Question

# Ferrous oxide has a cubic structure and each edge of the unit cell is 5.0${A^0}$. Assuming density of the oxide as 4.0$g - c{m^{ - 3}}$, then the number of $F{e^{2 + }}$ and ${O^{2 - }}$ions present in each unit cell will bea.) Four $F{e^{2 + }}$and two ${O^{2 - }}$b.) Four $F{e^{2 + }}$and four ${O^{2 - }}$c.) Two $F{e^{2 + }}$and four ${O^{2 - }}$d.) Two $F{e^{2 + }}$and two ${O^{2 - }}$

Hint: In the Ferrous oxide structure, both the ions are arranged so that they show a cubic arrangement. The number of $F{e^{2 + }}$ ions in each unit cell is equal to the number of ${O^{2 - }}$ ions. Further, one can also say that there is one ${O^{2 - }}$ ion and one $F{e^{2 + }}$ ion behind every three vertices.

First let us understand the cubic structure and then we will move step by step on the answer.
So, the cubic structure is one in which atoms are arranged so that they form a cube-like appearance with all sides equal.
Ferrous oxide is expected to have a cubic structure.
Now, in the question we are given -
Edge of the unit cell = 5.0${A^0}$= $5.0 \times {10^{ - 8}}$cm
The unit cell is cubic.
Density of the oxide is also given, = 4.0$g - c{m^{ - 3}}$
What we have to find is the number of $F{e^{2 + }}$ and ${O^{2 - }}$ ions present in each unit cell.
Let us move step by step to answer.
First we can find volume from the edge. We know that volume of a cubic unit cell is as-
Volume of unit cell = ${\left( {side} \right)^3}$
Thus, Volume of unit cell = ${\left( {5.0 \times {{10}^{ - 8}}} \right)^3}$
Volume of unit cell = $1.25 \times {10^{ - 22}}c{m^3}$
We have density also. So, we can find the mass of one unit cell.
We have the formula, mass = density $\times$ volume
Thus, mass of one unit cell = 4.0$g - c{m^{ - 3}}$ $\times$$1.25 \times {10^{ - 22}}c{m^3}$
mass of one unit cell = $5 \times {10^{ - 22}}g$
So, now we have the mass of the unit cell also.
Further, mass of one mole of Ferrous oxide = Mass of Fe + Mass of O
Thus, mass of one mole of Ferrous oxide = 55.8 + 16
mass of one mole of Ferrous oxide = 71.8g
Mass of one molecule of Ferrous oxide can also be found out as-
Mass of one molecule of Ferrous oxide = $\dfrac{{Mass{\text{ }}of{\text{ }}one{\text{ }}mole{\text{ }}of{\text{ }}Ferrous{\text{ }}oxide}}{{6.022 \times {{10}^{23}}}}$
Mass of one molecule of Ferrous oxide = $\dfrac{{71.8}}{{6.022 \times {{10}^{23}}}} = 1.192 \times {10^{ - 22}}g$
Now, we have the mass of the unit cell and mass of one molecule. So, by dividing the mass of the unit cell with the mass of one molecule of Ferrous oxide, we can find out the number of FeO formula units present.
Thus, Number of FeO formula units = $\dfrac{{Mass{\text{ of unit cell}}}}{{{\text{Mass of one molecule}}}}$
Number of FeO formula units = $\dfrac{{5 \times {{10}^{ - 22}}}}{{1.192 \times {{10}^{ - 22}}}} \simeq 4$
So, FeO formula units = 4.
Thus, number of $F{e^{2 + }}$ions = 4
and number of ${O^{2 - }}$ ions = 4

So, option b.) is the correct answer.

Note: Always check that all the values have the same units as in the question the edge length is mentioned in the ${A^0}$ while other values are in cm. So, we should convert and get one unit first. We know that 1 ${A^0}$= ${10^{ - 10}}m$ = ${10^{ - 8}}cm$.
Further, the term $6.022 \times {10^{23}}$corresponds to Avogadro number. It is denoted by ${N_A}$. We know that one mole of a substance always contains $6.022 \times {10^{23}}$number of molecules or ions or atoms.