Solid State

Metal $x$ forms FCC (edge length $L_x$ )Metal y forms BCC (edge length $L_y$ )Metal z forms SC (edge length $L z$ )Given $r_z = \frac{\sqrt{3}}{2} r_y$ and $r_y = \frac{8}{\sqrt{3}} r_x$ $\therefore \; r_z = \frac{\sqrt{3}}{2} \times \frac{8}{\sqrt{3}} r_x = 4 r_x$ $M_z = \frac{3}{2} M_y \; M_z = 3 M_x$ $\therefore M y = 2 M x$ Packing efficiency $\text{FCC} > \text{BCC} > \text{SC}$Packing efficiency unit cell $x > y > z$In FCC unit cell:- atoms along the face diagonals are in contact.$\therefore \sqrt{2} L_x = 4 r_x \Rightarrow L_x = 2 \sqrt{2} r_x$In BCC unit cell: atoms along the body diagonal are$L_y = \frac{32}{3} r_x$ In SC unit cell, atoms along the edge are in contact$\therefore \; Lz = 2 rz$$= 2 \times 4 r_x = 8 r x$$L_x = 2 \sqrt{2} r_x$$L_y = \frac{32}{3} r_x$$L_z = 8 r_x$$\therefore Ly_y > Lz_z > Lx$ Density of $x$ (Number of atoms of $x$ per unit cell $(z) = 4$ ) $d_x = \frac{zM_x}{(L_x)^3 N_A} = \frac{4 \times M_x}{(2 \sqrt{2} r_x)^3 \times N_A}$$= \frac{4 M_x}{16 \sqrt{2} r_x^3 N_A} = \frac{M_x}{4 \sqrt{2} r_x^3 N_A}$Density of $y$ : (Number of atoms of y per unit cell $(z) = 2$ )$\therefore$ Density of $x >$ density of $y$.