Question

Packing fraction of diamond if it has arrangement of carbon atoms similar to the arrangement in zinc sulphide is:
(A) 52%
(B) 90%
(C) 34%
(D) 68%

Answer 1

Packing fraction or the packing efficiency is the percentage of total space filled by the particles. Mathematically the formula of packing fraction is the ratio of all the spheres in the unit cell by the total volume of the unit cell. It is expressed in percentage. It can be calculated from the formula given below.
The equation for packing fraction being equal to:
Packing fraction = $\dfrac{{{\text{No}}{\text{. of atoms in the unit cell } \times \text{the volume of the atoms}}}}{{{\text{volume of the unit cell}}}}$

Complete Stepwise Solution:
The packing of the atoms referred to as here is called “diamond closed packing”.
A diamond has eight atoms in the unit cell,
Therefore,
Packing fraction = $\dfrac{{{\text{8 } \times \text{the volume of the atoms}}}}{{{\text{volume of the unit cell}}}}$
The volume of the atoms, considering that they are spherical in shape is = $\dfrac{4}{3}\pi {{\text{r}}^{\text{3}}}$ , where “r” is the radius of the sphere.
The volume of the unit cell, assuming that it has cubic shape = ${{\text{a}}^{\text{3}}}$
Putting these values in the equation for packing fraction we get,
Packing fraction = $\dfrac{{8 \times \dfrac{4}{3}\pi {{\text{r}}^{\text{3}}}}}{{{{\text{a}}^{\text{3}}}}}$
The radius of the ${\text{r = }}\dfrac{{\text{a}}}{{\text{8}}}\sqrt {\text{3}}$
Putting the value of the radius of the atom in the above equation for packing fraction we get,
Packing fraction = $\dfrac{{8 \times \dfrac{4}{3}\pi {{\text{r}}^{\text{3}}}}}{{{{\text{a}}^{\text{3}}}}}$ = $\dfrac{{8 \times \dfrac{4}{3}\pi {{\left( {\dfrac{{\text{a}}}{{\text{8}}}\sqrt {\text{3}} } \right)}^{\text{3}}}}}{{{{\text{a}}^{\text{3}}}}} = \sqrt 3 \dfrac{\pi }{{16}} = 0.34$ = 34%
Hence, the Packing fraction of diamond if it has an arrangement of carbon atoms similar to the arrangement in zinc sulphide is 34% and the correct answer is option C.

Note:
The packing fraction is the highest for the face centred cubic (FCC) or the cubic close packing (CCP) or the hexagonal close packing (HCP) structure and is equal to 74 %. That means only 26% of the space in the lattice remains vacant while in the diamond close packed arrangement, 66 % of the space remains vacant which has the lowest packing efficiency of atoms.