Proton cycling, buffering, and reaction stoichiometry in natural waters

Marine Chemistry 121 (2010) 246–255

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Marine Chemistry j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m a r c h e m

Proton cycling, buffering, and reaction stoichiometry in natural waters☆ A.F. Hofmann a,d,⁎, J.J. Middelburg a,b, K. Soetaert a, D. A.Wolf-Gladrow c, F.J.R. Meysman a,d a

Netherlands Institute of Ecology (NIOO-KNAW), Centre for Estuarine and Marine Ecology — P.O. Box 140, 4400 AC Yerseke, The Netherlands Faculty of Geosciences, Utrecht University, P.O. Box 80021, 3508 TA Utrecht, The Netherlands c Alfred Wegener Institute for Polar and Marine Research — P.O. Box 12 01 61, D-27515 Bremerhaven, Federal Republic of Germany d Laboratory of Analytical and Environmental Chemistry, Vrije Universiteit Brussel (VUB) — Pleinlaan 2, 1050 Brussel, Belgium b

a r t i c l e

i n f o

Article history: Received 10 November 2009 Received in revised form 14 May 2010 Accepted 21 May 2010 Available online 1 June 2010 Keywords: pH modeling Reaction stoichiometry Buffering Proton cycling Effects of biogeochemistry on proton cycling Stoichiometric coefficient for the proton Buffer factor Proton-cycling sensitivity Ionization fractions Acid-base chemistry

a b s t r a c t Ongoing acidification of the global ocean necessitates a solid understanding of how biogeochemical processes are driving proton cycling and observed pH changes in natural waters. The standard way of calculating the pH evolution of an aquatic system is to specify first how biogeochemical processes affect total alkalinity, followed by the solution of a nonlinear acid-base equilibrium equation system. This approach, however, does not explicitly reveal how individual biogeochemical processes contribute to the overall proton cycling in the system. Here, we provide an extension of the classical acid-base theory that explicitly quantifies the proton production/ consumption by a given process, showing that it can be calculated as the proton-cycling sensitivity times the rate of the biogeochemical process at hand. The proton-cycling sensitivity emerges as a central concept in acid-base chemistry of natural waters and can be further decomposed as the ratio of a stoichiometric coefficient for the proton over the buffer factor. The stoichiometric coefficient for the proton expresses how many moles of protons would be produced per mole of reaction if buffering was absent, and is obtained by bringing the reaction equation of the process into a specific form: the fractional reaction equation at ambient pH. The buffer factor quantifies how acid-base systems attenuate the proton production/consumption by biogeochemical processes, and is identified as the negative of the partial derivative of the total alkalinity with respect to the proton concentration. Applying this new concept to an acidification scenario for the future surface ocean, we illustrate its potential to analyze proton cycling in natural waters. Thereby we show that a reduced buffer factor due to anthropogenic carbon input makes the ocean more vulnerable to any process influencing the pH. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Anthropogenic release of carbon dioxide to the atmosphere leads to the acidification of the ocean (IPCC, 2007; Zeebe et al., 2008). This evolution is predicted to have an adverse effect on aquatic ecosystems (Orr et al., 2005; Kleypas et al., 2006; Gazeau et al., 2007; Guinotte and Fabry, 2008). In response, there is currently a substantial effort invested by the scientific community to better understand past, present and future changes in the pH of natural waters. One crucial task in this program is to accurately quantify the magnitude of those pH changes, as is done in several recent studies (e.g. Santana-Casiano et al., 2007; Wootton et al., 2008; Dore et al., 2009). In natural waters, protons are intensely cycled (Hofmann et al., 2009a), and observed pH changes are the result of an imbalance in proton cycling. Therefore, an

☆ The concepts introduced here have been implemented in a generic way in the software package AquaEnv (Hofmann et al., 2010) for the open-source programming language R (R Development Core Team, 2008). The AquaEnv package can be obtained from the websites: http://r-forge.r-project.org/projects/aquaenv/ and http://cran.rproject.org/package=AquaEnv. ⁎ Corresponding author. Current address: Monterey Bay Aquarium Research Institute (MBARI), 7700 Sandholdt Road, Moss Landing, CA 95039-9644, USA. E-mail address: [email protected] (A.F. Hofmann). 0304-4203/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.marchem.2010.05.004

