Question

A unit cell of iron crystal has an edge length of $288pm$ and density $7.86gc{m^{ - 3}}$ . Find the number of atoms per unit cell and the type of crystal lattice.
Given: molar mass of iron $ = 56g/mol$ , Avogadro’s number $ = 6.023 \times {10^{23}}$

Answer 1

Hint:
A unit cell is defined as the smallest repeating units having the full symmetry of the crystal structure. It shows the three dimensional pattern of the entire crystal. A uit cell has its own geometric shape. In order to find the number of atoms per unit cell we will use Avogadro’s number and number of moles.

Complete step by step answer:
Edge length of unit cell $\left( a \right) = 288pm$
$ \Rightarrow $Volume of unit cell $ = {a^3}$
$ \Rightarrow $Volume of unit cell $ = {\left( {288} \right)^3}$
$ \Rightarrow $Volume of unit cell $ = 2.389 \times {10^{ - 23}}c{m^{ - 3}}$
Density of iron $\left( \rho \right) = 7.86gc{m^{ - 3}}$
$\left( M \right) = \rho \times V$
Where, $\rho = $ density, $M = $ Mass of iron cell, $V = $ Volume.
Substituting the value of density and volume in the above formula we get,
$M = 7.86 \times 2.389 \times {10^{ - 23}}$
$M = 18.77 \times {10^{ - 23}}$
Molar mass of iron$ = 56g/mol$
Moles of iron in unit cell is given by the formula:
$n = \dfrac{M}{{MW}}$
Where, $n = $ number of moles of iron, $M = $ molar mass of iron, $MW = $ molecular weight of iron.
Substituting the value of molar mass and molecular weight of iron we get,
$n = \dfrac{{18.77 \times {{10}^{ - 23}}}}{{56}}$
$n = 3.351 \times {10^{ - 24}}$ .
In order to calculate the number of atoms we will use the formula given as follows:
Number of atoms$ = n \times {N_A}$
Where, $n = $ number of moles of iron, ${N_A} = $ Avogadro’s number.
Substituting the values we get,
Number of atoms$ = 3.351 \times {10^{ - 24}} \times 6.023 \times {10^{23}}$
Number of atoms$ = 2.018 \approx 2$ atoms per unit cell
The number of atoms per unit cell in an iron crystal is $2$ .
In a body centered cubic structure, the number of atoms per unit cell is $2.$
Therefore, an iron crystal is a body centered cubic structure.
Additional information:
There are three types of centered cubic unit cell:
-Body centered cubic structure
-It contains one constituent particle at the body center compared to its other corners.
-Face centered cubic structure:
-It contains one constituent particle at the center of each face.
-End centered cubic structure:
-It contains one constituent particle at the center of any two opposite faces .

Note:When a unit cell contains one or more constituents particles present at positions other than the corners are known as centered unit cells. Unit cells are known to be the building blocks of crystal lattice. Unlike crystal lattice, a unit cell has volume and a specific number of points.