Question

A metal crystallizes into two cubic phases, face centered cubic(fcc) and body centered cubic (bcc), whose unit cell lengths are $3.5\,{A^0}$ and $3.0\,{A^0}$ respectively. The ratio of densities of fcc and bcc is;
A) $1.259:1$
B) $1:1.259$
C) $3:2$
D) $1.142:1$

Answer 1

A body-centred cubic (bcc) unit cell has an atom at each of its corners and also one atom at its body centre. Thus, in a body-centered cubic (bcc) unit cell is 2. A face-centred cubic (fcc) unit cell contains atoms at all the corners and at the centre of all the faces of the cube. It can be seen that each atom located at the face-centre is shared between two adjacent unit cells and only 1/2 of each atom belongs to a unit cell. Thus, in a body-centered cubic (bcc) unit cell is 2.

Complete step by step answer:
Given that,
Cell length of face centred cubic is $3.5\,{A^0}$
Cell length of Body centred Cubic is $3.0\,{A^0}$
Now we have to calculate densities of the fcc and bcc.
As we know that
Density of a unit cell is given by the formula;
Density $\left( \rho \right) = \dfrac{{{Z_{eff}} \times {M_w}}}{{{a^3} \times {N_A}}}$
Where ${Z_{eff}}$ is the number of atoms present in one unit cell,${M_w}$is the mass, ${a^3}$ is the volume of a unit cell and ${N_A}$ is the number of molecule.
For face centred cubic, the number of atoms ${Z_{eff}}$ present in one unit cell is 4 while in case of body centred cubic the number of atoms ${Z_{eff}}$ present in one unit cell is 2.
For face centred cubic;
${\rho _{fcc}} = \dfrac{{4 \times {M_w}}}{{{{\left( {3.5 \times {{10}^{ - 8}}} \right)}^3} \times {N_A}}}$
For body centred cubic,
${\rho _{bcc}} = \dfrac{{2 \times {M_w}}}{{{{\left( {3 \times {{10}^{ - 8}}} \right)}^3} \times {N_A}}}$
$\begin{array}{c} \dfrac{{{\rho _{fcc}}}}{{{\rho _{bcc}}}} = \dfrac{{4 \times {{\left( 3 \right)}^3}}}{{{{\left( {3.5} \right)}^3} \times 2}} = \dfrac{{1.26}}{1}\\\ = 1.26:1 \end{array}$

Hence option A is correct.

Note: The efficiency of Packing in Body Centred Cubic Structures is 68% while in case of face centred cubic structures the packing efficiency is 74%.