# The Open Carbonate System In an open carbonate system, the aqueous solution is in chemical equilibrium with atmospheric CO2.1

Unlike the closed system, where the total amount of inorganic carbon (DIC) remains constant as the pH changes, the amount of DIC in an open system increases with increasing pH.

Henry’s Law

The relationship between dissolved carbon dioxide, CO2(aq), and carbon dioxide in the gas phase, CO2(g), is a simple proportionality given by

 (1a) Henry’s law: CO2(aq)  =  const ∙ CO2(g)

The proportionality factor is Henry’s constant. However, before you specify its value, you should be clear about the quantities (and units) on both sides of this equation. Let’s agree on two things: First, instead of CO2(g) we use the partial pressure PCO2 in atm. Second, instead of CO2(aq) we use the composite carbonic acid H2CO3*. This yields2

 (1b) {H2CO3*}  =  KH ∙ PCO2 with   KH = 10-1.47 M atm-1   (at 25 °C)

The partial pressure PCO2 is an input parameter and can be entered here.

Equilibrium Thermodynamics

The system is described by four equilibrium reactions and their associated equilibrium constants:3

 (2a) CO2(g) ⇔ H2CO3* log KH = -1.47 (2b) H2CO3* ⇔ H+ + HCO3- log K1 = -6.35 (2c) HCO3- ⇔ H+ + CO3-2 log K2 = -10.33 (2d) H2O ⇔ H+ + OH- log KW = -14.0

The chemical species are interrelated as follows: The four equilibrium reactions – expressed by the law of mass action – constitute the backbone of the mathematical description. This is our next step.

Algebraic System of Nonlinear Equations

The open CO2-H2O system is characterized by 6 species (or unknowns):

 CO2(g),   H2CO3*,   HCO3-,   CO3-2,   H+   and   OH- (or H2O)

Accordingly, we need 6 equations to solve for them:

 (3a) KH =   {H2CO3*} / PCO2 =   10-1.47 (3b) K1 =   {H+} {HCO3-} / {H2CO3*} =   10-6.35 (3c) K2 =   {H+} {CO3-2} / {HCO3-} =   10-10.33 (3d) Kw =   {H+} {OH-} =   10-14.0 (3e) CT =   [H2CO3*] + [HCO3-] + [CO3-2] (mass balance) (3f) 0 =   [H+] – [HCO3-] – 2 [CO3-2] – [OH-] (charge balance)

The first four equations are mass-action laws in 2a to (2d); the last two equations represent the mass and charge balance. Note the “asymmetry”: The mass-action laws are based on activities (indicated by curly braces) while the balance equations are based on molar concentrations (indicated by square brackets).

Note 1. Remove 3a, and you get the set of equations describing the closed system (based on only five equations).

Note 2. CT in 3e is the total inorganic carbon, usually abbreviated by DIC.

[More details on the three equilibrium constants (KH, K1, K2) and how they are implemented in the program’s thermodynamic database can be found here.]

Equilibrium Speciation in the Open CO2-H2O System

For a given partial pressure PCO2, the open CO2-H2O system is completely determined by the set of equations (3a) through (3f). Under normal atmospheric conditions (PCO2 = 0.00039 atm, 25), we get the following equilibrium speciation:4

 Input: pCO2 3.408 ( = – log PCO2 ) Output: pH 5.61 CO2 0.0133 mM ( = H2CO3* ) HCO3- 0.0024 mM CO3-2 4.7·10-8 mM DIC 0.0157 mM ( = CO2 + HCO3- + CO3-2 )

This is the composition of pristine rainwater.

Open vs Closed System

It is quite instructive to compare the above result with that of the closed CO2 system:

open system closed system
input     pCO2 = 3.408 DIC = 1 mM
pH     5.61 4.68
CO2 mM   0.0133 0.979
HCO3- mM   0.0024 0.021
CO3-2 mM   4.7·10-8 4.8·10-8
DIC mM   0.0157 1.000
pCO2     3.408 1.54

In an open system, you enter pCO2 (or CO2 partial pressure); in a closed system, you enter DIC. (You cannot specify both values at the same time.) However, you can formally reverse the roles by imitating:

 • an “open system in contact with atmosphere” by entering 0.0157 mM DIC in a closed system • a “closed system with (1 mM DIC)” by an “open system with pCO2 = 1.54”5 The concept of open/closed systems becomes particularly interesting when the solution is attacked by acids or bases:

• in a open system, the CO2 (or pCO2 value) remains constant
• in a closed system DIC remains constant (and CO2 changes) Example: Titration Calculation

The diagram below displays the results of a titration calculation (addition of HCl and NaOH to an open CO2 system). Note how DIC grows exponentially for pH > 5.6. The more alkaline the solution becomes, the more CO2 is sucked out of the atmosphere (which increases the DIC). This is exactly the opposite behavior of the closed CO2 system.

Remarks & Footnotes

1. More about the open and closed system (and the difference between the two) can be found here and here

2. Curly braces {..} denote activities while square brackets [..] molar concentrations

3. The equilibrium constants refer to 25

4. Start with pure water (button H2O and select “Open CO2 System” to enter the pCO2 value, then button Start. The carbonate speciation is displayed in table Ions

5. The value pCO2 ≈ 1.5 is typical for groundwater, where the hundredfold CO2 emerges from the degradation of organic matter.