Kinetics of the Carbonic Acid System
There are two types of carbonic acid:
(1a)  true carbonic acid:  H_{2}CO_{3} 
(1b)  composite (apparent) carbonic acid:  H_{2}CO_{3}^{*} = CO_{2}(aq) + H_{2}CO_{3} ^{1} 
Each of these two acids is characterized by its own equilibrium constant (for the first dissociation step):
carbonic acid  reaction formula  equilibrium constant  

(2a)  true  H_{2}CO_{3} = H^{+} + HCO_{3}^{}  K_{true} 
(2b)  composite (apparent)  H_{2}CO_{3}^{*} = H^{+} + HCO_{3}^{}  K_{1} 
The link between the reaction formula and the equilibrium constant is established by the law of mass action. It yields:
(3a)  K_{true}  =  {H^{+}} {HCO_{3}^{}} / {H_{2}CO_{3}}  
(3b)  K_{1}  =  {H^{+}} {HCO_{3}^{}} / ({CO_{2}(aq) · H_{2}O} + {H_{2}CO_{3}}) 
The aim is to unveil the kinetics behind this thermodynamic approach. It enables us to calculate the equilibrium constants K_{true} and K_{1}) exclusively from kinetic rates.
Reaction Kinetics
Three components, CO_{2}(aq), H_{2}CO_{3}, and HCO_{3}^{}, are involved in the carbonic acid formation:^{2}
Here, the chemical kinetics is completely determined by six individual rates, k_{12} to k_{32}. Translating this diagram into a set of differential equations yields:
(4a)  – d [ 1 ] / dt = (k_{12}+k_{13}) [ 1 ] – k_{21} [ 2 ] – k_{31} [ 3 ] 
(4b)  – d [ 2 ] / dt = (k_{21}+k_{23}) [ 2 ] – k_{12} [ 1 ] – k_{32} [ 3 ] 
(4c)  – d [ 3 ] / dt = (k_{31}+k_{32}) [ 3 ] – k_{13} [ 1 ] – k_{23} [ 2 ] 
with the abbreviations:  [ 1 ] = H^{+} + HCO_{3} 
[ 2 ] = H_{2}CO_{3}  
[ 3 ] = CO_{2}(aq) + H_{2}O 
From experiments we know that the rates for the slow reaction (k_{13}, k_{31}) and for the fast reaction (k_{12}, k_{21}) differ considerably from each other (by more than 5 orders of magnitude):
(5)  k_{13}, k_{31} ≪ k_{12}, k_{21} 
This allows us to neglect k_{13} and k_{31} in 4a. From the equilibrium condition for 4a, i.e. d[1]/dt = 0, then follows:
(6)  – d [ 1 ] / dt = k_{12} [ 1 ] – k_{21} [ 2 ] = 0  ⇒  k_{21} / k_{12} = [ 1 ] / [ 2 ] 
In fact, the ratio of rates on the righthand side defines the equilibrium constant for the true carbonic acid:
(7)  K_{true} = [ 1 ] / [ 2 ] = {H^{+}} {HCO_{3}^{}} / {H_{2}CO_{3}} 
Entering this relation, in form of [ 1 ] = K_{true} [ 2 ], into 4c yields:
(8)  – d [ 3 ] / dt = (k_{31}+k_{32}) [ 3 ] – (k_{13} K_{true} + k_{23}) [ 2 ] 
Usually the quantities in the round brackets are lumped together in two new rates:
(9a)  k_{a} = k_{13} K_{true} + k_{23}  
(9b)  k_{b} = k_{31} + k_{32} 
Again, assuming equilibrium, i.e. d[3]/dt = 0, a second equilibrium constant emerges:
(10)  K_{0} = k_{a} / k_{b} = [ 3 ] / [ 2 ] = {CO_{2}(aq) · H_{2}O} / {H_{2}CO_{3}} 
In this way, the above diagram (where three components are interrelated by 6 kinetic rates) reduces to a simpler diagram where the same components are related by no more than two equilibrium constants, K_{0} and K_{true}:
