9. Show that knowing total alkalinity (A) (as mg/L CaCO$_3$) and pH the following relations
can be derived for the bicarbonate and carbonate concentrations (as mg/L CaCO$_3$):
$\text{HCO}_3^- = \frac{50,000\left\{\frac{A}{50,000} + [H^+] - \frac{K_w}{[H^+]}\right\}}{1 + \frac{2K_2}{[H^+]}}$
$\text{CO}_3^{2-} = \frac{2K_2}{[H^+]}(HCO_3^-)$
Where, $K_w$= ionization constant of water=1×10$^{-14}$ at 25°C, and $K_2$= second
dissociation constant of carbonic acid =4.68×10$^{-11}$ at 25°C.
