The Open Carbonate System
In an open carbonate system, the aqueous solution is in chemical equilibrium with atmospheric CO_{2}.^{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, CO_{2}(aq), and carbon dioxide in the gas phase, CO_{2}(g), is a simple proportionality given by
(1a)  Henry’s law:  CO_{2}(aq) = const ∙ CO_{2}(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 CO_{2}(g) we use the partial pressure P_{CO2} in atm. Second, instead of CO_{2}(aq) we use the composite carbonic acid H_{2}CO_{3}^{*}. This yields^{2}
(1b)  {H_{2}CO_{3}^{*}} = K_{H} ∙ P_{CO2}  with K_{H} = 10^{1.47} M atm^{1} (at 25 °C) 
The partial pressure P_{CO2} 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)  CO_{2}(g) ⇔ H_{2}CO_{3}^{*}  log K_{H}  = 1.47  
(2b)  H_{2}CO_{3}^{*} ⇔ H^{+} + HCO_{3}^{}  log K_{1}  = 6.35  
(2c)  HCO_{3}^{} ⇔ H^{+} + CO_{3}^{2}  log K_{2}  = 10.33  
(2d)  H_{2}O ⇔ H^{+} + OH^{}  log K_{W}  = 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 CO_{2}H_{2}O system is characterized by 6 species (or unknowns):
CO_{2}(g), H_{2}CO_{3}^{*}, HCO_{3}^{}, CO_{3}^{2}, H^{+} and OH^{} (or H_{2}O) 
Accordingly, we need 6 equations to solve for them:
(3a)  K_{H}  = {H_{2}CO_{3}^{*}} / P_{CO2}  = 10^{1.47}  
(3b)  K_{1}  = {H^{+}} {HCO_{3}^{}} / {H_{2}CO_{3}^{*}}  = 10^{6.35}  
(3c)  K_{2}  = {H^{+}} {CO_{3}^{2}} / {HCO_{3}^{}}  = 10^{10.33}  
(3d)  K_{w}  = {H^{+}} {OH^{}}  = 10^{14.0}  
(3e)  C_{T}  = [H_{2}CO_{3}^{*}] + [HCO_{3}^{}] + [CO_{3}^{2}]  (mass balance)  
(3f)  0  = [H^{+}] – [HCO_{3}^{}] – 2 [CO_{3}^{2}] – [OH^{}]  (charge balance) 
The first four equations are massaction laws in 2a to (2d); the last two equations represent the mass and charge balance. Note the “asymmetry”: The massaction 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. C_{T} in 3e is the total inorganic carbon, usually abbreviated by DIC.
[More details on the three equilibrium constants (K_{H}, K_{1}, K_{2}) and how they are implemented in the program’s thermodynamic database can be found here.]
Equilibrium Speciation in the Open CO_{2}H_{2}O System
For a given partial pressure P_{CO2}, the open CO_{2}H_{2}O system is completely determined by the set of equations (3a) through (3f). Under normal atmospheric conditions (P_{CO2} = 0.00039 atm, 25), we get the following equilibrium speciation:^{4}
Input:  pCO2  3.408  ( = – log P_{CO2} )  
Output:  pH  5.61  
CO_{2}  0.0133  mM  ( = H_{2}CO_{3}^{*} )  
HCO_{3}^{}  0.0024  mM  
CO_{3}^{2}  4.7·10^{8}  mM  
DIC  0.0157  mM  ( = CO_{2} + HCO_{3}^{} + CO_{3}^{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 CO_{2} system:
open system  closed system  

input  pCO2 = 3.408  DIC = 1 mM  
pH  5.61  4.68  
CO_{2}  mM  0.0133  0.979  
HCO_{3}^{}  mM  0.0024  0.021  
CO_{3}^{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 CO_{2} 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 CO_{2} (or pCO2 value) remains constant
 in a closed system DIC remains constant (and CO_{2} changes)
Example: Titration Calculation
The diagram below displays the results of a titration calculation (addition of HCl and NaOH to an open CO_{2} system). Note how DIC grows exponentially for pH > 5.6.
The more alkaline the solution becomes, the more CO_{2} is sucked out of the atmosphere (which increases the DIC). This is exactly the opposite behavior of the closed CO_{2} system.
Remarks & Footnotes

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

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

The equilibrium constants refer to 25. ↩

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. ↩

The value pCO2 ≈ 1.5 is typical for groundwater, where the hundredfold CO_{2} emerges from the degradation of organic matter. ↩