The unit cell of aluminum is a cube with an edge length of 405 pm. The density of aluminum is $2.70gc{m}^{-3}$. What is the structure of the unit cell of aluminum?

The correct option is B

Face-centered cubic cell

The explanation for correct option:-

b) Face-centered cubic cell

Density of unit cell = $\rho$= $2.70gc{m}^{-3}$

The formula of density of unit cell = $\rho$= $\mathrm{Z}\times \mathrm{M}/{\mathrm{N}}_{\mathrm{a}}\times {\mathrm{a}}^3$

where $\rho$= density = $2.70gc{m}^{-3}$

Z is no of atoms in the unit cell

M is the molar mass

N_a = Avogadro's number

a³ is the volume of unit cell

a = edge length = 405 pm = $405\times {10}^{-10}$$cm$

The molar mass of Aluminum = $26g/mol$

N_a = $6.022\times {10}^{23}$

$\mathrm{Z}=\mathrm{\rho }\times {\mathrm{N}}_{\mathrm{a}}\times {\mathrm{a}}^3/\mathrm{M}$

$\mathrm{Z}=2.70\times (6.022\times {10}^{23})\times {\left(405\right)}^3\times {10}^{-30}/26$

$Z=4$

For a face-centered cubic cell, no of atoms in a unit cell (Z) is equal to 4.

The explanation for incorrect options:-

a) Body-centered cubic cell

• In the case of a body-centered cubic cell, the no of atoms in a unit cell (Z) is equal to 2. So this option is incorrect.

c) End-centered cubic cell

• In the case of an end-centered cubic cell, the no of atoms in a unit cell (Z) is equal to 2. This option is also incorrect

d) Simple cubic cell

• In the case of a simple cubic cell, the no of atoms in a unit cell (Z) is equal to 1. This option is also incorrect.

Therefore the correct option is b) face-centered cubic cell.