I've always written it as
CO2 + H2O ==> H2CO3 but I wouldn't count yours wrong.
Then the two ionizations are
H2CO3 --> H^+ + HCO3^- and
HCO3^- ==> H^+ + CO3^2-
Here is a great article about CO2 which includes how it dissolves in water. Most of it is simply dissolved CO2.
http://en.wikipedia.org/wiki/Carbon_dioxide
Is the equation of the chemical reaction that occurs when you blow your breath through boiling water?
CO2 + H2O -> H + HCO3
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