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

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:

& \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).