The Open Carbonate System

In an open carbonate system the aqueous solution is in chemical equilibrium with the CO₂ 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, CO₂(aq), and carbon dioxide in the gas phase, CO₂(g), is simple proportionality expressed by

(1a)

Henry’s law:

CO2(aq)  =  const ∙ CO2(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 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)

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)	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.

Nonlinear System of 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]	(mole balance)
(3f)	0	=   [H+] – [HCO3-] – 2 [CO3-2] – [OH-]	(charge balance)

The first four equations are mass-action laws taken from 2a to (2d); the last two equations represent the mole and charge balance. Please note the “asymmetry”: The mass-action laws are based on activities (denoted by curly braces) while the mole- and charge-balance 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, K₁, K₂), and how they are implemented in the program’s thermodynamic database, are given here.]

Equilibrium Speciation of 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) to (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’s quite instructive to compare the above result with 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 the pCO2 (or CO₂ 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 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 from the atmosphere (which increases the DIC). That’s just the opposite behavior of the closed CO₂ system.

Remarks