The Open Carbonate System
In an open carbonate system the aqueous solution is in chemical equilibrium with the CO_{2} of the atmosphere.^{1}
In contrast to the closed system, where the total amount of inorganic carbon (DIC) remains constant when 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 simple proportionality expressed by
(1a)  Henry’s law:  CO_{2}(aq) = const ∙ CO_{2}(g) 
The proportionality factor is Henry’s constant. But before specifying its value be aware of the quantities (and units) on both sites of this equation. Let’s agree about 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) 
Partial pressure P_{CO2} is an input parameter and can be entered here.
Equilibrium Thermodynamics
Altogether, the system is described by four equilibrium reactions and their corresponding 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.
Nonlinear System of 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}]  (mole balance) 
(3f)  0  = [H^{+}] – [HCO_{3}^{}] – 2 [CO_{3}^{2}] – [OH^{}]  (charge balance) 
The first four equations are massaction laws taken from 2a to (2d); the last two equations represent the mole and charge balance. Please note the “asymmetry”: The massaction laws are based on activities (denoted by curly braces) while the mole and chargebalance equations rely on molar concentrations (denoted by square brackets).
Note 1. Remove 3a, and what you get is the set of equations that describes the closed system (based on five equations only).
Note 2. C_{T} in 3e is the total inorganic carbon, usually abbreviated by DIC.
[More details about the three equilibrium constants (K_{H}, K_{1}, K_{2}), and how they are implemented in the program’s thermodynamic database, are given here.]
Equilibrium Speciation of 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) to (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’s quite instructive to compare the above result with 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 the pCO2 (or CO_{2} partial pressure); in a closed system you enter DIC. (You cannot enter both values independently.) However, you can interchange the roles formally 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 especially relevant 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 from the atmosphere (which increases the DIC). That’s just the opposite behavior of the closed CO_{2} system.
Remarks

More about the open_ and closed systems (and the difference between them) is given here and as PowerPoint. ↩

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

Here 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. ↩

pCO2 ≈ 1.5 is typical for groundwater, where the hundred times larger CO_{2} emerges from degradation of organic matter. ↩