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In the leaching method, bauxite ore is digested with a concentrated solution of $NaOH$that produces $'X'$. When $C{O_2}$ gas is passed through the aqueous solution of $'X'$, a hydrated compound$'Y'$is precipitated.$'X'$and $'Y'$respectively are:
A. $Na\left[ {Al{{\left( {OH} \right)}_4}} \right]$and $A{l_2}{\left( {C{O_3}} \right)_3}.x{H_2}O$
B. $Al{\left( {OH} \right)_3}$and $A{l_2}{O_3}.x{H_2}O$
C.$NaAl{O_2}$and $A{l_2}{\left( {C{O_3}} \right)_3}.x{H_2}O$
D. $Na\left[ {Al{{\left( {OH} \right)}_4}} \right]$and $A{l_2}{O_3}.x{H_2}O$

Answer
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Hint: To solve this question, you must recall the Bayer process for refining of Aluminium metal from its ore. Leaching involves the treatment of the ore with a suitable reagent in order to make it soluble while insoluble impurities are separated.

Complete step by step answer:
Bayer process is an industrial means of refining bauxite ore to produce aluminium oxide (alumina)
The main ore of aluminium is bauxite\[\left( {A{l_2}{O_3}.2{H_2}O} \right)\]which contains impurities like ferric oxides, titanium oxide and silica. Bauxite ore contains $30 - 50\% $ aluminium oxide (alumina), the rest are impurities. For the refining of aluminium metal, the aluminium oxide must be purified.
Finely powdered bauxite is reacted with caustic soda solution at around ${150^0}C$under pressure. Since alumina is an amphoteric oxide it can be separated from the other oxides, namely ferric oxide and titanium oxide. As a result, alumina dissolves forming soluble sodium aluminate.
\[A{l_2}{O_3}(s) + 2NaOH(aq) + 3{H_2}O(l) \to 2Na\left[ {Al{{\left( {OH} \right)}_4}} \right](aq)\]
The impurities are unaffected by the action of sodium hydroxide and are separated as insoluble red mud which is then filtered off. The filtrate is diluted and carbon dioxide gas is passed through it.
$2Na\left[ {Al{{\left( {OH} \right)}_4}} \right](aq) + C{O_2}(g) \to A{l_2}{O_3}.x{H_2}O(s) + 2NaHC{O_3}(aq)$
By the action of carbon dioxide gas, sodium aluminate forms pure hydrated aluminium oxide (alumina).
Thus, we can conclude that, $'X'$and $'Y'$are $Na\left[ {Al{{\left( {OH} \right)}_4}} \right]$and $A{l_2}{O_3}.x{H_2}O$ respectively.

Therefore, the correct option is D.

Note:
In Bayer’s process, the waste product produced during the digestion of bauxite ore by caustic soda (sodium hydroxide) is known as red mud. Red mud has very high calcium and sodium hydroxide content and has a complex chemical composition containing a variety of impurities. As a result, it is very corrosive and a source of pollution.
The amount of red mud produced is significantly high but there are no considerable uses for it. Thus, it becomes a health risk and poses a great danger to the environment.