unit cells. Since it is not shared by any other unit cell therefore, there are two atoms in a body centered cubic cell.
The density of a unit cell is given by the formula:
\[\begin{gathered}
 & \alpha =\dfrac{\text{Mass of unit cell}}{\text{Volume of unit cell}} \\
 & \alpha =\left( \dfrac{Z\times M}{{{a}^{3}}\times Na} \right)gc{{m}^{-3}} \\
\end{gathered}\]
Where Z = no of atoms in a unit cell
M = mass of each atom
Na = Avogadro’s no.
a = edge of the unit cell

Complete Solution :
We add the atomic mass of \[Cs\]and \[Br\]to get total mass.
\[\Rightarrow CsBr=133+80=213gmo{{l}^{-1}}\]

Density of a unit cell is calculated as:
\[\Rightarrow \alpha =\left( \dfrac{Z\times M}{{{a}^{3}}\times Na} \right)gc{{m}^{-3}}\]
Since, this is a body centred cubic cell, no. of atoms in unit cell (Z) is 2. Mass (M) is \[213gmo{{l}^{-1}}\]. Length of unit cell (a) is 436.6 pm and Avogadro’s number (Na) is \[6.02\times {{10}^{23}}mo{{l}^{-1}}\].

Substituting these values in the formula for density, we get:
\[Density=\dfrac{2\times 213}{{{\left( 436.6\times {{10}^{-10}} \right)}^{3}}\left( 6.022\times {{10}^{23}} \right)}\]
 \[\Rightarrow \dfrac{426}{{{10}^{-7}}\times 2628.52}\]
\[\Rightarrow 8.5g/c{{m}^{3}}\]
So, the correct answer is “Option C”.

Note: In a body centred unit cell, one atom is present at the centre of the structure and every corner of the cube has atoms.
The number of atoms in a BCC cell can be calculated as follows:
- \[8\text{ corners}\times \text{1/8 per corner atom = 8}\times \text{1/8}=1\text{ atom}\]
- 1 body centre atom = \[\text{1}\times \text{1=1 atom}\]
Therefore, total number of atoms per unit cell = 1 + 1 = 2 atoms.

Some other types of unit cell are:
- Primitive cubic unit cell (Atoms are present only at corners).
- Face centred unit cell (Atoms are present in all corners and an atom is present at the centre of every face of the cube).