Question

A metal crystallises into two cubic phases, face center cubic ($fcc$) and body centered cubic ($bcc$), whose unit cell lengths are $3.5\mathring{A}$ and $3.0\mathring{A}$, respectively. The ratio of densities of ($bcc$) is:

• A. 1.259 : 1
• B. 1 : 1.259
• C. 3 : 2
• D. 1.142 : 1

Answer 1

The correct option is A 1.259 : 1

Theoretical density of crystal, $\rho =\frac{nM}{N_o{a}^3}g/c{m}^3$

For fcc, $n=4;a=3.5\times {10}^{-8}cm$

So, ${\rho }_1=\frac{4\times M}{N_o\times (3.5{)}^3\times ({10}^{-}8{)}^3}g/c{m}^3$.....(i)

For bcc, $n=2;a=3\times {10}^{-8}cm$

So, ${\rho }_2=\frac{2\times M}{N_o\times (3{)}^3\times ({10}^{-}8{)}^3}g/c{m}^3$.....(ii)

So $\left(i\right)/\left(ii\right)$

$\Rightarrow \frac{\rho \left(fcc\right)}{\rho \left(bcc\right)}=\frac{4\times (3{)}^3}{2\times (3.5{)}^3}=\frac{2\times 27}{42.875}=\frac{1.259}{1}$

Hence, Option "A" is the correct answer.