Appendix B. Solubility of carbon dioxide in seawater

Carbon dioxide does not just dissolve in water, rather, it reacts with water to form carbonic acid:

1. CO₂ + H₂O ↔ H₂CO₃

The double arrow indicates that the CO₂ dissolution reaction is reversible; it can go forward to carbonic acid or backwards to CO₂ and water. In time, reversible reactions will reach equilibrium, a point where the forward and backward rates are exactly equal and no net change occurs. Acid-base reactions like those discussed here are very fast and are essentially always at equilibrium. For reactions at equilibrium the concentrations of products and reactants are related by a simple relation:

2. [H₂CO₃] /([CO₂] [H₂O]) = constant

The brackets refer to concentration of the species within them. The concentration of water remains essentially unchanged by the dissolution of small amounts of solutes and it is customary to incorporate this constant value into the constant to obtain

3. [H₂CO₃] / [CO₂] = another constant

It is customary to present the concentration of a gas in terms of its partial pressure (P_CO2).

4. [H₂CO₃] / [P_CO2] = K_H → [H₂CO₃] = K_H·P_CO2 = K_H·(ppm CO₂)/1000000

The constant K_H is called the Henry’s Law constant. The Henry’s Law relation shows that carbonic acid concentration is directly proportional to atmospheric CO₂ partial pressure. Equation 4 can be expressed in terms of the atmospheric concentration of CO₂ in ppm rather than partial pressure. Because atmospheric pressure is one atmosphere (by definition) the partial pressure of a certain ppm level of CO₂ is simply that level divided by one million and multiplied by one (atmosphere) as shown in equation 4.

Once in solution carbonic acid dissociates into hydrogen (H⁺) and bicarbonate (HCO₃⁻) ions:

5. H₂CO₃ ↔ H⁺ + HCO₃⁻

Dissociating into hydrogen ions is what makes carbonic acid an acid. Acidity is usually measured by pH which is defined as the negative common logarithm of the hydrogen ion concentration or pH = −log[H⁺]. Bicarbonate ion is itself an acid and can also dissociate into hydrogen and carbonate (CO₃²⁻) ions:

6. HCO₃⁻ ↔ H⁺ + CO₃²⁻

Equilibrium relations can be written for reactions 5 and 6 as follows:

7. [H⁺] [HCO₃⁻] / [H₂CO₃] = K₁

8. [H⁺] [CO₃^2-] / [HCO₃⁻] = K₂

Here K₁ and K₂ stand for the first and second dissociation constants of carbonic acid. When carbon dioxide dissolves in water it forms carbonic acid, which then reacts further to form bicarbonate and carbonate ions. The total amount of CO₂ that dissolves will be the sum of carbonic acid, bicarbonate and carbonate species. Equations 7 and 8 relate the amounts of each of these relative to each other as a function of [H⁺] or pH. Equations 7 and 8 can be rearranged as follows:

9. [HCO₃⁻] = K₁ [H₂CO₃] / [H⁺]

10. [CO₃^2-] = K₂ [HCO₃⁻] / [H⁺] = K₁·K₂·[H₂CO₃] / [H⁺]²

The sum of carbonic acid, bicarbonate and carbonate species is then given by:

11. dissolved CO₂ = [H₂CO₃] + [HCO₃⁻] + [CO₃^2-] = [H₂CO₃]∙(1 + K₁/ [H⁺] + K₁·K₂/ [H⁺]²)
= K_H·(ppm CO₂)∙(1 + K₁/ [H⁺] + K₁·K₂/ [H⁺]²)/1000000

Equation 11 gives the solubility of CO₂ in seawater as a function of [H⁺] and ppm CO₂ in the atmosphere for a given set of constant values. Since the constants themselves are functions of temperature, CO₂ solubility also depends on temperature through temperature’s effect on the constants.

The pH of the oceans is about 8.1. Thus [H⁺] = 10^−8.1 = 7.94 x 10^-9 moles/liter. This value was put into equation 11 with constants K₁ and K₂ obtained from Roy¹ et al. 1993, and K_H obtained from Weiss² (1974). A value of 0.0022 mol/kg (2.3 millimolar) was obtained for CO₂ solubility in seawater at 20º C. This value is close to 2.1-2.3 millimolar value given in the literature.^3-4 For my purposes, I am interested in how CO₂ solubility changed with rising CO₂ levels over the last 50 years, the exact values themselves are not so critical, but rather how rising CO₂ level has affected solubility and hence the ability of the oceans to absorb CO₂. To use equation 11 to answer this question it is necessary to determine how changes in dissolved CO₂ affect [H⁺] (pH). Once this is known then calculation of solubility is straightforward.

