Question

Ferrous oxide has a cubic structure with an edge length of the unit cell equal to 5.0 A˚. Assuming the density of ferrous oxide to be $3.84\dfrac{g}{{c{m^3}}}$ , the number of $F{e^{2 + }}$ and ${O^{2 - }}$ ions present in each unit cell will be respectively, (use ${N_A} = {\text{ }}6{\text{ }} \times {\text{ }}{10^{23}}$)
A) 4 and 4
B) 2 and 2
C) 1 and 1
D) 3 and 4

Answer 1

In the ferrous oxide structure, both ions ( $F{e^{2 + }}$ and ${O^{2 - }}$ ) are arranged in such a way that they show a cubic arrangement. Since the edge length of the unit cell and the density of ferrous oxide is given in the question, and we know the expression of density, $d = \dfrac{{nM}}{{V{N_A}}}$. Substituting the values, we will calculate the number of $F{e^{2 + }}$ and ${O^{2 - }}$ ions present in each unit cell.

Complete step by step answer:
Given in the question is that ferrous oxide has cubic structure with all sides equal.
The expression for density (d) is as follows,
​
$d = \dfrac{{nM}}{{V{N_A}}} \\\ \Rightarrow n = \dfrac{{dV{N_A}}}{M} \\\$
Here, ‘a’ is the edge length of the unit cell, M is the molecular mass and V is the volume of a unit cell.

Given in the question are,
$A = 5{A^0} = 5 \times {10^{ - 8}}cm$,
$d = 3.84\dfrac{g}{{c{m^3}}}$
Let the units of ferrous oxide in a unit cell =n
Molecular weight of Ferrous Oxide (FeO) = 56 + 16 = 72 $\dfrac{g}{{mol}}$
Volume of one unit = cube of length of corner $= {\left( {5 \times {{10}^{ - 8}}} \right)^3}$
Now, substituting all the values in the expression of density,

$$n = \dfrac{{dV{N_A}}}{M} \\\ \Rightarrow n = \dfrac{{3.84 \times {{\left( {5 \times {{10}^{ - 8}}} \right)}^3} \times 6.023 \times {{10}^{23}}}}{{72}} \\\ \Rightarrow n = 4 \\\$$

So, the number of 4 $F{e^{2 + }}$ and 4 ${O^{2 - }}$ ions present in each unit cell will be respectively.

Therefore, the correct answer is option (A).

Note: The term ${N_A}$ is the Avogadro number which corresponds to $6{\text{ }} \times {\text{ }}{10^{23}}$ . We know that one mole of a substance contains ${N_A}$ i.e. $6{\text{ }} \times {\text{ }}{10^{23}}$ number of molecules or ions or atoms.