Statements (b) and (c) are correct whereas statements (a) and (d) are incorrect.

(a) Process of precipitating colloidal sol is known as coagulation.

(b) Concentration of colloidal solutions is small due to large molar mass and thus, their colligative properties are very small compared to true solutions. Hence, these solution freezes at higher temperature.

(c) Micelles are formed at the critical micelle concentration (CMC) which depends to temperature.

(d) Micelles and macromolecular colloids are two different types of colloid.

(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.

Option (A) : Correct. Adsorption is a surface phenomenon in which the concentration of the adsorbate increases only on the surface. It is always accompanied by a decrease in the entropy ($$\Delta$$S). Since adsorption is an exothermic process, the enthalpy change ($$\Delta$$H) of the system is also negative.

Option (B) : Correct. Easily liquefiable gases, that is, gases with high critical temperatures, are more readily adsorbed because van der Waals forces are stronger near the critical temperature. Thus, the adsorption of ethane will be more than that of nitrogen.

Option (C) : Cloud is an aerosol in which liquid is dispersed phase and gas is dispersion medium. Whereas, emulsion is liquid in liquid colloidal system.

Option (D) : Incorrect. Brownian movement varies with the size of the particles and viscosity of the solution. Smaller particles with lower viscosity show faster movement.

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

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