An initial attempt to deal with this issue is to consider the case of CO₂ dissolution in pure water to form H₂CO₃. Some of the H₂CO₃, will disassociate into H⁺ and HCO₃⁻ according to equation 5. The amount of each will be equal, and we can represent the concentration of each as x. With this equation 7 can be written as follows

12. x² / [H₂CO₃] = K₁ → x = (K₁ [H₂CO₃] )^½ = [H⁺]

[H₂CO₃] can be calculated from ppm CO2 using equation 4 and the result substituted into equation 12 to give:

13. [H⁺] = (K_1∙∙K_H∙ppm CO₂)^½/1000

Equation 13 can be used to calculate [H⁺] which is then used in equation 11 to determine CO₂ solubility. This was done and the results appear as the red line in Figure 1. The solubility of CO₂ in pure water is very low, more than 100 times less than the solubility in seawater. This is because in pure water, the acidity of carbonic acid is sufficiently strong to drive the pH well into the acid range. Using constants for pure water and 25º C, equation 13 gives pH 5.6 for the present-day CO₂ level of 380 ppm, well below the ocean pH of 8.1. Indeed the pH of rainwater in clean air is about 5.6, reflecting dissolution of atmospheric carbon dioxide. Polluted air containing oxides of sulfur or nitrogen, which upon dissolution form acids much stronger than carbonic acid, creates the phenomenon known as acid rain, in which the pH of rainwater can fall to 3 or even lower.

Acid rain occurs because pure water has no buffering. That is, it contains no alkaline substances that react with dissolved acids to counteract their acidity. The ocean contains such buffering, which is why its pH is high (8.1) in the presence of dissolved carbon dioxide. Obviously the simple case of CO₂ dissolution in water is not a useful model for determining the effect of CO₂ levels on seawater pH. Another simple approach would be to assume that the buffering is strong enough to keep the pH from changing because of the dissolution of a weak acid like CO₂. In this case CO₂ solubility will be directly proportional to ppm CO2 in the atmosphere as shown by the blue line in Figure 1. In actuality, pH has fallen about 0.075 unit since the beginning of the industrial revolution largely because of increasing atmospheric CO₂.^5-7 Apparently, the ocean is not strongly buffered. Looking at the composition of seawater, two buffering agents are present.³ These agents must govern the pH of seawater. They are calcium carbonate (CaCO₃) and the dihydrogen borate ion (H₂BO₃⁻) which react with carbonic acid according to the following reactions:

14. H₂CO₃ + CaCO₃ ↔ Ca²⁺ + 2 HCO₃⁻

15. H₂CO₃ + H₂BO₃⁻ ↔ H₃BO₃ + HCO₃⁻

Equilibrium expressions can be written for these reactions as before:

16. [Ca²⁺]∙[HCO₃⁻]² / [H₂CO₃] = K_C = K₁ K_SP / K₂

17. [HCO₃⁻]∙[H₃BO₃] / ([H₂BO₃⁻]∙[H₂CO₃]) = K_B = K₁ / ^BK₁

The equilibrium constants are related to the carbonic acid dissociation constants K₁ and K₂, ^BK₁ (the first boric acid dissociation constant) and the calcium carbonate solubility constant K_SP. The solubility constant is the equilibrium constant for the dissolution of calcium carbonate. Values for K_SP and ^BK₁ were obtained from Kloosterman⁸ and Dickson.⁹ I know that at present the pH of the ocean is 8.1. I used a CO₂ level of 370 ppm (the value in 2000) for the present value (the value associated with pH 8.1). With these two pieces of information I can calculate [H₂CO₃] and [HCO₃⁻] from equations 4 and 7. With these [Ca²⁺] is then calculated from equation 16. At pH 8.1 virtually all borate species will exist either as the acid H₃BO₃ or as the monovalent ion H₂BO₃⁻. Given that the total amount of borate species in seawater is 0.47 mmol/kg,³ it follows that [H₂BO₃⁻] = 0.00047 − [H₃BO₃]. With this equation 17 was solved for [H₃BO₃], from which [H₂BO₃⁻] was obtained from the mass-balance relation. I define these values as the base values, that is, the values for the base case conditions of 370 ppm CO₂.

I desire to calculate [H⁺] as CO₂ levels change from the standard level of 370 ppm. Equation 4 gives [H₂CO₃] directly. Given a value for [HCO₃⁻] equation 9 can be solved for [H⁺]. Thus the problem becomes determining how the simultaneous equilibria expressed in equation 16 and 17 affect [HCO₃⁻] as [H₂CO₃] changes. This value can be expressed in terms of deviations from the standard values caused by the buffering reactions 14 and 15. If I denote x as the extent to which reaction 14 moves forward and y as the extent to which reaction 15 moves forward, I can write equations 16 and 17 as:

18. ([Ca²⁺]₀ + x)∙([HCO₃⁻]₀ + 2x + y)² / [H₂CO₃] = K_C

19. ([HCO₃⁻]₀ + 2x + y)∙([H₃BO₃]₀ + y) / {([H₂BO₃⁻]₀ − y)∙[H₂CO₃]} = K_B

Here the subscript 0 refers to the base values for the various concentrations, which were previously calculated for 370 ppm CO₂ and pH 8.1. No subscript appears for [H₂CO₃] because this concentration is determined from the atmospheric CO₂ level (which is a given) according to equation 4. Any H₂CO₃ consumed or produced by the reactions 14 and 15 would quickly be made up by CO₂ exchange with the atmosphere according to reaction 1. With this, all parameters in equations 18 and 19 except x and y are known and these equations can be solved for x and y. This was done by first solving equation 18 algebraically for y in terms of x. With this, for a given value of x, a value of y was determined and this value, plus the assumed x value was used in equation 19. Using the Microsoft Excel goal seek function, the value x need to solve equation 19 can be found readily. With x and y known, so is [HCO₃⁻], from which [H⁺] is obtained from equation 7. With [H⁺], CO₂ solubility is calculated according to equation 11. The result appears in Figure 1 as the weak buffering line.

As a test of the validity of this model, pH values were calculated for the CO₂ level in the year 1751 (277 ppm) and for 1994 (359 ppm). A temperature of 14º C was used for the 1751 value (as compared to 15º C for today) to account for the roughly one degree of warming since then. Values of 8.173 and 8.108 were obtained for pH, compared to literature values of 8.179 and 8.104.^5-7 This close correspondence is a good indication that the model presented here is valid.

With a valid model it is instructive to use it to examine a number of issues raised in the global warming debate. One of these is the “soda pop analogy”. Some global warming skeptics like to argue that higher CO₂ levels today are the result of higher temperatures and not the other way around. They like to use the analogy that, just as warm soda goes flat (loses its dissolved CO₂), the oceans have lost some of their vast reserves of dissolved CO₂ as a result of the roughly 1º C degree of warming since the beginning of the industrial revolution. Using equations 18 and 19 and the goal seek function, it is a straightforward matter to calculate CO₂ solubility as a function of CO₂ for various temperatures (see Figure 2). From Figure 2 it is clear that the solubility at 380 pm CO₂ (today’s value) at a temperature of 20º C (5º C higher than today) is higher than the solubility at 300 ppm CO₂ (the value a century ago) at a temperature of 10º C (more than 4º C lower than then). That is, even if the temperature rise were ten times greater than it actually was, they would still be no net release of CO₂ from the oceans over the last century, but rather a net absorption of CO₂.

In actual fact, only about half of the CO₂ released to the atmosphere by human activity over the past century is still in the atmosphere. If the missing CO₂ was removed by the oceans, we can use the solubility model presented here to determine the amount of seawater (i.e. the depth of the surface layer of the ocean) necessary to absorb the missing CO₂.

References

1. Roy, R.N.L., N. Roy, M. Vogel, C. P. Moore, T. Pearson, C. E. Good, F. J. Millero, and D. J. Campbell, (1993) “Determination of the ionization constants of carbonic acid in seawater”, Marine Chemistry, 44 249-268. (www.ugamp.nerc.ac.uk/um/source_p/umpl/EQ_CONST_dk.htm)

2. Weiss, R.F. (1974) "Carbon dioxide in water and seawater: the solubility of a non-ideal gas" Mar. Chem. 2: 203-2 15. (www.ugamp.nerc.ac.uk/um/source_p/umpl/EQ_CONST_dk.htm)

3. Anthoni, J F., “Composition of seawater”, (www.seafriends.org.nz/oceano/seawater.htm)

4. Classroom Bermuda Atlantic Time Series Study, “How much CO2 can sea water hold?”, (www.coexploration.org/bbsr/classroombats/html/co2_in_the_sea.html)

5. “Oceanic acidification” Wikipedia (http://en.wikipedia.org/wiki/Ocean_acidification)

6. Orr, J. C. et al. (2005). “Anthropogenic ocean acidification over the twenty-first century and its impact on calcifying organisms” Nature 437, 681-686.

7. Key, R.M., Kozyr, A., Sabine, C.L., Lee, K., Wanninkhof, R., Bullister, J., Feely, R.A., Millero, F., Mordy, C. and Peng, T.-H. (2004). A global ocean carbon climatology: Results from GLODAP. Global Biogeochemical Cycles 18, GB4031.

8. Klostermann, A F. “The Calcium Question”, (www.natureef.com/calcium1.htm)

9. Dickson, A.E. (1990) “Thermodynamics of the dissociation of boric acid in syntheic sea water from 273.15 to 298.15 K”, Deep-Sea Research, 37, 755-766. (www.ugamp.nerc.ac.uk/um/source_p/umpl/EQ_CONST_dk.htm)

10. World Resources Institute Earthtrends database (http://earthtrends.wri.org/searchable_db/index.php?theme=3)