Chemistry - Solid State Question with Solution | TestHub

Edge Length ($a$): From the body-centered contact (Ti⁴⁺ and O²⁻), $a=2(r_{T{i}^{4+}}+r_{O^{2-}})=2(1.3+1.0)=4.6\mathring{A}$, giving a unit cell volume $V_{cell}=a^3=(4.6{)}^3=97.4{\mathring{A}}^3$.

Occupied Volume ($V_{occ}$): The cell contains $1B{a}^{2+}$, $1T{i}^{4+}$, and $3O^{2-}$ ions, so:

$$V_{occ}=\frac{4}{3}\pi [1(2.2{)}^3+1(1.3{)}^3+3(1.0{)}^3]=\frac{4}{3}\pi [10.6+2.2+3]=\frac{4}{3}\pi (15.8)\approx 66.2{\mathring{A}}^3$$

Packing & Void Fraction: The packing fraction is $PF=\frac{V_{occ}}{V_{cell}}=\frac{66.2}{97.4}\approx 0.68$, leading to a void fraction of $1-0.68=0.32$.