By Roger Cohen and Will Happer

We are often told that CO2 released from burning fossil fuels will cause dangerous ocean acidification.  Actually the oceans will remain comfortably alkaline and hospitable to life for the foreseeable future. This brief note is a quantitative review of the physical chemistry of ocean pH.  High school chemistry and algebra should provide enough background to follow the discussion.  An excellent introduction to the chemistry of the oceans can be found in the book: Seawater: Its Composition, Properties and Behavior, by Wright and Colling.  More details on ocean pH can be found in a recent review by Tans, which can be read below.

1 Introduction

We are often told that the CO2 released from burning fossil fuels will cause dangerous ocean acidification. Actually the oceans will remain comfortably alkaline and hospitable to life for the foreseeable future. This brief note is a quantitative review of the physical chemistry of ocean pH. High school chemistry and algebra should provide enough background to follow the discussion. An excellent introduction to the chemistry of the oceans can be found in the book: Seawater: Its Composition, Properties and Behavior, by Wright and Colling[1]. More details on ocean pH can be found in a recent review by Tans [2].

2 Alkalinity of the Unbuffered Ocean

Ocean water is salty because of the weathering of the earth’s rocks by rainwater, and because of salts dissolved from the ocean floor, especially near plate-spreading boundaries. The salt water contains positive ions (cations), mainly sodium Na+, magnesium Mg2+, calcium Ca2+ and potassium K+. For clarity, we can think of the ions as coming from the strong bases NaOH, and potassium hydroxide, KOH (lye) and much less soluble magnesium hydroxide, Mg(OH)2 and calcium hydroxide Ca(OH)2 (slaked lime). The ocean also contains negative ions (anions), mainly chloride Cl− and sulfate SO2− 4 , which we can think of the anions as coming from the strong acids, hydrochloric acid, HCl and sulfuric acid, H2SO4. For want of a better word, we will refer to the cations of strong bases, and anions of strong acids as pH-independent ions. The concentration of pH-independent ions is unaffected by normal changes in the pH of water.

The electrical charge of the pH-independent positive ions slightly exceeds that of pHindependent negative ions. The excess charge is called the alkalinity [A]. In ocean water with a typical salinity of 35 g kg−1 (grams of salt per kg of water) the alkalinity is about [A] ≈ 2.3 × 10−3M (1)

For ocean chemistry the unit of alkalinity M (or equivalent) is usually taken to be one mole of elementary charge (one Faraday) per kg of water.

If there were nothing but pH-independent ions of strong acids and strong bases in the ocean, the excess positive charge would be compensated the negative charge of hydroxyl ions, 1 OH−. Recall that the H2O molecules of liquid water can dissociate into positive hydrogen ions H+ and negative hydroxyl ions OH− in the reaction H2O ⇋ H + + OH−. (2)

The concentrations, in the traditional oceanographic unit of moles per kilogram (M), are related by [H+][OH−] = Kw, where Kw = 1.00 × 10−14 M2 . (3)

Some of the equilibrium “constants” can very substantially over the range of temperatures and pressures of the oceans. For example, the dissociation coefficient of water (3) increases slowly with temperature, because it is easier to spontaneously break up H2O molecules in warmer water. Here and elsewhere we will use a water temperature of 25 C, and we will neglect any pressure dependence of the equilibrium coefficients. This will reasonably characterize the first 500 m of ocean waters in low to mid latitudes where most of the earth’s oceans and much of its aquatic life are found. We will ignore small difference between the concentration of the ion, for example [H+], and the thermodynamic activity a(H+), the quantity that rigorously goes into laws of mass action like (3).

The concentration of H+ ions in solutions is traditionally written in terms of a dimensionless number, the pH, as [H+] = 10−pH M, or pH = − log ([H+]/M) (4)

For charge neutrality we must have [A] + [H+] = [OH−]. (5)

Using (3) in (5) we find the quadratic equation for [H+], [H+] 2 + [A][H+] − Kw = 0. (6)

The quadratic equation (6) for [H+] has an unphysical solution where [H+] is a negative real number, and the solution of interest to us [H+] = 1 2 ( p 4Kw + [A] 2 − [A]) = 4.35 × 10−12M. (7)

The representative alkalinity (1) was used to evaluate (7). Substituting (7) into (4) we find that the pH-independent ions would give the ocean a pH of pH = 11.4. (8)

This is about the same pH as household ammonia, and considerably more alkaline than California’s celebrated Mono Lake (pH = 9.8). But bacteria that can thrive in waters of pH=11.4 are known to exist.

