Kinetics of the Carbonic Acid System

There are two types of carbonic acid:

 (1a) true carbonic acid: H2CO3 (1b) composite (apparent) carbonic acid: H2CO3*  =  CO2(aq) + H2CO3 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 H2CO3   = H+ + HCO3- Ktrue
(2b) composite (apparent) H2CO3* = H+ + HCO3- K1

The link between the reaction formula and the equilibrium constant is established by the law of mass action. It yields:

 (3a) Ktrue = {H+} {HCO3-} / {H2CO3} (3b) K1 = {H+} {HCO3-} / ({CO2(aq) · H2O} + {H2CO3})

The aim is to unveil the kinetics behind this thermodynamic approach. It enables us to calculate the equilibrium constants Ktrue and K1) exclusively from kinetic rates.

Reaction Kinetics

Three components, CO2(aq), H2CO3, and HCO3-, are involved in the carbonic acid formation:2

Here, the chemical kinetics is completely determined by six individual rates, k12 to k32. Translating this diagram into a set of differential equations yields:

 (4a) – d [ 1 ] / dt   =   (k12+k13) [ 1 ] – k21 [ 2 ] – k31 [ 3 ] (4b) – d [ 2 ] / dt   =   (k21+k23) [ 2 ] – k12 [ 1 ] – k32 [ 3 ] (4c) – d [ 3 ] / dt   =   (k31+k32) [ 3 ] – k13 [ 1 ] – k23 [ 2 ]
 with the abbreviations: [ 1 ]  =  H+ + HCO3 [ 2 ]  =  H2CO3 [ 3 ]  =  CO2(aq) + H2O

From experiments we know that the rates for the slow reaction (k13, k31) and for the fast reaction (k12, k21) differ considerably from each other (by more than 5 orders of magnitude):

 (5) k13, k31   ≪   k12, k21

This allows us to neglect k13 and k31 in 4a. From the equilibrium condition for 4a, i.e. d[1]/dt = 0, then follows:

 (6) – d [ 1 ] / dt  =  k12 [ 1 ] – k21 [ 2 ]  =  0 ⇒ k21 / k12 = [ 1 ] / [ 2 ]

In fact, the ratio of rates on the right-hand side defines the equilibrium constant for the true carbonic acid:

 (7) Ktrue  =  [ 1 ] / [ 2 ]  =  {H+} {HCO3-} / {H2CO3}

Entering this relation, in form of [ 1 ] = Ktrue [ 2 ], into 4c yields:

 (8) – d [ 3 ] / dt  =  (k31+k32) [ 3 ] – (k13 Ktrue + k23) [ 2 ]

Usually the quantities in the round brackets are lumped together in two new rates:

 (9a) ka  =  k13 Ktrue + k23 (9b) kb  =  k31 + k32

Again, assuming equilibrium, i.e. d[3]/dt = 0, a second equilibrium constant emerges:

 (10) K0  =  ka / kb  =  [ 3 ] / [ 2 ]  =  {CO2(aq) · H2O} / {H2CO3}

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, K0 and Ktrue:

Composite Equilibrium Constant K1

Conventionally, the two species CO2(aq) and H2CO3 are treated together as if they were one substance (denoted by H2CO3*). Just this was done in 1b above. The corresponding equilibrium constant is given in 3b, i.e.

 (11) K1  =  {H+} {HCO3-} / ({CO2(aq) · H2O} + {H2CO3})

The main idea is now to express the carbonic acid problem (characterized by two equilibrium constants Ktrue and K0 in the scheme above) by a single equilibrium constant, namely K1. We obtain the desired relation in two steps:

First, we use 7 to replace {H+}{HCO3-} in the nominator of 11:

 (12) K1 = Ktrue {H+} {H2CO3} / ({CO2(aq) · H2O} + {H2CO3}) = Ktrue / ({CO2(aq) · H2O}/{H2CO3} + 1)

Second, we insert 10 and get the final result:

 (13) K1  =  Ktrue / ( K0 + 1 )

The composite acidity constant of carbonic acid is then given by:

 (13b) pKa  =  – log K1

The interrelation between the three equilibrium constants (K1, K0, Ktrue) can be portrayed as follows:

Kinetic and Thermodynamic Parameters

For the kinetic rates the following estimates are given (at standard conditions: 25, 1 atm):

 k12 = 5·1010 M-1 s-1 Pocker & Blomkquist3 k21 = 1·107 s-1 Pocker & Blomkquist3 ka = kH2CO3 ≈ 18 s-1 Stumm & Morgan4 kb = kCO2 ≈ 0.04 s-1 Stumm & Morgan4

[Note: For most rates a parameter range – rather than a definite value – is given in literature. For example, in Stumm & Morgan we find kH2CO3 = 10 … 20 s-1 and kCO2 = 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 (k12, k21) and slow reactions (ka, kb). Since ka is an upper bound of k23 and kb is an upper bound of k32 the assumption in 5 is fully justified.

Based on these four kinetic parameters we are able to calculate the three equilibrium constants:5

 (14a) K0 = ka/kb = 450 ⇔ log K0 = 2.65 (14b) Ktrue = k21/k12 = 2.0·10-4 M ⇔ log Ktrue = -3.7 (14c) K1 = Ktrue / ( K0 + 1 ) = 4.4·10-7 M ⇔ log K1 = -6.35

In fact, these are the “official” values of Ktrue and K1 presented already here.

Notes and References

1. This is not a reaction formula for H2CO3*; it is only an abbreviation for H2CO3*

2. A PowerPoint presentation of this topic is here

3. Pocker, Y., D. Bjorkquist: Stopped-flow 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

4. 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

5. Please note the different units for K0 on one hand and for Ktrue and K1 on the other hand.