A five$-$stage countercurrent extraction extraction battery is used to extract the sludge from the reaction

$N{a}_{2}C{O}_3+$CaO$+H_{2}OCaC{O}_3+2$NaOH

The $CaC{O}_3$ carries with it $1\mathrm{.}5$ times its weight of solution in flowing from one unit to another. It is desired to recover $99$ percent of the

NaOH. The products from the reaction enter the first unit with no excess reactants but with $0\mathrm{.}6kg$ of $H_{2}O$ per kilogram of $CaC{O}_3.$

$($a$)$ How much wash water must be used per kilogram of calcium carbonate?

$($b$)$ What is the concentration of the solution leaving each unit, assuming that $CaC{O}_3$ is completely insoluble?