In which of the following cases there is a change in oxidation number?

$SO_2$ gas is passed through acidified $Cr_2O_7^{2-}$ solution

The correct answer is Option 2. $SO_2$ gas is passed through acidified $Cr_2O_7^{2-}$ solution.

Let us examine each case to determine if there is a change in the oxidation number of chromium (Cr):

(1) Aqueous solution of $CrO_4^{2-}$ is acidified. In this case, the chromate ion (\(CrO_4^{2-}\)) is present. The oxidation state of chromium in chromate is +6. When the solution is acidified, chromate ions are converted to dichromate ions (\(Cr_2O_7^{2-}\)). The oxidation state of chromium in dichromate is also +6. There is no change in the oxidation state of chromium.

(2) $SO_2$ gas is passed through acidified $Cr_2O_7^{2-}$ solution. In acidified \(Cr_2O_7\) solution, dichromate ions (\(Cr_2O_7^{2-}\)) are present with chromium in the +6 oxidation state. When \(SO_2\) gas is passed through, it is oxidized to sulfuric acid (\(H_2SO_4\)), and chromium is reduced to chromium(III) ions (\(Cr^{3+}\)). The oxidation state of chromium changes from +6 to +3.

(3) $Cr_2O_7^{2-}$ solution is made alkaline. In an alkaline solution, dichromate ions (\(Cr_2O_7^{2-}\)) are reduced to chromate ions (\(CrO_4^{2-}\)). The oxidation state of chromium remains at +6, so there is no change in the oxidation state.

(4) $CrO_2Cl_2$ is dissolved in $NaOH$. In this case, chromyl chloride (\(CrO_2Cl_2\)) is present, where chromium is in the +6 oxidation state. When dissolved in \(NaOH\), it undergoes hydrolysis to form chromate ions (\(CrO_4^{2-}\)) and chloride ions (\(Cl^-\)). The oxidation state of chromium remains +6.

So, the case where there is a change in the oxidation number of chromium is (2) \(SO_2\) gas is passed through acidified \(Cr_2O_7\) solution.