Overview Activity Chem Equilibrium Acids Carbonate System Mineral Phases

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 K_true 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 the above 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 one knows 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 ] = K_true [ 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 K_true:

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 K_true 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, 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):

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 specific 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 (k₁₂, k₂₁) and slow reactions (k_a, k_b). Since k_a is an upper bound of k₂₃ and k_b 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 K_true and K₁ presented already here.

Notes and References