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Polonium crystallizes in a simple cubic unit cell. It has atomic mass \[{\text{209}}\] and density \[{\text{91}}{\text{.5kg}}{{\text{m}}^{{\text{ - 3}}}}\]. What is the edge length of its unit cell?

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Last updated date: 21st Jul 2024
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Answer
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Hint: In matters are divided into three types. There are solids, liquids and gas. Solids are divided into types. They are crystalline and amorphous. The crystalline is further divided into ionic solids, covalent solids, molecular solids and metallic solids. In crystalline solid, the basic repeating structural unit is called a unit cell.
Formula used:
Number of atoms in a simple cubic unit cell is dependent on the number of atoms in the corner of the unit cell.
Number of atoms in a simple cubic unit cell is equal to the number of atoms in the corner of the unit cell divided by eight, because each corner is divided by eight unit cells.
\[{\text{number of atoms in a simple cubic unit cell = }}\dfrac{{{\text{Nc}}}}{{\text{8}}}\]
Here, Nc is represent as the number of atoms in the corner of the unit cell
Density of the unit cell,
\[{\text{Density of the unit cell = }}\dfrac{{{\text{nM}}}}{{{{\text{a}}^{\text{3}}}{{\text{N}}_{\text{A}}}}}\]
Volume of unit cell is a3
Here,
M is molar mass.
NA is Avogadro number.
a is the edge length of its unit cell
n is the number of atoms in a unit cell.

Complete answer:
Polonium crystallizes in a simple cubic unit cell.
Polonium has atomic mass, M is \[{\text{209}}\] .
Density of polonium is \[{\text{91}}{\text{.5kg}}{{\text{m}}^{{\text{ - 3}}}}\].
Density of polonium is also write as \[{\text{0}}{\text{.0915gc}}{{\text{m}}^{{\text{ - 3}}}}\].
There are eight atoms in a simple cubic unit cell in the corner of the unit cell. Each corner is divided by eight unit cells.
\[{\text{number of atoms in a simple cubic unit cell = }}\dfrac{{{\text{Nc}}}}{{\text{8}}}\]
Here, Nc is represent as the number of atoms in the corner of the unit cell
\[ = \dfrac{8}{8} = 1\]
One only atom present in each unit cell of a simple cubic crystal.
\[{\text{n = 1}}\]
\[{\text{Density of the unit cell = }}\dfrac{{{\text{nM}}}}{{{{\text{a}}^{\text{3}}}{{\text{N}}_{\text{A}}}}}\]
Now we can substitute the known values we get,
$0.0915$ = $\dfrac{{1} \times {209}}{{6} \times {10^{23}} \times {a^3}}$
On simplification we get,
\[{\text{a = 15}}{\text{.6}}{{\text{A}}^0}\]
Polonium crystallizes in a simple cubic unit cell. It has atomic mass \[{\text{209}}\] and density \[{\text{91}}{\text{.5kg}}{{\text{m}}^{{\text{ - 3}}}}\]. The edge length of its unit cell is \[{\text{15}}{\text{.6}}{{\text{A}}^{\text{o}}}\].

Note:
We have to remember that there are seven types of primitive crystal system. There are cubic, tetragonal, orthorhombic, hexagonal, monoclinic, triclinic and rhombohedral. This classification of crystal is based on the crystallographic nature of angles, unit cell, primitive lattice and axes. These cubic crystals are further divided into three types. There are simple cubic crystals, face centered cubic crystals and body centered cubic crystals. This classification is based on the atoms arrangement in the cubic structure in the unit cell.