How is Bauxite purified by Serpeck's process?

Hint: Finely powdered bauxite ($Al_2O_3$ ) is heated with $\text{coke}$ and $N$ to at $1800-2000K$. Alumina forms aluminium nitride and silica is reduced to $Si$ which is volatile. The hydrolysis of aluminium nitride gives aluminium hydroxide, when we heat it, it gives pure alumina.

Complete step by step answer:
In this question we have asked about the purification of Bauxite by Serpeck's reaction. So, first of all, what is bauxite? Bauxite is a form of sedimentary rock and is the principal source of the popular metal aluminium. The chemical formula of bauxite is $Al_2O_3.2H_2O$. This process is used when silica is present in a considerable amount of bauxite ore. This process is used for the purification of bauxite ore containing silica ($SiO_2$) as the main impurity and when Silica is the main impurity it is called white bauxite. The powdered bauxite is mixed with carbon and heated up to 1800℃ and a current of nitrogen is passed, aluminum nitride is obtained.
$Al_2O_3. 2H_2O + 3C + N_2 $ → $2AI N + 3CO+ 2H_2O$
  Aluminium nitride thus obtained is hydrolysed with water to get a precipitate of $Al(OH)_3$.
$2AlN + 6H_2O$ → $2NH_3 + Al(OH)_3$
The precipitate of $Al(OH)_3$ is filtered, washed and dried. The silica present as an impurity in bauxite is reduced to silicon which is volatile at high temperature. So, it is removed easily.
 $SiO_2 + 2C$ → $Si + 2CO$
Note:
Bauxite can also be purified by the Hall Heroult's process,which is the major industrial process for smelting aluminium (bauxite) and by Bayer process, which is the principal industrial means of refining bauxite to produce alumina (aluminium oxide).