Calculate the oxidation number of sulfur,chromium,and nitrogen in $H_2SO_5$,$Cr_2O_7^{2-}$,and $NO_3^-$. Suggest the structures of these compounds and explain the fallacy in the direct calculation of the oxidation number for $H_2SO_5$.

(N/A) $(i)$ $H_2SO_5$ (Caro's acid):
Applying the standard rule: $2(+1) + S + 5(-2) = 0 \implies S = +8$.
Since the maximum oxidation state of sulfur is $+6$,this result is a fallacy.
The structure contains a peroxide linkage $(-O-O-)$. Thus,two oxygen atoms have an oxidation state of $-1$ and three have $-2$.
Calculation: $2(+1) + S + 2(-1) + 3(-2) = 0 \implies 2 + S - 2 - 6 = 0 \implies S = +6$.
$(ii)$ $Cr_2O_7^{2-}$ (Dichromate ion):
$2(Cr) + 7(-2) = -2 \implies 2Cr - 14 = -2 \implies 2Cr = +12 \implies Cr = +6$.
$(iii)$ $NO_3^-$ (Nitrate ion):
$N + 3(-2) = -1 \implies N - 6 = -1 \implies N = +5$.