$(50$ points$)$ A $20$L batch of hard water is treated to remove with a strong acid cation ion$-$exchange resins

$($made of polystyrene with $8\%$ DVB crosslinking$).$ The hard water initially contains $0\mathrm{.}1MN{a}^{+}$and $0\mathrm{.}1$

$MM{g}^{2+}.$ The capacity of the resins is $1\mathrm{.}5e\frac{q}{L}$ and the resins are initially charged with $N{a}^{+}.($a$)$ What

is the volume of the resins that is needed to remove $90\%M{g}^{2+}$ in the solution? $($b$)$ After the ion$-$

exchange process is completed, regeneration of the resin is achieved by transferring the resins into a

$10$L highly concentrated $1M$ sodium chloride solution. Calculate the final concentrations of $M{g}^{2+}$ and

$N{a}^{+}$in the solution, and the concentrations of $Mg{R}_2$ and NaR in the resin.

Relative$-$Molar$-$Selectivity Coefficients $K$ for Polystyrene

Cation and Anion Exchangers with $8\%$ DVB Cross$-$Linking