A typical ocean pH is 8.1, much less than the value (8) that would be produced by the pH-independent ions. The reason is the presence of weak acids, notably carbonic acid H2CO3, boric acid B(OH)3 and silicic acid Si(OH)4. Of these, carbonic acid is by far the most important. If we ignore the other weak acids, we find that the pH of ocean water in contact with the atmosphere depends on the concentration of atmospheric CO2 as shown in Fig. 1. The pH values were calculated with Eq. (20), which we discuss below, and (4). Also shown in Fig. 1 is the pH of pure rain water, uncontaminated by the oxides of sulfur or nitrogen. Current CO2 concentrations would give pure rainwater a pH of about 5.4, mildly acidic (ordinary vinegar has pH ≈ 3).

Buffering by Weak Acids

For conceptual clarity, we begin by imagining an atmosphere free of CO2 and an ocean with only pH-independent ions, and therefore with the high pH = 11.4 of (8). Now imagine adding CO2 g molecules to the atmosphere, and maintaining a fixed atmospheric level by adding still more CO2 g molecules to replace the large number that dissolve into the ocean according to the reaction CO2 g → CO2 aq (9)

A small fraction of the dissolved CO2 aq molecules will attach water molecules to become molecules of undissociated carbonic acid, H2CO3. The ratio of the concentrations of the two uncharged species of dissolved inorganic carbon will be independent of pH, but will depend on temperature. For simplicity, we will let the symbol [CO2] describe the total concentration of uncharged, inorganic carbon molecules in solution [CO2] = [CO2 aq] + [ H2CO3] (10)

The concentration of uncharged inorganic carbon molecules in the surface water will be proportional to the partial pressure PCO2 (in atmospheres) of the CO2 molecules in the gas phase [CO2] = α PCO2 , where α = 2.84 × 10−2 M atm−1 . (11)

The numerical value of the coefficient α for Henry’s law (11) is from the work of Weiss [3] for a temperature of 20 C.

A dissolved carbonic acid molecule can dissociate into a bicarbonate ion and a free proton H2CO3 ⇋ H+ + HCO− 3 . (12)

We can write the equilibrium concentrations of dissolved inorganic carbon species associated with the reaction (12) as [H+][HCO− 3 ] = K1[CO2], where K1 = 10.00 × 10−7 M. (13)

The numerical value for the equilibrium coefficient K1 is from the work of Merbach [4] for 20 C. Using (13) and (11) we find that the concentration of bicarbonate ions is [HCO− 3 ] = K1 [H+] α PCO2 .

(14) A bicarbonate ion can dissociate to a positive hydrogen ion and a carbonate ion HCO− 3 ⇋ H+ + CO2− 3 . (15)

The equilibrium concentrations are related by [H+][CO2− 3 ] = K2[HCO− 3 ], where K2 = 7.69 × 10−10 M. (16) 4

The bicarbonate dissociation coefficient K2 is several orders of magnitude smaller than the carbonic acid dissociation coefficient K1 because energy needed to remove a proton from H2CO3 in water is much less than the energy to remove a proton from HCO− 3 . Using (16) with (15) and (14) we find [CO2− 3 ] = K1K2 [H+] 2 α PCO2 . (17)

The total concentration of dissolved inorganic carbon is [C] = [CO2] + [HCO− 3 ] + [CO2− 3 ]. (18)

The dissolved CO2 molecules will release bicarbonate and carbonate ions, in accordance with the reactions (12) and (15), so the equation of charge neutrality (5) must be modified to give [A] + [H+] = [OH−] + [HCO− 3 ] + 2[CO2− 3 ]. (19)

Substituting (17) and (14) into (19) and using (3) we find [H+] 3 + [A][H+] 2 − (Kw + K1α PCO2 )[H+] − 2K2K1α PCO2 = 0. (20)

For positive alkalinity [A] > 0 and partial pressure PCO2 > 0, the cubic equation (20) for [H+] has three solutions, two un-physical solutions where [H+] is either a negative real number or has an imaginary part, and the solution of interest to us, where [H+] is a positive real number. The partitioning of dissolved inorganic carbon in the ocean between uncharged species, bicarbonate and carbonate ions is shown in Fig. 2, which was calculated with (20), (17), (14) and (11). At current atmospheric levels of CO2, about 400 ppm, most of the dissolved inorganic carbon consists of bicarbonate ions. As atmospheric CO2 levels increase, the concentration of bicarbonate ions will slowly increase, and the concentration of carbonate ions will slowly decrease.

