Question

Chromium metal crystallizes as a body-centered cubic lattice. The length of the unit cell edge is found to be $287pm$ . What would be the density (in $gc{m^{ - 3}}$ ) of chromium?
A. $3.40$
B. $5.20$
C. $12.10$
D. $7.30$

Answer 1

Hint: A unit cell which has a lattice point at the body centre in addition to the lattice point at every corner is known to be the body centered unit cell. Here, in this case, the particles that are present on the body diagonal touch each other.

Complete step by step answer:

In a body- centered cubic (BCC) crystal lattice structure, the nearest distance between two atoms is $\dfrac{{a\sqrt 3 }}{2}$ .
In a BCC crystal lattice structure, there are 8 atoms present at each corner of the unit cell with the contribution of each atom in the unit cell being one-eighth part. Along with it, there is one atom present at the center of the unit cell. Thus, the total number of atoms present in the BCC crystal lattice structure is:
$z = \left( {8 \times \dfrac{1}{8}} \right) + \left( {1 \times 1} \right) = 2$
The volume of the unit cell is calculated by taking the cube of the length of each side of the unit cell. The total mass of the unit cell is calculated by the product of the total mass of all the atoms that are present in the unit cell. Thus, the density can be related as:
$d = \dfrac{{z \times M}}{{{N_A} \times {a^3}}}$ …(i)
Where, $d = $ density ($g/c{m^3}$ )
$M = $ Molar mass
$a = $ length of the side of the unit cell or edge length
${N_A} = 6.023 \times {10^{23}}$ particles
As per the questions, the values of the parameters provided are as follows:
$M = 52g$
$z = 287pm = 2.87 \times {10^{ - 8}}cm$
Substituting these values in equation (i), we have:
$d = \dfrac{{2 \times 52}}{{6.023 \times {{10}^{23}} \times {{(2.87 \times {{10}^{ - 8}})}^3}}}$
Thus, on solving, we have the density of the chromium atoms as:
$d = 7.30g/c{m^3}$
Thus, the correct option is D. $7.30$ .

Note:
The ratio of volume occupied by the atoms in the BCC crystal lattice to that of the volume of the unit cell is quite intermediate in between that of the simple cubic and face-centered cubic crystal lattice. The packing efficiency of the BCC crystal structure is found to be $68\% $ which means the remaining $32\% $ is empty space or void inside the unit cell.