Solid State

1. Atomic radii relations :

Given,

$ r_{y} = \frac{8}{\sqrt{3}} r_{x} \tag{1} $ ......(1)

$ r_{z} = \frac{\sqrt{3}}{2} r_{y} \tag{2} $ ........(2)

Substituting Eq. (1) into Eq. (2), we get :

$ r_{z} = \frac{\sqrt{3}}{2} \times \frac{8}{\sqrt{3}} r_{x} = 4 r_{x} \tag{3} $ .......(3)

2. Edge lengths of the unit cells :

The edge lengths (L) of the unit cells for x, y, and z are given by :

$ L_x = 2 \sqrt{2} r_x \quad \text{(for FCC)} \tag{4} $ .......(4)

$ L_y = 4 r_y = \frac{32}{\sqrt{3}} r_x \quad \text{(for BCC, using Eq. (1))} \tag{5} $ .......(5)

$ L_z = 2 r_z = 8 r_x \quad \text{(for simple cubic, using Eq. (3))} \tag{6} $ .......(6)

3. Comparing the edge lengths :

From Eq. (4), (5) and (6), we find :

$ L_y > L_z > L_x \tag{7} $

This means Option B : $L_y > L_z$ is correct.

4. Packing Efficiencies :

The packing efficiencies for FCC, BCC, and simple cubic are calculated as follows :

- FCC :

$ PE_{fcc} = \frac{4 \times \frac{4}{3} \pi r_x^3}{(L_x)^3} = \frac{\pi}{\sqrt{2}} \tag{8} $

- BCC :

$ PE_{bcc} = \frac{2 \times \frac{4}{3} \pi r_y^3}{(L_y)^3} = \frac{3 \pi \sqrt{3}}{32} \tag{9} $

- Simple Cubic :

$ PE_{sc} = \frac{1 \times \frac{4}{3} \pi r_z^3}{(L_z)^3} = \frac{\pi}{6} \tag{10} $

Comparing these packing efficiencies, we find :

$ PE_{fcc} > PE_{bcc} > PE_{sc} \tag{11} $

This means Option A : Packing efficiency of unit cell of $x > y > z$ is correct.

5. Calculating Densities :

The densities of x and y are given by the formula $d = \frac{Z \cdot M}{N_a \cdot L^3}$, where $Z$ is the number of atoms in the unit cell, $M$ is the molar mass of the metal, $N_a$ is Avogadro's number, and $L$ is the edge length of the unit cell.

- For x (FCC), $Z=4$ :

$ d_x = \frac{4 M_x}{N_a (L_x)^3} \tag{12} $

- For y (BCC), $Z=2$ :

$ d_y = \frac{2 M_y}{N_a (L_y)^3} \tag{13} $

$M_z=\frac{3}{2} M_y$ and $M_z=3 M_x$. This implies that $M_x=\frac{M_y}{2}$. Thus, $2M_x=M_y$, we can write :

$ \frac{d_x}{d_y} = \frac{2 M_x}{M_y} \times \left(\frac{L_y}{L_x}\right)^3 \tag{14} $ .......(7)

Substituting the given relation $2 M_x = M_y$ into Eq. (7), we get :

$ \frac{d_x}{d_y} = 2 \times 1 \times \left(\frac{32/3}{2 \sqrt{2}}\right)^3 \tag{15} $ ..........(8)

After simplifying Eq. (8), we find $d_x > d_y$.

This means Option D : Density of $x > y$ is correct.

So, the correct answers are Options A, B, and D.

(a) The empirical formula of the compound :

Contribution of M and X :

${M_{\left( {2 \times {1 \over 2}} \right)}}{X_{\left( {4 \times {1 \over 4}} \right)}} \Rightarrow MX$

(b) Coordination number of both M and X is 8.

(c) Distance between M and X

$ = \sqrt {{{{a^2}} \over 4} + {{{a^2}} \over 2}} = \sqrt {{3 \over 4}} a = {{\sqrt 3 } \over 2}a$

$ \Rightarrow $ 0.866 a

(d) ${r_M}\,:\,{r_X} = (\sqrt 3 - 1)\,:\,1 \Rightarrow 0.732:1$, thus statement (d) is incorrect.

Coordination number cannot be 12, for any atom in the topmost layer, as there is no layer above it. Thus, each atom is in contact with six atoms in the same layer and three atoms from the layer below it.

For cubic close packing, we have

Packing fraction = ${{Volume\,of\,four\,spheres\,in\,the\,unit\,cell} \over {Total\,volume\,of\,the\,unit\,cell}}$

$ = 4 \times {{(4/3)\pi {r^3}} \over {16\sqrt 2 {r^3}}} = {\pi \over {3\sqrt 2 }} = 0.74 = 74\% $

In fcc unit cell, the effective number of atoms is

One atom at each corner = 8 corner atoms $ \times {1 \over 8} = 1$

Atoms at each of the six face centres :

6 face centered atoms $ \times {1 \over 2} = 3$.

Number of octahedral voids = 4.

Number of tetrahedral voids = 8.

Therefore, per atom, there is one octahedral void and two tetrahedral voids.

In fcc (or ccp), the unit edge length is given by

$\sqrt 2 a = 4r \Rightarrow a = {{4r} \over {\sqrt 2 }} = 2\sqrt 2 r$

Frenkel defect occurs in compounds in which the anions are much larger in size than cations. It is a dislocation effect. The presence of Schottky defects lowers the density of the crystal, it hence affects the physical properties of the crystal. In the metal excess defects, the electrons trapped in anion vacancies are known as F-centres.