In Fig. 3 we show the partitioning of dissolved inorganic carbon between uncharged species, bicarbonate and carbonate ions in uncontaminated rain water. Unlike the ocean, where most of the inorganic carbon consists of bicarbonate ions, most of the dissolved inorganic carbon in pure rainwater is the uncharged species CO2 of (10).

3.1 Boron

Although most of the buffering of ocean alkalinity is due to dissolved CO2, other weak acids make a minor contribution. Consider boric acid, B(OH)3, the next most important buffering species in the ocean after dissolved CO2. A boric acid molecule can attach an OH− from a water molecule and release a free proton B(OH)3 + H2O ⇋ H+ + B(OH)− 4 . (21)

We can write the equilibrium concentrations of dissolved boron species associated with the reaction (21) as [H+][B(OH)− 4 ] = KB[B(OH)3], where KB = 2.512 × 10−9 M. (22)

The dissociation coefficient[5] of (22) is intermediate between that of carbonic acid (13) and the bicarbonate ion (16). In analogy to (18), we write the total concentration[5] of boron-containing molecules in sea water as [B] = [B(OH)3] + [B(OH)− 4 ] ≈ 0.42 × 10−3M. (23)

Combining (23) with (22) we find, in analogy to (14) and (17) that the concentration of borate ions is [B(OH)− 4 ] = KB[B] KB + [H+] . (24)

If the boric acid were the only buffer for ocean alkalinity, the equation (19) of charge neutrality would become [A] + [H+] = [OH−] + [B(OH)− 4 ]. (25)

Substituting (3) and (24) into (25), we see that the hydrogen-ion concentration is given by the solution of the cubic equation [H+] 3 + ([A] + KB)[H+] 2 − (Kw + {[B] − [A]}KB)[H+] − KwKB = 0. (26)

Like (20), the cubic equation (26) for [H+] has three solutions, two un-physical solutions where [H+] is either a negative real number or has an imaginary part, and the solution of interest to us, where [H+] is a positive real number. Solutions of (26) for boron concentrations comparable to those in the oceans are shown in Fig. 4. There is not enough boron in the oceans to make much change to the pH. For otherwise pure water with [B] = 0.42 mM of dissolved boric acid we can use (26) with [A] = 0 to find [H+] = 1.031 × 10−6 or pH = 5.99. Adding the concentration (23 of boric acid to the CO2-free ocean only decreases the pH for about 11.36 to 11.27, still harshly alkaline. A solution of pure boric acid at the concentration of (23) has about the same acidity as rainwater at around 30 ppm of atmospheric CO2. There is not much boric acid in the oceans and it is a weak acid to begin with.

For an ocean buffered by both boric acid and dissolved CO2, the equation (19) of charge neutrality would become [A] + [H+] = [OH−] + [HCO− 3 ] + 2[CO2− 3 ] + [B(OH)− 4 ]. (27)

Substituting (24), (17) and (14) into (19) and using (3) we find [H+] 4 + ([A] + KB)[H+] 3 − (Kw + {[B] − [A]}KB + K1α PCO2 )[H+] 2 −(KwKB + {KB + 2K2}K1α PCO2 )[H+] − 2KBK2K1α PCO2 = 0. (28)

The quartic equation (28) for [H+] has four solutions, three un-physical solutions where [H+] is either a negative real number or has an imaginary part, and the solution of interest to us, where [H+] is a positive real number. Solutions of (28) are shown in Fig. for [B] = 0.42 mM, approximately the current ocean value, and for a range of atmospheric CO2 concentrations. Boron has much less effect on ocean pH than dissolved CO2.

4 Equilibrium CO2 in Atmosphere and Ocean

In hydrostatic equilibrium the atmospheric pressure p (in dynes cm−2 ) at altitude z is given by the barometric equation dp dz = −gmp kBT . (29)

Here m is the mean mass of an atmospheric molecule, T is the absolute atmospheric temperature in K at altitude z, and kB is Boltzmann’s constant. We neglect the weakening of the acceleration of gravity g with altitude and set g = 981 cm s−1 . Then we can use the ideal gas law p = NkT to write (29) as dp = −gmNdz. (30)

which has the solution

Z ∞ z N ′ dz′ = p(z) gm . (31)

