Question

The unit cell of MgO is shown below.

Does MgO have a structure like that of NaCl or ZnS? Ifthe density of MgO is 3.58 g/cm3, estimate the radius (incentimeters) of the ${\mathbf{O}}^{\mathbf{2}\mathbf{-}}$anions and the ${\mathbf{Mg}}^{\mathbf{2}\mathbf{+}}$cations.

Short answer

The given molecule has a structure of NaCl compound, which is fcc.

The radius (in centimeters) of ${\mathrm{O}}^{2-}\mathrm{and}{\mathrm{Mg}}^{2+}$ions islocalid="1663957592782" $2.65\mathrm{\AA }$.

Step-by-step solution

GIven Information

The density of MgO is 3.58 g/cm3

Concept Introduction

The crystal structure of any compound defines the symmetry positions of atoms/ions in the unit cell.

Describing the crystalline structure of MgO

MgO has the${\mathrm{Mg}}^{2+}$ions while compared to ${\mathrm{Zn}}^{2+}$ions, which means MgO has the crystal structure as ZnO, but it is not.

The MgO compound has the face cubic-centered structure (fcc), the same as the NaCl compound.

Describing the estimation of the radius of Mg2+ and O2- ions

The density of MgO is $3.98\mathrm{gm}/{\mathrm{cm}}^3$

The molecular weight of MgO is $\mathrm{M}=40.3044\mathrm{gm}/\mathrm{mole}$

The number of atoms per unit cell Z=4

The density of MgO can be expressed as,

$$\begin{gathered} \mathrm{d}=\frac{\mathrm{Z}\times \mathrm{M}}{{\mathrm{N}}_{\mathrm{A}}\times {\mathrm{a}}^3} \\ {\mathrm{a}}^3=\frac{\mathrm{Z}\times \mathrm{M}}{{\mathrm{N}}_{\mathrm{A}}\times \mathrm{d}} \\ \mathrm{a}=\sqrt[3]{\frac{\mathrm{Z}\times \mathrm{M}}{{\mathrm{N}}_{\mathrm{A}}\times \mathrm{d}}} \\ \mathrm{a}=\sqrt[3]{\frac{4\times 40.3044\mathrm{gm}/\mathrm{mole}}{6.023\times {10}^{23}\mathrm{atoms}/\mathrm{mole}\times 3.98\mathrm{gm}/{\mathrm{cm}}^3}} \\ \mathrm{a}=7.47\times {10}^{-8}\mathrm{cm} \\ \mathrm{a}=7.47\mathrm{\AA } \end{gathered}$$

Now, we know that the structure is fcc, therefore,

$$\begin{gathered} \mathrm{a}=2\sqrt{2}\mathrm{r} \\ \mathrm{r}=\frac{\mathrm{a}}{2\sqrt{2}} \\ \mathrm{r}=\frac{7.47\mathrm{\AA }}{2\sqrt{2}} \\ \mathrm{r}=2.65\mathrm{\AA } \end{gathered}$$

The radius of ${\mathrm{Mg}}^{2+}\mathrm{cations}\mathrm{and}{\mathrm{O}}^{2-}\mathrm{anions}\mathrm{is}2.65\mathrm{\AA }.$