equally important task is to better understand how biogeochemical processes are driving proton cycling and thus the observed pH changes. This latter aspect is the topic of this communication. In essence, the issue of proton cycling within aquatic systems comes down to the question of how many protons are consumed or produced by individual biogeochemical processes. Although the theoretical investigation of the acid-base chemistry of naturals waters already has a long tradition (e.g. Morel and Hering, 1993; Stumm and Morgan, 1996; Zeebe and Wolf-Gladrow, 2001), the problem has not been explored to its full potential. This is because the current theoretical framework for acid-base chemistry in aquatic systems strongly focuses on alkalinity changes, while, actually, the problem is an issue of proton changes. Although, indeed, alkalinity and pH are tightly related, the effect of biogeochemical processes on protons and thus on pH is only implicitly dealt with in the current alkalinitycentred framework. So, here, we propose an approach that explicitly quantifies how biogeochemical processes and their stoichiometric representations are linked to proton consumption and production. 2. Problem statement The question of how many protons a given biogeochemical process actually produces or consumes is not as straightforward as might appear

A.F. Hofmann et al. / Marine Chemistry 121 (2010) 246–255

at first sight. This can be illustrated by analyzing the example of calcium carbonate precipitation. If RP represents the rate of CaCO3 precipitation (expressed as μmol per kg of solution per unit of time), then we can write the rate of change of protons associated with CaCO3 precipitation as
 d Hþ = SP RP dt

ð1Þ

This proton balance assumes that carbonate precipitation is the only process acting: later, we will show how to account for the simultaneous effect of multiple processes. The modulating factor SP determines how strongly carbonate precipitation is translated into proton production, and hence, we refer to it as the proton-cycling sensitivity of CaCO3 precipitation. Processes that have a larger sensitivity will induce larger pH changes for the same reaction rate. The challenge is now to determine these proton-cycling sensitivities for different processes. In theory, the production rate of a particular chemical species due to a chemical reaction should equal the reaction rate times the stoichiometric coefficient of that chemical species in the reaction equation. Accordingly, the net proton production rate due to CaCO3 precipitation would become
þ d H P = νHþ RP dt

ð2Þ

where νHP+ represents the stoichiometric coefficient of protons in a suitable reaction equation for CaCO3 precipitation. Yet, when trying to apply this, one is immediately confronted with the problem of the non-uniqueness of stoichiometric reaction equations. Indeed, there are several alternatives to write a reaction equation for CaCO3 precipitation, the most frequently used being 2þ

Ca

2þ

Ca

2þ

Ca

ð3Þ

−

ð4Þ

þ 2HCO3 →CaCO3 þ CO2 þ H2 O

2þ

Ca

2−

þ CO3 →CaCO3

−

þ HCO3 →CaCO3 þ H

þ þ

þ H2 O þ CO2 →CaCO3 þ 2H

ð5Þ ð6Þ

One simplistic way to interpret this would be to say that the proton production due to carbonate precipitation “depends on the carbon source”. In other words, the reaction Eqs. (3) and (4) would not produce any protons (SP = νHP+= 0), while the reactions Eqs. (5) and (6) would produce protons (νHP+ would be 1 or 2, respectively). This reasoning is clearly incorrect. The amount of protons produced by calcium carbonate precipitation at any given time must not depend on the particular reaction equation used. Therefore, none of the reaction schemes listed above seems to be able to tell the true story: none of them specifies how many moles of protons are actually produced by the precipitation of one mole of CaCO3 at ambient conditions. 3. The conventional approach A first step towards the resolution of this problem is to realize that the non-uniqueness of reaction Eqs. (3)–(6) results from the presence of acid-base reactions, in particular, the two dissociation reactions of the carbonate system. −

þ

CO2 þ H2 O⇌HCO3 þ H −

2−

þ

HCO3 ⇌CO3 þ H

247

and the two dissociation reaction Eqs. (7) and (8). The presence of the acid-base reactions generates thus arbitrariness in the way reaction equations are written, and as a result, there is uncertainty about the actual rate of proton production. This problem has been known for a long time in acid-base theory, and a resolution has also been provided: rather than focusing on proton changes, one should look at so-called total quantities, like total alkalinity (TA) and dissolved inorganic carbon (DIC) (Dickson, 1981; Morel and Hering, 1993; Stumm and Morgan, 1996; Zeebe and WolfGladrow, 2001; Wolf-Gladrow et al., 2007). Here, TA and DIC can be defined as −

