Question

Close packing is maximum in the crystal lattice of
A. Face-centred cubic
B. Body-centred cubic
C. Simple cubic
D. Primitive cube

Answer 1

Hint: The close packing of the elements means there is a little or insignificant amount of space between the atoms. The small amount of space or close packing is observed in the case of solid-state matter and that proves the importance of the packing structure as the compound is present in the crystalline state as a result of the packing.

Complete step-by-step answer:Close packing depends on the packing efficiency of the specific crystal lattice. There are three types of lattices, the simple cubic lattice, face-centred lattice and body-centred lattice. These three lattices have their specific sides of each cubic structure according to atoms present in the vertices of the cube and specific location inside the three-dimensional structure of the lattice.
Simple cubic lattice is the one in which each of the sides of the cube denoted by $a$ is twice the radius of the atoms involved. This proves that the side will be $a = 2r$. The volume of each atom can be calculated as $\dfrac{4}{3}\pi {r^3}$ and the volume of the single cubic cell is $8{r^3}$. Therefore, the packing efficiency for the simple cubic lattice is $\dfrac{{\dfrac{4}{3}\pi {r^3}}}{{8{r^3}}} \times 100$ and that is equal to $52.4\% $.
Face-centred cubic lattice is formed as the specific lattice structures in which there is an atom at the centre of each of the faces of the cube. The side of the cube is denoted by $a$ and the radius is related to each side of the $FCC$ lattice as \[r = \dfrac{1}{{\sqrt 8 }}a\]. The volume of each of the atoms can be calculated as $\dfrac{8}{3}\pi {r^3}$ and the volume of the single cubic cell is $8\sqrt 8 {r^3}$. This is why the packing efficiency of the face-centred cubic lattice will be $\dfrac{{\dfrac{8}{3}\pi {r^3}}}{{8\sqrt 8 {r^3}}} \times 100$ which will be equal to $74\% $.
Body-centred cubic lattice is formed as the specific lattice structure in which there is a single atom at the centre of the whole unit cubic cell. The side of cube is denoted by $a$ and the radius is related to each side of the $BCC$ lattice as $r = \dfrac{{\sqrt 3 }}{4}a$ , The volume of each of the atom can be calculated as $\dfrac{8}{3}\pi {r^3}$ and the volume of the single cubic cell is $\dfrac{{64{r^3}}}{{3\sqrt 3 }}$. This is why the packing efficiency of the body-centred cubic lattice will be $\dfrac{{\dfrac{8}{3}\pi {r^3}}}{{\dfrac{{64{r^3}}}{{3\sqrt 3 }}}} \times 100$ which will be equal to $68.04\% $.
Therefore, the highest packing efficiency is observed in the case of the face-centred cubic lattice structure which has the close packing of around $74\% $. This is why the crystal lattice with the highest close packing is face-centred lattice structure.

Hence the correct option is (A).

Note:The packing efficiency of the lattice differs according to the placement of the atoms in the crystal structure. The proper packing is responsible for the structuring of the crystals when atoms are bound together in the vertices and specific regions in the cubic lattice structure of the crystal.