1) mathrm(C)_(2) mathrm(O)_(2)+mathrm(Na)_(2) mathrm(NO)_(3)+mathrm(NaOH) arrow mathrm(Na)_(2) mathrm(CrO)_(4)+mathrm(NaNO)_(2)+mathrm(H)_(2) mathrm(O) 2) mathrm(K)_(2) mathrm(~S)^+ mathrm(KC) mathrm(MnO)_(4)+mathrm(H)_(2) mathrm(O) arrow mathrm(MnO)_(2)+mathrm(S)+mathrm(KOH)

Let's correct and balance the chemical equations step by step.<br /><br />1) \( \mathrm{Cr}_{2} \mathrm{O}_{3} + 3 \mathrm{NaNO}_{3} + 4 \mathrm{NaOH} \rightarrow 2 \mathrm{Na}_{2} \mathrm{CrO}_{4} + 3 \mathrm{NaNO}_{2} + 2 \mathrm{H}_{2} \mathrm{O} \)<br /><br />2) \( 3 \mathrm{K}_{2} \mathrm{S} + 2 \mathrm{KMnO}_{4} + 4 \mathrm{H}_{2} \mathrm{O} \rightarrow 3 \mathrm{MnO}_{2} + 3 \mathrm{S} + 6 \mathrm{KOH} \)<br /><br />Now, let's balance the equations:<br /><br />1) \( \mathrm{Cr}_{2} \mathrm{O}_{3} + 3 \mathrm{NaNO}_{3} + 4 \mathrm{NaOH} \rightarrow 2 \mathrm{Na}_{2} \mathrm{CrO}_{4} + 3 \mathrm{NaNO}_{2} + 2 \mathrm{H}_{2} \mathrm{O} \)<br /><br />2) \( 3 \mathrm{K}_{2} \mathrm{S} + 2 \mathrm{KMnO}_{4} + 4 \mathrm{H}_{2} \mathrm{O} \rightarrow 3 \mathrm{MnO}_{2} + 3 \mathrm{S} + 6 \mathrm{KOH} \)<br /><br />These equations are now balanced correctly.