A water, pH $5\mathrm{.}5,$ is dosed with $80$mg$/$ liter of alum. $C_{T,C{O}_3}=1{10}^{-4}.$

$($a$)$ How much hydrated lime in $($mol$/$L$),Ca(OH{)}_2,$ is needed so that the $pH$ does not change during coagulation?

$($b$)$ How much hydrated lime in $($mol$/$L$)$ must be added to produce $$ water with $pH$ of $7\mathrm{.}5$ after coagulation?

Please both answers need to be in mol$/$L