Composite Equilibrium Constant K_{1}
Conventionally, the two species CO_{2}(aq) and H_{2}CO_{3} are treated together as if they were one substance (denoted by H_{2}CO_{3}^{*}). Just this was done in 1b above. The corresponding equilibrium constant is given in 3b, i.e.
(11)  K_{1} = {H^{+}} {HCO_{3}^{}} / ({CO_{2}(aq) · H_{2}O} + {H_{2}CO_{3}}) 
The main idea is now to express the carbonic acid problem (characterized by two equilibrium constants K_{true} and K_{0} in the scheme above) by a single equilibrium constant, namely K_{1}. We obtain the desired relation in two steps:
First, we use 7 to replace {H^{+}}{HCO_{3}^{}} in the nominator of 11:
(12)  K_{1}  =  K_{true} {H^{+}} {H_{2}CO_{3}} / ({CO_{2}(aq) · H_{2}O} + {H_{2}CO_{3}}) 
=  K_{true} / ({CO_{2}(aq) · H_{2}O}/{H_{2}CO_{3}} + 1) 
Second, we insert 10 and get the final result:
(13)  K_{1} = K_{true} / ( K_{0} + 1 ) 
The composite acidity constant of carbonic acid is then given by:
(13b)  pK_{a} = – log K_{1} 
The interrelation between the three equilibrium constants (K_{1}, K_{0}, K_{true}) can be portrayed as follows:
Kinetic and Thermodynamic Parameters
For the kinetic rates the following estimates are given (at standard conditions: 25, 1 atm):
k_{12}  =  5·10^{10} M^{1} s^{1}  Pocker & Blomkquist^{3}  
k_{21}  =  1·10^{7} s^{1}  Pocker & Blomkquist^{3}  
k_{a}  =  k_{H2CO3}  ≈  18 s^{1}  Stumm & Morgan^{4} 
k_{b}  =  k_{CO2}  ≈  0.04 s^{1}  Stumm & Morgan^{4} 
[Note: For most rates a parameter range – rather than a definite value – is given in literature. For example, in Stumm & Morgan we find k_{H2CO3} = 10 … 20 s^{1} and k_{CO2} = 0.025 … 0.04 s^{1}.]
The numerical values of the kinetic rates in the table reveal the clear separation (or broad gap) between fast reactions (k_{12}, k_{21}) and slow reactions (k_{a}, k_{b}). Since k_{a} is an upper bound of k_{23} and k_{b} is an upper bound of k_{32} the assumption in 5 is fully justified.
Based on these four kinetic parameters we are able to calculate the three equilibrium constants:^{5}
(14a)  K_{0}  =  k_{a}/k_{b}  =  450  ⇔  log K_{0}  =  2.65  
(14b)  K_{true}  =  k_{21}/k_{12}  =  2.0·10^{4} M  ⇔  log K_{true}  =  3.7  
(14c)  K_{1}  =  K_{true} / ( K_{0} + 1 )  =  4.4·10^{7} M  ⇔  log K_{1}  =  6.35 
In fact, these are the “official” values of K_{true} and K_{1} presented already here.
Notes and References

This is not a reaction formula for H_{2}CO_{3}^{*}; it is only an abbreviation for H_{2}CO_{3}^{*}. ↩

Pocker, Y., D. Bjorkquist: Stoppedflow studies of carbon dioxide hydration and bicarbonate dehydration in H2O and D2O acid–base and metal ion catalysis. J. Am. Chem. Soc. 99, 6537–6543, 1977 ↩ ↩^{2}

W. Stumm and J.J. Morgan: Aquatic Chemistry, Chemical Equilibria and Rates in Natural Waters, 3rd ed. John Wiley & Sons, Inc., New York, 1996 ↩ ↩^{2}

Please note the different units for K_{0} on one hand and for K_{true} and K_{1} on the other hand. ↩