Question

The incorrect statement for cubic close packed (ccp) three dimensional structure is:

• A. The number of the nearest neighbours of an atom present in the topmost layer is 12.
• B. The packing efficiency of atom is $74\%$.
• C. The number of octahedral and tetrahedral voids per atom are $1$ and $2$, respectively.
• D. The unit cell edge length is $2\sqrt{2}$ times the radius of the atom

Answer 1

The correct option is A The number of the nearest neighbours of an atom present in the topmost layer is 12.


(a) Nearest neighbour in the topmost layer of ccp structure is 9 thus, incorrect.

(c) Tetrahedral voids = $2$
Octahedral voids = $1$ per atom thus, correct

(d) Edge length, $a=\frac{4}{\sqrt{2}}r=2\sqrt{2}r$
Explanation: Edge length = $a$,
Radius = $r$
$A{C}^2=A{B}^2+B{C}^2$
$(4r{)}^2=a^2+a^2=2a^2$
$4r=\sqrt{2}a$
$\Rightarrow$ $r=\frac{\sqrt{2}a}{4}=\frac{a}{2\sqrt{2}}$
$\Rightarrow$ $a=2\sqrt{2}r$

(b) In ccp structure, number of spheres is $4$.
Hence, volume of $4$ spheres = $4\left(\frac{4}{3}\pi r^3\right)$
Total volume of unit cell = $a^3$ = $(2\sqrt{2}r{)}^3$
$\%$ of packing efficiency:
= $\frac{Volumeof4spheres}{Volumeofunitcell}$ = $\frac{4\left(\frac{4}{3}\pi r^3\right)}{(2\sqrt{2}r{)}^3}\times 100$
= $74.05\%=74\%$.
Thus, (b) is correct.