Suppose that the reactions between carbon dioxide, hydrogen ions, bicarbonate and carbonate ions, all dissolved in water, are as follows:
Note that reaction  is actually the combination of two elementary reactions:
:  +     H + 
Assuming the mass action law applies to these two reactions, that water is abundant, and that the different chemical species are always well-mixed, their rate equations are
where  denotes concentration, t is time, and K and k are appropriate proportionality constants for reaction , called respectively the forwards and reverse rate constants for this reaction. (Similarly K and k for reaction .)
, the concentrations are unchanging, hence the left hand sides of these equations are zero. Then, from the first of these four equations, the ratio of reaction s rate constants equals the ratio of its equilibrium concentrations, and this ratio, called K', is called the equilibrium constant for reaction , i.e.
where the subscript eq denotes that these are equilibrium concentrations.
Similarly, from the fourth equation for the equilibrium constant K for reaction ,
Rearranging  gives
and rearranging , then substituting in , gives
The total concentration of dissolved inorganic carbon in the system is given by substituting in  and :
Re-arranging this gives the equation for :
The equations for  and  are obtained by substituting this into  and .