Atoms of metals x ,   y and z form face-centred cubic (fcc) unit cell of edge length L x , body-centred…

Metal $\mathrm{x}$ forms FCC (edge length ${\mathrm{L}}_{\mathrm{x}}$ )

Metal y forms BCC (edge length ${\mathrm{L}}_{\mathrm{y}}$ )

Metal $\mathrm{z}$ forms SC (edge length ${\mathrm{L}}_{\mathrm{z}}$ )

Given ${\mathrm{r}}_{\mathrm{z}}=\frac{\sqrt{3}}{2}{\mathrm{r}}_{\mathrm{y}}$ and ${\mathrm{r}}_{\mathrm{y}}=\frac{8}{\sqrt{3}}{\mathrm{r}}_{\mathrm{x}}$

$\therefore {\mathrm{r}}_{\mathrm{z}}=\frac{\sqrt{3}}{2}\times \frac{8}{\sqrt{3}}{\mathrm{r}}_{\mathrm{x}}=4{\mathrm{r}}_{\mathrm{x}}$

${\mathrm{M}}_{\mathrm{z}}=\frac{3}{2}{\mathrm{M}}_{\mathrm{y}} {\mathrm{M}}_{\mathrm{z}}=3{\mathrm{M}}_{\mathrm{x}}$

$\therefore {\mathrm{M}}_{\mathrm{y}}=2{\mathrm{M}}_{\mathrm{x}}$

Packing efficiency $\mathrm{FCC} > \mathrm{BCC} > \mathrm{SC}$

Packing efficiency unit cell $\mathrm{x}>\mathrm{y}>\mathrm{z}$

In FCC unit cell:- atoms along the face diagonals are in contact.

$\therefore \sqrt{2} {\mathrm{L}}_{\mathrm{x}}=4{\mathrm{r}}_{\mathrm{x}}\Rightarrow {\mathrm{L}}_{\mathrm{x}}=2\sqrt{2}{\mathrm{r}}_{\mathrm{x}}$

In BCC unit cell: atoms along the body diagonal are

$\therefore \sqrt{3}{\mathrm{L}}_{\mathrm{y}}=4{\mathrm{r}}_{\mathrm{y}}\Rightarrow {\mathrm{L}}_{\mathrm{y}}=\frac{4}{\sqrt{3}}{\mathrm{r}}_{\mathrm{y}}=\frac{4}{\sqrt{3}}\times \frac{8}{\sqrt{3}}{\mathrm{r}}_{\mathrm{x}}=\frac{32}{3}{\mathrm{r}}_{\mathrm{x}}$

${\mathrm{L}}_{\mathrm{y}}=\frac{32}{3}{\mathrm{r}}_{\mathrm{x}}$

In SC unit cell, atoms along the edge are in contact

$\therefore {\mathrm{L}}_{\mathrm{z}}=2{\mathrm{r}}_{\mathrm{z}}$

$=2\times 4{\mathrm{r}}_{\mathrm{x}}=8{\mathrm{r}}_{\mathrm{x}}$

${\mathrm{L}}_{\mathrm{x}}=2\sqrt{2}{\mathrm{r}}_{\mathrm{x}}$

${\mathrm{L}}_{\mathrm{y}}=\frac{32}{3}{\mathrm{r}}_{\mathrm{x}}$

${\mathrm{L}}_{\mathrm{z}}=8{\mathrm{r}}_{\mathrm{x}}$

$\therefore {\mathrm{L}}_{\mathrm{y}}>{\mathrm{L}}_{\mathrm{z}}>{\mathrm{L}}_{\mathrm{x}}$

Density of $\mathrm{x}$ (Number of atoms of $\mathrm{x}$ per unit cell $\left(\mathrm{z}\right)=4)$

${\mathrm{d}}_{\mathrm{x}}=\frac{{\mathrm{zM}}_{\mathrm{x}}}{{\left({\mathrm{L}}_{\mathrm{x}}\right)}^3 {\mathrm{N}}_{\mathrm{A}}}=\frac{4\times {\mathrm{M}}_{\mathrm{x}}}{{\left(2\sqrt{2}{\mathrm{r}}_{\mathrm{x}}\right)}^3\times {\mathrm{N}}_{\mathrm{A}}}$

$=\frac{4{\mathrm{M}}_{\mathrm{x}}}{16\sqrt{2}{\mathrm{r}}_{\mathrm{x}}^3{\mathrm{N}}_{\mathrm{A}}}=\frac{{\mathrm{M}}_{\mathrm{x}}}{4\sqrt{2}{\mathrm{r}}_{\mathrm{x}}^3{\mathrm{N}}_{\mathrm{A}}}$

Density of $\mathrm{y}$ : (Number of atoms of y per unit cell $\left(\mathrm{z}\right)=2$ )

${\mathrm{d}}_{\mathrm{y}}=\frac{{\mathrm{zM}}_{\mathrm{y}}}{{\left({\mathrm{L}}_{\mathrm{y}}\right)}^3{\mathrm{N}}_{\mathrm{A}}}=\frac{2\times 2{\mathrm{M}}_{\mathrm{x}}}{{\left(\frac{32}{3}{\mathrm{r}}_{\mathrm{x}}\right)}^3{\mathrm{N}}_{\mathrm{A}}}=\frac{108{\mathrm{M}}_{\mathrm{x}}}{32768{\mathrm{r}}_{\mathrm{x}}^3{\mathrm{N}}_{\mathrm{A}}}$

$\therefore$ Density of $\mathrm{x}>$ density of $\mathrm{y}$.