Lets say I want to model a water system as the following set of equilibria:

$\ce{H2O <=> OH- + H3O+}$ | $\ce{A=[OH^{-}][H3O+]}$

$\ce{CO2 + H2O <=> H2CO3}$ | $\ce{B=\frac{[H2CO3]}{[CO2]}}$

$\ce{H2CO3 + H2O <=> HCO3- + H3O+}$ | $\ce{C=\frac{[HCO3-][H3O+]}{[H2CO3]}}$

$\ce{HCO3 + H2O <=> CO3^{2-} + H3O+}$ | $\ce{D=\frac{[CO3^{2-}][H3O+]}{[HCO3-]}}$

The whole system starts off as just $\ce{H2O}$ and $\ce{CO2}$ from the atmosphere. Now consider that calcium carbonate ($\ce{CaCO3}$) from limestone is going to react with carbonic acid in the water ($\ce{H2CO3}$) to cause the dissolution of calcium ions, increasing the water hardness (We will ignore the small affect of calcium carbonate dissolving on its own very slowly). Thus the following equilibrium also exists:

$\ce{CaCO3 + H2CO3 <=> Ca^{2+} + 2HCO3-}$ | $\ce{E=\frac{[Ca^{2+}][HCO3-]^2}{[H2CO3]}}$

Using basic math, I can find the relationship between hydronium and calcium concentration:

$\ce{\frac{(E[H3O+]^2)}{(C^2 B[CO2] )}=[Ca^{2+}]}$

This can be linked to pH:

$\ce{\frac{E}{(C^2 B[CO2 ](10^{2pH}) )}=[Ca^{2+}]}$

My question is, can I change this to take into consideration the change in pH over time caused by the change in Carbonic Acid etc. without a computer?


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