How is alumina separated from silica in the bauxite ore associated with silica ? Give equations.

Bauxite ore is concentrated by the method of leaching or chemical separation.
a) This method involves the digesting of powdered bauxite ore with concentrated NaOH at 473-523 K and 35-36 bar pressure. Al₂O₃ is leached out as sodium aluminate (and SiO₂ too as sodium silicate) leaving the impurities behind.
${\mathrm{Al}}_2{\mathrm{O}}_{3\left(\mathrm{S}\right)}+2{\mathrm{NaOH}}_{\left(\mathrm{aq}\right)}+3{\mathrm{H}}_2{\mathrm{O}}_{\left(\mathrm{I}\right)}$
From bauxite      $\underset{\mathrm{Sodium} \mathrm{aluminate}}{\to 2\mathrm{Na}{\left[\mathrm{Al}(\mathrm{OH}{)}_4\right]}_{\left(\mathrm{aq}\right)}}$
b) Now CO₂ gas is passed through the solution and small amount of freshly prepared hydrated aluminium oxide is also added in order to induce the precipitation.
$2\mathrm{Na}{\left[\mathrm{Al}(\mathrm{OH}{)}_4\right]}_{\left(\mathrm{aq}\right)}+{\mathrm{CO}}_{2\left(\mathrm{aq}\right)}\to \underset{\mathrm{Hydrated} \mathrm{aluminium} \mathrm{oxide} \mathrm{ppt}}{{\mathrm{Al}}_2{\mathrm{O}}_3\cdot {\mathrm{XH}}_2{\mathrm{O}}_{\left(\mathrm{s}\right)}+2{\mathrm{NaHCO}}_{3\left(\mathrm{aq}\right)}}$
c) The sodium silicate remains in the solution and the insoluble hydrated alumina is filtered
dried and heated to give pure Al₂O₃.
${\mathrm{Al}}_2{\mathrm{O}}_3\cdot {\mathrm{xH}}_2{\mathrm{O}}_{\left(\mathrm{s}\right)}\stackrel{1470 \mathrm{K}}{\to } {\mathrm{Al}}_2{\mathrm{O}}_{3\left(\mathrm{s}\right)}+{\mathrm{xH}}_2{\mathrm{O}}_{\left(\mathrm{g}\right)}$