Question

Barium crystallizes in a body-centered cubic unit cell with an edge length of \(5.025{A‌^o}\)

(a) What is the atomic radius of barium in this structure?

(b) Calculate the density of barium

Short answer

a.Atomic radius of Barium in structure is \(2.176{A‌^o}\).

b.The density of barium is \(4.42\;{\rm{g}}/{\rm{c}}{{\rm{m}}^3}\) with body centered packed structure.

Step-by-step solution

Define the density of the solid

In the body centred cubic unit cell there are 2 atoms in each unit cell. The atomic radius of body centered cubic unit cell is equal to \(\dfrac{{\sqrt 3 }}{4}a\). Here ' \(a\) ' indicates the edge length. Density can be calculated the below formula.

Density of solid \( = \dfrac{{{\rm{Z}} \times {\rm{M}}}}{{{{\rm{a}}^3}\;{{\rm{N}}_{\rm{a}}}}}\)

The edge length of face-centered cubic unit cell is directly related to atomic radius. Edge length can be further used to calculate the density of so

To determine the atomic radius of barium metal in body-centredcubic unit cell.

Atomic radius \( = \dfrac{{\sqrt 3 }}{4}a\)

Given, edge length \( = 5.025{A‌^o}\)

Atomic radius

\(\begin{aligned}{}& = \dfrac{{\sqrt 3 }}{4} \times 5.025{A‌^o}\\ &= 2.176{A‌^o}\end{aligned}\)

The atomic radius of body centered packed structure is \(\dfrac{{\sqrt 3 }}{4}\) of edge length. Hence, the atomic radius of Barium in structure is\(2.176{A‌^o}\).

Step 3:Determine the density of barium

Density of solid \( = \dfrac{{{\rm{Z}} \times {\rm{M}}}}{{{{\rm{a}}^3}\;{{\rm{N}}_{\rm{a}}}}}\)

\({\rm{Z}} = 2\); for body centred packed structure

\({\rm{M}}\)molar mass \( = 137.32\,{\rm{g/mol}}\)

\({\rm{a}} = \) edge length \( = 2.176{A‌^o} = 2.176 \times {10^{ - 8}}\;{\rm{cm}}\)

\({{\rm{N}}_{\rm{a}}} = \)Avogadro number \( = 6.023 \times {10^{23}}\)

Density \( = \dfrac{{2 \times 137.32\;{\rm{g}}/{\rm{mol}}}}{{{{\left( {2.176 \times {{10}^{ - 8}}\;{\rm{cm}}} \right)}^3} \times 6.023 \times {{10}^{23}}}} = 4.42\;{\rm{g}}/{\rm{c}}{{\rm{m}}^3}\)

The density of barium is \(4.42\;{\rm{g}}/{\rm{c}}{{\rm{m}}^3}\) with body centered packed structure.