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An inorganic Lewis acid \[\left[ {\text{X}} \right]\] gives gelatinous white ppt. with \[{\text{N}}{{\text{H}}_{\text{4}}}{\text{OH}}\] in presence of \[{\text{N}}{{\text{H}}_{\text{4}}}{\text{Cl}}\]. \[\left[ {\text{X}} \right]\] will respond to which of the following characteristics:
This question has multiple correct options:
(A) \[\left[ {\text{X}} \right]\] fumes in moist air
(B) \[\left[ {\text{X}} \right]\] on heating with solid \[{{\text{K}}_{\text{2}}}{\text{C}}{{\text{r}}_{\text{2}}}{{\text{O}}_{\text{7}}}\] and conc. \[{{\text{H}}_{\text{2}}}{\text{S}}{{\text{O}}_{\text{4}}}\] gives deep red or orange red fumes
(C) \[\left[ {\text{X}} \right]\] in addition to excess \[{\text{NaOH}}\] gives white ppt.
(D) \[\left[ {\text{X}} \right]\] on heating with \[{\text{N}}{{\text{a}}_{\text{2}}}{\text{C}}{{\text{O}}_{\text{3}}}\] and nitrate gives a blue bead in oxidising flame

Answer
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Hint: We can detect the presence of the group 3 cations with help of \[{\text{N}}{{\text{H}}_{\text{4}}}{\text{OH}}\] in presence of \[{\text{N}}{{\text{H}}_{\text{4}}}{\text{Cl}}\].
Group 3 cations are precipitated as hydroxides. From the color of the precipitate, you can tell which group 3 cation is present.

Step by step answer: An inorganic Lewis acid \[\left[ {\text{X}} \right]\] gives gelatinous white ppt. with \[{\text{N}}{{\text{H}}_{\text{4}}}{\text{OH}}\] in presence of \[{\text{N}}{{\text{H}}_{\text{4}}}{\text{Cl}}\]. The reagent \[{\text{N}}{{\text{H}}_{\text{4}}}{\text{OH}}\] in presence of \[{\text{N}}{{\text{H}}_{\text{4}}}{\text{Cl}}\] is the reagent used for the analysis of cations of group 3. The white gelatinous ppt is of aluminium hydroxide \[{\text{Al}}{\left( {{\text{OH}}} \right)_3}\] . Hence, the inorganic Lewis acid \[\left[ {\text{X}} \right]\] will contain \[{\text{A}}{{\text{l}}^{3 + }}\] ions. So the inorganic Lewis acid \[\left[ {\text{X}} \right]\] is aluminum chloride \[{\text{AlC}}{{\text{l}}_3}\] .
\[{\text{AlC}}{{\text{l}}_3}\] fumes in moist air. \[{\text{AlC}}{{\text{l}}_3}\] reacts with moisture from air to form aluminum hydroxide and hydrogen chloride gas. The fumes are of hydrogen chloride gas.
\[{\text{AlC}}{{\text{l}}_3} + {\text{3}}{{\text{H}}_2}{\text{O }} \to {\text{ Al}}{\left( {{\text{OH}}} \right)_3}{\text{ + 3HCl}} \uparrow \]
Hence, the option (A) is the correct option.
When you heat \[{\text{AlC}}{{\text{l}}_3}\] with solid \[{{\text{K}}_{\text{2}}}{\text{C}}{{\text{r}}_{\text{2}}}{{\text{O}}_{\text{7}}}\] and conc. \[{{\text{H}}_{\text{2}}}{\text{S}}{{\text{O}}_{\text{4}}}\] you obtain deep red or orange red fumes of chromyl chloride \[{\text{Cr}}{{\text{O}}_{\text{2}}}{\text{C}}{{\text{l}}_{\text{2}}}\]
Hence, the option (B) is the correct option.

When you heat \[{\text{AlC}}{{\text{l}}_3}\] with \[{\text{NaOH}}\] you get a white ppt of \[{\text{Al}}{\left( {{\text{OH}}} \right)_3}\] that dissolves in excess \[{\text{NaOH}}\] .
\[{\text{AlC}}{{\text{l}}_3} + {\text{3NaOH }} \to {\text{ Al}}{\left( {{\text{OH}}} \right)_3} \downarrow \left( {{\text{white ppt}}} \right){\text{ + 3NaCl}}\]
\[{\text{Al}}{\left( {{\text{OH}}} \right)_3} + {\text{NaOH + 2 }}{{\text{H}}_2}{\text{O}} \to {\text{ Na}}\left[ {{\text{Al}}{{\left( {{{\text{H}}_2}{\text{O}}} \right)}_2}{{\left( {{\text{OH}}} \right)}_4}} \right]\]
Hence, the option (C) is the incorrect answer.
When you heat \[{\text{AlC}}{{\text{l}}_3}\] with \[{\text{N}}{{\text{a}}_{\text{2}}}{\text{C}}{{\text{O}}_{\text{3}}}\] and nitrate you obtain a blue bead in oxidising flame. This is a cobalt nitrate charcoal test. This test is used to detect the presence of aluminium. Hence, the option (D) is also correct.

Hence, the options (A), (B) and (D) are correct options.

Note: Keep in mind while solving such questions as Group 3 cations include \[{\text{C}}{{\text{r}}^{3 + }},{\text{A}}{{\text{l}}^{3 + }}\] and \[{\text{F}}{{\text{e}}^{3 + }}\].
Aluminium hydroxide is in the form of gelatinous white precipitate. Chromium hydroxide is in the form of blue-green precipitate.Ferric hydroxide is in the form of rust red precipitate