A compound has a rock type structure . The formula weight of $AB$ is $6.023Y$ ,and closest $A-B$ distance is ${{Y}^{\dfrac{1}{3}}}$ nanometre,where $Y$ is an arbitrary number,find the density of the lattice.
a.) $5\,Kg/{{m}^{3}}$
b.) $5\,\times {{10}^{-3}}Kg/{{m}^{3}}$
c.) ${{10}^{-3}}Kg/{{m}^{3}}$
d.) $15\,\times {{10}^{-2}}\,Kg/{{m}^{3}}$

Hint: This question involves knowledge of properties of rock salt structure of solids.Its edge length,number of atoms per unit cell in rock salt type structure or face centered cubic lattice is to be known.
The edge length in rock salt structure is twice the closest distance between the constituent atoms of the lattice.
The number of atoms per unit cell is $4$.

Complete Solution :
Density of a lattice is defined as the mass of the unit cell divided by volume of the unit length.
Mass of unit cell = $\dfrac{n\times formula\,\,weight\,of\,lattice}{{{N}_{A}}}$,where $n$ is number of particles in one unit cell and ${{N}_{A}}$ is Avogadro’s number = $6.023\times {{10}^{23}}$.
Density = $\dfrac{mass\,of\,unit\,cell}{edge\,length}$
Formula weight = $6.023Y$
closest $A-B$ distance is ${{Y}^{\dfrac{1}{3}}}$ nanometre
Edge length in rock salt structure = $2$${{Y}^{\dfrac{1}{3}}}$

Putting the values in the formula of density,
$\Rightarrow \dfrac{4\times 6.023Y}{6.023\times {{10}^{23}}}\times \dfrac{1}{{{\left( 2\times {{Y}^{\dfrac{1}{3}}}\times {{10}^{-9}} \right)}^{3}}}$
$\Rightarrow \dfrac{4\times Y}{{{10}^{23}}}\times \dfrac{1}{8\times Y\times {{10}^{-27}}}$
$\Rightarrow \dfrac{1}{2\times {{10}^{-4}}}$
$\Rightarrow 5kg/{{m}^{3}}$
So, the correct answer is “Option A”.

Note: Face centered cubic is also known as cubic close packing.
Other than this there are other types of packing structures and lattice and their unit cells like simple cubic close packing and body centered cubic and hexagonal close packing of crystal structures.
- When metal atoms are arranged with spheres in one layer directly above or below spheres in another layer, the lattice structure is called simple cubic.
- Some metals crystallize in an arrangement that has a cubic unit cell with atoms at all of the corners and an atom in the center, as shown in This is called a body-centered cubic solid.
- The unit cell of a lattice consists of both cation and anion arranged in different manner to fulfill pace efficiency and gain stability.