1. THE RELATIONSHIP BETWEEN H2CO3* AND HCO3-
We can rearrange the expression for K1 to obtain:
This equation shows that, when pH = pK1, the activities of carbonic acid and bicarbonate are equal.
We can also rearrange the expression for K2 to obtain:
This equation shows that, when pH = pK2, the activities of bicarbonate and carbonate ion are equal.
Simple rearrangement of the equilibrium constants for the first and second dissociation reactions of H2CO3* leads to the equations shown in this slide. If we take the logarithm of these two equations and rearrange them further (make sure you can do this!), we obtain:
pH = pK1 + log (aHCO3-/aH2CO3*)
and
pH = pK2 + log (aCO32-/aHCO3-)
These equations demonstrate that when pH = pKa, the activities of the two species involved are equal. If pH < pKa, then the acidic (more protonated) species has a higher activity (is dominant), whereas if pH > pKa, the more basic (less protonated) species has a higher activity (is dominant).
For example, at 25°C, if pH = pK1 = 6.36, the activities of carbonic acid (H2CO3*) and bicarbonate (HCO3-) are equal. At pH < 6.35, carbonic acid is predominant, and at pH > 6.35, bicarbonate is the dominant species.

2. In most natural waters, bicarbonate is the dominant carbonate species!
Bjerrum plot showing the activities of inorganic carbon species as a function of pH for a value of total inorganic carbon of 10-3 mol L-1.
Although Bjerrum plots can be constructed rigorously by solving the combined mass-action and mass-balance expressions in the system for the concentrations of each of the species, there is a faster, approximate route to the construction of these diagrams. Once the total carbonate concentration (CT) is chosen and the pK values are known, the first step is to plot points with pH coordinates equal to the pK values, and concentration coordinates equal to log CT At pH = pK, the concentrations of two species are equal, and therefore equal to CT/2, the log of which is log CT For example, at pH = pK1 = 6.35, the concentrations of H2CO3* and HCO3- are equal to one another and to CT/2. Likewise, at pH = pK2 = 10.33, the concentrations of HCO3- and CO32- are equal to one another and to CT/2. The points where species concentrations are equal are called cross-over points. At pH < pK1 = 6.35, H2CO3* accounts for more than 99% of CT, so the concentration of H2CO3* plots as a horizontal line with a Y-intercept of log CT. As pH nears pK1, the line must bend down to intersect the HCO3- line at the first cross-over point. The HCO3- line extends from the first cross-over point towards lower pH with a slope of +1. At pK1 < pH < pK2, HCO3- accounts for the bulk of CT, so its concentration now plots as a horizontal line. In this pH range, the H2CO3* line descends away from the cross-over point towards higher pH with a slope of -1. As pH approaches pK2, the HCO3- line drops down to the second cross-over point. At pH > pK2, CO32- is the predominant species, so its concentration now plots as a horizontal line at log CT, and the HCO3- line descends from the second cross-over point towards higher pH with a slope of -1. In the range pH > pK2, the H2CO3* line now descends towards higher pH with a slope of -2. As the CO32- line passes through the second cross-over point towards lower pH into the region where pK1 < pH < pK2, it descends with a slope of +1. When this same line crosses under the first cross-over point into the region where pH < pK1, its slope changes to +2.
In most natural waters, bicarbonate is the dominant carbonate species!

3. THE CO2-H2O SYSTEM - I
Carbonic acid is a weak acid of great importance in natural waters. The first step in its formation is the dissolution of CO2(g) in water according to:
CO2(g)  CO2(aq)
At equilibrium we have:
Once in solution, CO2(aq) reacts with water to form carbonic acid:
CO2(aq) + H2O(l)  H2CO30
The equilibrium constant KCO2 is a measure of the solubility of CO2(g) in water. It is sometimes referred to as the Henry’s Law constant. The reaction of CO2(aq) to H2CO30 is relatively slow; it is sufficiently slow that an enzyme is required to accelerate the reaction in the bloodstream.
Note that true H2CO30 is a much stronger acid that CO2(aq). It may not be apparent that CO2(aq) is an acid at all, at least in the Bronsted sense, because there are no protons in its formula. However, one must keep in mind that CO2(aq) is surrounded by water molecules, which do have protons that can be donated. In any given solution, the majority of CO2 is present as CO2(aq) and only a relatively small amount is present as true carbonic acid (H2CO30). For example, the equilibrium constant at 25°C for the reaction:
H2CO30  CO2(aq) + H2O(l)
is on the order of 650, indicating that the equilibrium lies far to the right, and CO2(aq) is the dominant species.

