sc ⇒ 2r = a
⇒ r = a/2
bcc ⇒ 4r = \( \sqrt{3a} \) ⇒ r = \(\frac { \sqrt{3a}} {4}\)
fcc ⇒ 4r ⇒  \( \sqrt{2a} \) ⇒ r = \(\frac { \sqrt{2a}} {4} = \frac { a} {2\sqrt{2a}}\)
\(\left( \frac { a }{ 2 } :\frac { \sqrt { 3 } }{ 4 } a:\frac { a }{ 2\sqrt { 2 } } \right) \)

(i) X-Ray diffraction analysis is the most powerful tool for the determination of crystal structure. The inter planar distance (d) between two successive planes of atoms can be calculated using the following equation form the X-Ray diffraction data
2dsinθ = nλ 
(ii) The above equation is known as Bragg's equation.
Where,
\(\lambda\) is the wavelength of X-ray used for diffraction.
θ is the angle of diffraction
n is the order of diffraction.

In this process, the crude metal is heated to form fusible liquid and allowed to flow on a sloping surface. The impure metal is placed on sloping hearth of a reverberatory furnace and it is heated just above the melting point of the metal in the absence of air, the molten pure metal flows down and the impurities are left behind. The molten metal is collected and solidified.

(i) Propane-1, 2-diol
(ii) Ethane-1,2 - diol
(iii) 2-Hydroxy propanoic acid

When ethylene reacts with alkaline KMnO₄ solution (Baeyer's reagent), it forms glycol (diol).

(i) Conversion of ores into oxides.
(ii) Reduction of metal oxides.

The structure of an ionic compound depends upon the stoichiometry and the size of the ions. 
The ratio of radius of cation and anion \(\left(\frac{\mathrm{r}_{\mathrm{C}}^{+}}{\mathrm{r}_{\mathrm{A}}^{-}}\right)\) plays an important role in determining the structure.

The cubic close packing is based on the face centered cubic unit cell. 
\(\mathrm{AC}=4 \mathrm{r} ; \quad 4 \mathrm{r}=\mathrm{a} \sqrt{2} ; \quad \mathrm{r}=\mathrm{a} \frac{\sqrt{2}}{4}\)
In ΔABC
\(A C^{2}=A B^{2}+B C^{2}  \)
\(A C=\sqrt{A B^{2}+B C^{2}}  \)
\(A C=\sqrt{a^{2}+a^{2}}=\sqrt{2 a^{2}}=\sqrt{2} a\)
Volume of the sphere with radius ' r ' \(=\frac{4}{3} \pi\left(\frac{\sqrt{2} a}{4}\right)^{3}\)
\(=\frac{4}{3} \pi\left(\frac{2 \sqrt{2} \mathrm{a}}{64}\right)^{3}=\frac{\sqrt{2} \pi \mathrm{a}^{3}}{24} \)
Total number of spheres belongs to a single fcc unit cell is 4.
\(\therefore\) The volume of all spheres in a fcc unit cell \(=4 \times\left(\frac{\sqrt{2} \pi \mathrm{a}}{24}\right)^{3}=\left(\frac{\sqrt{2} \pi \mathrm{a}^{3}}{6}\right)\)
\(\text {Packing efficiency } =\frac{\left(\frac{\sqrt{2} \pi a^{3}}{6}\right)}{\left(a^{3}\right)} \times 100=\frac{\sqrt{2} \pi}{6} \times 100 \\ \)
\(=\frac{1.414 \times 3.14 \times 100}{6}=74 \%\)

Glycerol occurs in many natural fats and it is also found in long chain fatty acids in the form of glyceryl esters (Triglycerides). The alkaline hydrolysis of these fats gives glycerol and the reaction is known as saponification.

Acetaldehyde - CH₃-CHO

 

(i) Metallic oxides such as Cr₂O₃ can be reduced by an aluminothermic process.
(ii) In this process, the metal oxide is mixed with aluminum powder and placed in a fire clay crucible.
(iii)To initiate the reduction process, an ignition mixture (usually magnesium and barium peroxide is used 
\({ BaO }_{ 2 }+Mg\longrightarrow Bao+MgO\)
(iv) During the above reaction a large amount of heat is evolved (temperature up to 2400°C, is generated and the reaction enthalpy is: 852 kJ mol^-1 which facilitates the reduction of Cr₂O₃ by aluminium power.
\({ Cr }_{ 2 }{ O }_{ 3 }+2Al\overset { \Delta }{ \longrightarrow } 2Cr+{ Al }_{ 2 }{ O }_{ 3 }\)

Mechanism:
This mechanism undergoes Saytzeff rule.
During intramolecular dehydration, if there is a possibility to form a carbon - carbon double bond at different locations, the preferred location is the one that gives the more (highly) substituted alkene i.e., the stable alkene.

(i) The defects in ionic solids is by adding impurity ions.
(ii) If the impurity ions are in different valance state from that of host, vacancies are created in the crystal lattice of the host.
(iii) For example, addition of CdCl₂ to silver chloride yields solid solutions where the divalent cation Cd²⁺ occupies the position of Ag⁺.
(iv) This will disturb the electrical neutrality of the crystal.
(v) In order to maintain the same, proportional number of Ag⁺ ions leaves the lattice.
(vi) This produces a cation vacancy in the lattice, such kind of crystal defects are called impurity defects.