Overview Activity Chem Equilibrium Acids Carbonate System Mineral Phases

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 P_CO2 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 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)		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. C_T in 3e is the total inorganic carbon, usually abbreviated by DIC.

[More details on the three equilibrium constants (K_H, 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 P_CO2, the open CO₂-H₂O 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 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