4. THE CO2-H2O SYSTEM - II
In practice, CO2(aq) and H2CO30 are combined and this combination is denoted as H2CO3*. It’s formation is dictated by the reaction:
CO2(g) + H2O(l)  H2CO3*
For which the equilibrium constant at 25°C is:
Most of the dissolved CO2 is actually present as CO2(aq); only a small amount is actually present as true carbonic acid H2CO30.
For convenience, both forms of dissolved CO2 are usually lumped together and denoted as H2CO3*. Once again, from a thermodynamic point of view, the exact nature of dissolved carbon dioxide is not important because thermodynamics only deals with macroscopic properties. However, when reaction rates or mechanisms are of interest to us, then the exact species is important. In this course, when we are clearly dealing with thermodynamic considerations and total dissolved CO2 (i.e., H2CO3*) it is permissible to drop the asterisk (*).
A mentioned previously, the reaction:
CO2(aq) + H2O(l)  H2CO30
is comparatively slow. It exhibits first-order kinetics with a rate constant at 0°C of k = 2.010-3 sec-1. This means that t½ = 0.693/(2.010-3 sec-1) = sec = min. In other words, CO2(aq) has a half-life of nearly 6 minutes at this temperature. After 6 minutes, half of the original amount of CO2(aq) will have been converted to H2CO30.

5. THE CO2-H2O SYSTEM - III
Carbonic acid (H2CO3*) is a weak acid that dissociates according to:
H2CO3*  HCO3- + H+
For which the dissociation constant at 25°C and 1 bar is:
Bicarbonate then dissociates according to:
HCO3-  CO32- + H+
Although H2CO3* is quite a weak acid, it is important to keep in mind that true carbonic acid, H2CO30, is a much stronger acid. For example,
H2CO30  HCO3- + H+
has a pKa value of , as opposed to 6.35 for H2CO3*. The reason the latter is a much weaker acid is because CO2(aq) is a much weaker acid, it is present in much higher quantity than H2CO30, and the kinetics of conversion of CO2(aq) to H2CO30 are quite slow.

6. In most natural waters, bicarbonate is the dominant carbonate species!
Bjerrum plot showing the activities of inorganic carbon species as a function of pH for a value of total inorganic carbon of 10-3 mol L-1.
Although Bjerrum plots can be constructed rigorously by solving the combined mass-action and mass-balance expressions in the system for the concentrations of each of the species, there is a faster, approximate route to the construction of these diagrams. Once the total carbonate concentration (CT) is chosen and the pK values are known, the first step is to plot points with pH coordinates equal to the pK values, and concentration coordinates equal to log CT At pH = pK, the concentrations of two species are equal, and therefore equal to CT/2, the log of which is log CT For example, at pH = pK1 = 6.35, the concentrations of H2CO3* and HCO3- are equal to one another and to CT/2. Likewise, at pH = pK2 = 10.33, the concentrations of HCO3- and CO32- are equal to one another and to CT/2. The points where species concentrations are equal are called cross-over points. At pH < pK1 = 6.35, H2CO3* accounts for more than 99% of CT, so the concentration of H2CO3* plots as a horizontal line with a Y-intercept of log CT. As pH nears pK1, the line must bend down to intersect the HCO3- line at the first cross-over point. The HCO3- line extends from the first cross-over point towards lower pH with a slope of +1. At pK1 < pH < pK2, HCO3- accounts for the bulk of CT, so its concentration now plots as a horizontal line. In this pH range, the H2CO3* line descends away from the cross-over point towards higher pH with a slope of -1. As pH approaches pK2, the HCO3- line drops down to the second cross-over point. At pH > pK2, CO32- is the predominant species, so its concentration now plots as a horizontal line at log CT, and the HCO3- line descends from the second cross-over point towards higher pH with a slope of -1. In the range pH > pK2, the H2CO3* line now descends towards higher pH with a slope of -2. As the CO32- line passes through the second cross-over point towards lower pH into the region where pK1 < pH < pK2, it descends with a slope of +1. When this same line crosses under the first cross-over point into the region where pH < pK1, its slope changes to +2.
In most natural waters, bicarbonate is the dominant carbonate species!

7. SPECIATION IN OPEN CO2-H2O SYSTEMS - I
In an open system, the system is in contact with its surroundings and components such as CO2 can migrate in and out of the system. Therefore, the total carbonate concentration will not be constant.
Let us consider a natural water open to the atmosphere, for which pCO2 = atm. We can calculate the concentration of H2CO3* directly from KCO2:
Note that M H2CO3* is independent of pH!
Many natural waters are actually closer to being open systems with respect to CO2, including, according to Kehew (2001), many ground waters. In an open system, CO2 is free to pass into or out of the system. Thus, there is no guarantee that the total inorganic carbon concentration will remain constant. In other words, we lose our mass-balance constraint on the system. On the other hand, open systems are generally in equilibrium with a CO2-containing atmosphere, which can, over short time scales, be considered to be relatively constant. We therefore gain a Henry’s Law constraint as shown in this slide. If pCO2 is fixed, then the concentration of H2CO3* is fixed, independent of pH. However, just because the concentration of H2CO3* is fixed, this does not mean that the total carbonate concentration is fixed, because variable amounts of dissociated forms of carbonate, i.e., HCO3- and CO32- will be present depending on the pH. However, if we know the pH and the pCO2 for an open CO2-H2O system, we can calculate the concentrations of all relevant species in the system from the mass-action expressions. These mass-action expressions include the Henry’s law constraint (KCO2) shown in this slide, the dissociation constants of carbonic acid (K1, K2) and the dissociation constant of water (Kw).

