$(5)$ Limestone is a type of sedimentary rock mostly composed of calcium carbonate

$(CaC{O}_3).$ The presence of limestone in certain geographical regions can be crucial

in determining several natural water chemistry parameters including alkalinity, $pH,$

and dissolved calcium concentration. The reaction between calcium carbonate and

dissolved $C{O}_2$ is shown in the diagram below. All other reactions including $CaC{O}_3($s$)$

precipitation$/$dissolution and open$-$carbonate system have already been introduced

to you.

Equilibrium Constants:

$p{K}_{sp}CaC{O}_3(s)$ precipitation $=8\mathrm{.}48$

$pK{a}_1=6\mathrm{.}35$

$pK{a}_2=10\mathrm{.}33$

$K{H}_{H^{\text{'}}}={10}^{-3\mathrm{.}5}\frac{M}{a}tmat25C$

$P_{CO2}={10}^{-1\mathrm{.}5}$atm

a$)$ Determine an expression for $K^{\text{'}}($see diagram above$)$ in terms of equilibrium

$K_{sp},K_{a1}$ and $K_{a2}.$ Write down all the equilibrium reactions involved and their

associated equilibrium constant expressions. What is the value of $K^{\text{'}}?(8$

points$)$

b$)$ Determine the lake's $pH$ and the molar concentrations of $C{O}_{2(aq)}$

$C{a}^{2+},$

$HC{O}_3^{-}$

$H^{+}$

$O{H}^{-},$ and $C{O}_3^{2-}$ if this lake is in equilibrium with calcium

carbonate and atmospheric $C{O}_2($at $25C).($bonus$)5$ points