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Explain: A green solution potassium manganate turns purple when $C{O_2}$ gas is passed to through the solution.

Answer
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Hint:In acidic medium the oxidation of magnesium in potassium manganate in the reactant changes from +6 to +7 to form the product after disproportionation reaction.

Complete answer:The carbon dioxide reacts with water to form carbonic acid.
$C{O_2} + {H_2}O \to {H_2}C{O_3}$
${H_2}C{O_3}$ dissociates into its constituent ions.
Carbonic acid is a weak acid, therefore it will not completely ionize into ions.
The dissociation reaction is shown below.
${H_2}C{O_3} \to {H^ + } + HCO_3^ -$
The carbonic acid dissociates into hydrogen ion and bicarbonate ion.
When manganate ion is treated with an acidic medium, it undergoes disproportionation reaction.
The disproportionation reaction is defined as the reaction in which the atom present in the compound carries an intermediate oxidation state disproportionate to generate two new products, one with higher oxidation state and one with lower oxidation state.
The disproportionation reaction is also known as dismutation. The disproportionation reaction is a type of redox reaction.
The disproportionation reaction of permanganate ion is shown below.
$3M{n^{ + 6}}O_4^{2 - } + 4{H^ + } \to 2M{n^{ + 7}}O_4^ - + M{n^{ + 4}}{O_2} + 2{H_2}O$
In this reaction the oxidation state manganate ion is +6 which on reacting with hydrogen ion forms permanganate ion in which the oxidation state of magnesium is +7, manganese dioxide in which the oxidation state of the magnesium is +4 with the by-product water.
As the oxidation state changes from +6 to +7, the green color solution of manganate changes to purple solution of permanganate.

Note:The overall reaction shows that the green color solution of potassium manganate reacts with carbon dioxide to form a green colour solution of potassium permanganate, a brown precipitate of manganese dioxide and potassium carbonate.