8. SPECIATION IN OPEN CO2-H2O SYSTEMS - II
The concentration of HCO3- as a function of pH is next calculated from K1:
but we have already calculated M H2CO3*: so
In this slide, we calculate the concentration of bicarbonate ion based on the value of the concentration of carbonic acid determined in slide 10, and the first dissociation constant for carbonic acid. Note that, on a plot of log concentration vs. pH, the concentration of bicarbonate increases linearly with pH. The resultant straight line has a slope of +1 and an intercept of log (K1KCO2pCO2).

9. SPECIATION IN OPEN CO2-H2O SYSTEMS - III
The concentration of CO32- as a function of pH is next calculated from K2:
but we have already calculated M HCO3- so:
and
In this slide, we calculate the concentration of carbonate ion based on the value of the concentration of bicarbonate determined in slide 11, and the second dissociation constant for carbonic acid. On a plot of log concentration vs. pH, the concentration of carbonate also increases linearly with pH, but the resultant straight line has a slope of +2 and an intercept of log (K2K1KCO2pCO2).

10. SPECIATION IN OPEN CO2-H2O SYSTEMS - IV
The total concentration of carbonate CT is obtained by summing:
Now, to get the total carbonate concentration CT, we sum the concentrations of all three carbonate species (remember to sum the concentrations of the species, not the logarithm of the concentrations), and take the logarithm of the total carbonate concentration so we can plot it on a log concentration vs. pH plot.

11. Plot of log concentrations of inorganic carbon species H+ and OH-, for open-system conditions with a fixed pCO2 = atm.
The diagram in this slide is a plot of log concentration vs. pH for an open CO2-H2O system with pCO2 = atm. Note that this plot yields similar information as the Bjerrum plot for closed systems. It shows us the pH ranges in which the three different carbonate species are predominant. However, this diagram also shows us the total carbonate for the system (the heavy curve), which we see changes with pH, unlike in the closed system. We see that, at pH < pK1, H2CO3* is the predominant species. Because it accounts for almost all the total carbonate in this pH range, the curve for CT is practically coincident with the line showing the concentration of H2CO3*. Moreover, because the concentration of H2CO3* is independent of pH, so is CT in this pH range. At pH = pK1, H2CO3* and HCO3- are present at equal concentrations, so the total carbonate concentration is double the concentration of either of these species. Hence, the CT curve rises log units (= log 2) above the crossover point at pH = pK1, with carbonate ion contributing negligibly to CT at this pH. At pK1 < pH < pK2, HCO3- is the predominant species, and its concentration is therefore approximately equal to CT. Therefore, the CT curve is nearly coincident with the line for HCO3- in this pH range, and CT increases one log unit for every unit increase in pH. At pH = pK2, HCO3- and CO32- are present in equal concentrations, with H2CO3* contributing negligibly. Thus, the CT curve rises log units above the crossover point at this pH. Finally, at pH > pK2, CO32- is the predominant species, the CT curve follows the CO32- line, and CT increases 2 log units for every unit increase in pH.

12. Plot of log concentrations of inorganic carbon species H+ and OH-, for open-system conditions with a fixed pCO2 = atm.
The diagram in this slide is a plot of log concentration vs. pH for an open CO2-H2O system with pCO2 = atm. Compared to the previous plot at lower partial pressure of CO2, CT is higher at every pH. However, the ranges of pH in which each species is predominant have not changed. In other words, the cross-over points are exactly the same; they are independent of pCO2. In effect, the CT curve and the lines for each of the carbonate species have shifted upward by 1.5 log units, but their horizontal positions have not changed. Note also that, as expected, the positions of the lines representing the concentrations of H+ and OH- have not changed.
The figures in this slide and the previous one illustrate that, in an open system, the solubility of CO2 increases dramatically with pH, once pH has increased beyond pK1. At low pH, CO2 solubility is independent of pH. These diagrams explain why it is unwise to leave concentrated base solutions, such as NaOH or KOH, exposed to the atmosphere. Such solutions will absorb large quantities of CO2 over time, which will neutralize some of the OH- in the solution, changing its concentration, and add carbonate impurities.
We are now in a position to add another dimension to the system. In our next lecture, we will discuss equilibria involving CO2, H2O and carbonate minerals such as calcite and dolomite.

13. Calcite Solubility?
CaCO3 -> Ca2+ + CO32-
Log K=8.48