Explain three-dimensional close packing from two-dimensional square close packed layers.

A metal $\mathrm{M}$ forms hexagonal close-packed structure. The total number of voids in $0.02 \mathrm{mol}$ of it is _____ $\times {10}^{21}$ (Nearest integer)

(Given ${\mathrm{N}}_{\mathrm{A}}=6.02\times {10}^{23}$)

In a face centered cubic (fcc) lattice, the position which represents an octahedral hole is

The vacant space in bcc lattice unit cell is:

A two dimensional solid is made by alternating circles with radius $\mathrm{a}$ and $\mathrm{b}$ such that the sides of the circles touch. The packing fraction is defined as the ratio of the area under the circles to the area under the rectangle with sides of length $\mathrm{x}$ and $\mathrm{y}.$

The ratio $\mathrm{r}=\mathrm{b}/\mathrm{a}$ for which the packing fraction is minimized is closest to: