A water $(pH7\mathrm{.}7,$ alk $=185m\frac{g}{L}$ as $CaC{O}_3)$ has a calcium hardness of $195m\frac{g}{L}$ as $CaC{O}_3$ and a magnesium hardness of $65m\frac{g}{L}$ as $CaC{O}_3.$

a$.$ Convert the calcium hardness, magnesium hardness, and alkalinity to units of $me\frac{q}{L}.$

b$.$ Using the expression for alkalinity, calculate the $C_{t,CO3}(M)$ for this water. Note that the carbonate system has $p{K}_{a1}=6\mathrm{.}35$ and $p{K}_{a2}=10\mathrm{.}33.$

c$.$ Draw the bar diagram for the water using units of $me\frac{q}{L}.($Your bar diagram should include $4$ species relevant for softening.$)$

d$.$ For this water, determine the amount of carbonic acid, calcium carbonate hardness $(**$ only the fraction for which lime needs to be added to increase the $pH),$ calcium noncarbonate hardness, magnesium carbonate hardness, magnesium noncarbonate hardness, and the total lime and soda ash required for complete softening $($i$.$e$.,$ excess lime$-$soda softening$).$ Show all calculations, and fill in a table like the one shown below with your final answers.

$\backslash$table$[[$Constituent$,\backslash$table$[[$Concentration$],[(me\frac{q}{L})$