$($a$)$ Write all reactions and equations for an open carbonate system.

$($b$)$ Derive the equations for $log[H_{2}C{O}_3^{**}],log[HC{O}_3^{-}],$ and $log[C{O}_3^{2-}]$ written with $pH,p{K}_{al},p{K}_{a2},K_H,$

and$/$or $p_{CO2}.$

$($c$)$ Assume that we fail to control $C{O}_2$ emission $($which is unfortunate but very likely$)$ and as a result,

$p_{CO2}$ in the atmosphere increases to $0\mathrm{.}31$atm. Draw a $logC-pH$ diagram and find the resulting rainwater

$pH.$