Question

A certain element crystallizes in a $\mathrm{bcc}$ lattice of unit cell edge length $27\stackrel{\circ }{\mathrm{A}}.$ If the same element under the same conditions crystallises in the $\mathrm{fcc}$ lattice, the edge length of the unit cell in $\stackrel{\circ }{\mathrm{A}}$ will be __________. (Round off to the nearest Integer).

(Assume each lattice point has a single atom)

(Assume $\sqrt{3}=1.73,\sqrt{2}=1.41$)

Answer 1

Answer: $\mathbf{33}\stackrel{\mathbf{\circ }}{\mathbf{A}}$

$$\begin{gathered} \mathrm{For}\mathrm{bcc}\mathrm{structure} \\ 4\mathrm{r}=\sqrt{3\mathrm{a}} \\ \mathrm{r}=\frac{\sqrt{3\mathrm{a}}}{4}\dots \dots \dots \dots \dots \left(1\right) \\ \mathrm{For}\mathrm{fcc}\mathrm{structure} \\ 4\mathrm{r}=\sqrt{2\mathrm{a}} \\ \mathrm{r}=\frac{\sqrt{2\mathrm{a}}}{4}\dots \dots \dots \dots \dots \left(2\right) \\ \mathrm{From}\left(1\right)\mathrm{and}\left(2\right) \\ \frac{\sqrt{3}\times \mathrm{a}\left(\mathrm{bcc}\right)}{4}=\frac{\sqrt{2}\times \mathrm{a}\left(\mathrm{fcc}\right)}{4} \\ \mathrm{a}\left(\mathrm{fcc}\right)=\frac{\sqrt{3}\times \mathrm{a}\left(\mathrm{bcc}\right)}{\sqrt{2}} \\ \sqrt{3}=1.732 \\ \sqrt{2}=1.414 \\ \mathrm{a}\left(\mathrm{bcc}\right)=27\stackrel{\circ }{\mathrm{A}} \\ =\frac{1.732\times 27}{1.414} \\ \mathrm{a}\left(\mathrm{fcc}\right)=\mathbf{33}\stackrel{\mathbf{\circ }}{\mathbf{A}} \end{gathered}$$