Here N′ = N(z ′ ) is the number density of molecules at altitude z above the surface. For a sea-level (z = 0) pressure of P(0) = 1 atm or p(0) = 1.013 × 106 dyne cm−2 , we find the total column density above the surface is n(0) = p(0) gm = 2.15 × 1025 cm−2 . (32)

The molar faction of CO2 has the same numerical value as the partial pressure in atmospheres, PCO2 , of CO2 at sea level, so the column density of CO2 molecules is nCO2 = n(0)PCO2 (33)

The earth’s radius is re = 6371 km, so its area is Se = 4πr2 e = 5.10 × 1018cm2 . (34)

The total number of moles of CO2 in the atmosphere is approximately Nat = βatPCO2 , (35)

where the coefficient is βat = Sen(0) NA = 1.82 × 1020moles atm−1 . (36)

Avagadro’s number is NA = 6.02 × 1023 molecules mole−1 . The number of moles of CO2 in the oceans is Noc = Moc[C], (37)

where the concentration of inorganic carbon [C] was given by (18), and where the mass of ocean water is approximately Moc = 1.36 × 1021 kg. (38)

The molar amounts, Nat and Noc, of CO2 in the atmosphere and ocean are shown in Fig. 6.

The rate of increase, dNoc/dNat, of CO2 in the ocean with CO2 in the atmosphere is shown in Fig. 7. Because of the long time (centuries) needed for additional CO2 molecules dissolved in the ocean surface to reach the deeper ocean, the currently observed rate, dNoc/dNat ≈ 1, is about three times smaller than the predicted equilibrium rate dNoc dNat =  Moc βat d[C] dPCO2 , (39) plotted in Fig. 7. 10

The estimates above underestimate the amount CO2 dissolved in the ocean since they assume a uniform temperature of 25 C, while most of the deeper layers of the ocean are much colder and can hold more CO2. And because of the biological pump, the CO2 levels can increase substantially with depth. Finally the dissolution calcium carbonate sinking down from the surface to more CO2 to the deeper ocean below the lysocline.

5 Summary

This minimalist discussion already shows how hard it is to scare informed people with ocean acidification, but, alas, many people are not informed. For example:

• The oceans would be highly alkaline with a pH of about 11.4, similar to that of household ammonia, if there were no weak acids to buffer the alkalinity. Almost all of the buffering is provided by dissolved CO2, with very minor additional buffering from boric acid, silicic acid and other even less important species.
• As shown in Fig. 1, doubling atmospheric CO2 from the current level of 400 ppm to 800 ppm only decreases the pH of ocean water from about 8.2 to 7.9. This is well within the day-night fluctuations that already occur because of photosynthesis by plankton and less than the pH decreases with depth that occur because of the biological pump and the dissolution of calcium carbonate precipitates below the lysocline.
• As shown in Fig. 2, doubling atmospheric CO2 from the current level of 400 ppm to 800 ppm only decreases the carbonate-ion concentration, [CO2− 3 ], by about 30%. Ocean surface waters are already supersaturated by several hundred per cent for formation of CaCO3 crystals from Ca2+ and CO2− 3 . So scare stories about dissolving carbonate shells are nonsense.
• As shown in Fig. 7, the ocean has only absorbed 1/3 or less of the CO2 that it would eventually absorb when the concentrations of CO2 in the deep oceans came to equilibrium with surface concentrations. Effects like that of the biological pump and calcium carbonate dissolution below the lysocline allow the ocean to absorb substantially more than the amount that would be in chemical-equilibrium with the atmosphere.
• Over most of the Phanerozoic, the past 550 million years, CO2 concentrations in the atmosphere have been measured in thousands of parts per million, and life flourished in both the oceans and on land. This is hardly surprising, given the relative insensitivity of ocean pH to large changes in CO2 concentrations that we have discussed above, and given the fact that the pH changes that do occur are small compared to the natural variations of ocean pH in space and time.

References

[1] John Wright and Angela Colling, Seawater: Its Composition, Properties and Behavior Butterworth-Heinemann,Oxford (1995). 11

[2] P.Tans, Oceanography 22, 26 (2009). http://dx.doi.org/10.5670/oceanog.2009.94

[3] R.F. Weiss, Marine Chemistry 2, 203 (1974).

[4] C. Mehrbach et al., Limnology and Oceanography 18, 897 (1973).

[5] R.E. Zeebe, A. Sanyal, J. D. Ortiz and D. A. Wolf-Gladrow, Marine Chemistry 73, 113 (2001).

Download: Fundamentals of Ocean pH