2−

þ

TA ¼ ½HCO3  þ 2½CO3 −½H  −

2−

DIC ¼ ½CO2  þ ½HCO3  þ ½CO3 

ð9Þ ð10Þ

Technically, these total quantities function as reaction invariants with respect to the acid-base reactions (e.g. Olander, 1960; Aris and Mah, 1963; DiToro, 1976; Boudreau, 1987; Morel and Hering, 1993; Saaltink et al., 1998; Meysman, 2001; Hofmann et al., 2008). This implies that their rate of change becomes independent of the acidbase reactions: all four of the above reaction equations describing CaCO3 precipitation consume 1 unit of DIC and 2 units of TA for each unit of CaCO3 precipitated. Accordingly, the associated rates of change can be written as1 dTA = −2RP dt

ð11Þ

dDIC = −RP dt

ð12Þ

Tracking changes in total concentrations forms the basis of classical pH modeling approaches (see Hofmann et al., 2008, for a discussion of pH modeling methods). The resulting balance equations for alkalinity and DIC are supplemented with the equilibrium mass action equations of the acid-base systems. Here, for the sake of simplicity, we only consider the carbonate system. The respective equilibrium mass action equations can be written as functions of the proton concentration [H+] forming expressions for the ionization fractions (Skoog and West, 1982; Stumm and Morgan, 1996)
þ 2   H ½CO2  c þ = þ2 α0 ½H  = DIC ½H  + K1⁎ ½Hþ  + K1⁎ K2⁎

ð13Þ

⁎
þ   K1 H ½HCO− c þ 3  = þ2 α1 ½H  = DIC ½H  + K1⁎ ½Hþ  + K1⁎ K2⁎

ð14Þ

  ½CO2− K1⁎ K2⁎ c þ 3  = þ2 α2 ½H  = DIC ½H  + K1⁎ ½Hþ  + K1⁎ K2⁎

ð15Þ

where K1⁎ and K2⁎ represent the time-invariant2 stoichiometric equilibrium constants of the two dissociation reactions (7) and (8). Using the ionization fractions, the alkalinity expression (9) can be reformulated as        þ c þ c þ þ TA = f DIC; ½H  = α1 ½H  + 2α2 ½H  DIC−½H 

ð16Þ

ð7Þ ð8Þ

The “additional” reaction Eqs. (4)–(6) can be obtained by making suitable linear combinations of the “basal” CaCO3 precipitation Eq. (3)

1 Note that, while these equations are only valid for our simple example, Hofmann et al. (2008) detail a procedure to derive mass balance equations for TA, DIC, and other total quantities in an arbitrary system. 2 Note that Hofmann et al. (2009a) detail a method to investigate proton cycling which allows for time variable dissociation constants.

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Together, the TA and DIC mass balances (Eqs. (11) and (12)) and the alkalinity Eq. (16) form a set of three equations in three unknowns (TA, DIC and [H+]). Changes in the proton concentration (i.e. the time evolution of pH) are obtained by iteratively solving this nonlinear equation system (Ben-Yaakov, 1970; Culberson, 1980; Luff et al., 2001; Follows et al., 2006). However, the proton production issue is not entirely resolved. Clearly, the conventional TA and DIC approach does allow for a calculation of pH changes as a result of CaCO3 precipitation. However, it does not enable us to directly calculate how many moles of protons are generated per mole of CaCO3 precipitated. This is because the effect of CaCO3 precipitation is only specified in terms of TA and DIC, and the effect of CaCO3 precipitation on protons remains invisibly “buried” within the numerical pH calculation procedure. In summary, the conventional TA and DIC approach does not explicitly quantify the proton-cycling sensitivity SP of CaCO3 precipitation. Accordingly, when multiple biogeochemical processes are acting concurrently, only the combined effect of all processes on pH can be calculated, one cannot quantify how strongly individual processes influence proton cycling. In the next section, we propose a procedure to explicitly arrive at the proton-cycling sensitivity of biogeochemical processes defined by arbitrary stoichiometric representations. 4. Fractional stoichiometry and proton cycling The reaction Eqs. (3)–(6) are only a few of the many possibilities to write a consistent reaction equation for CaCO3 precipitation. In fact, any equation of the form 2þ

Ca

þ c0 CO2 þ

− c1 HCO3

þ

2− c2 CO3

þ c0 H2 O→CaCO3 þ ð2c0 þ c1 ÞH

þ

ð17Þ is stoichiometrically correct, provided that the coefficients are constrained by c0 + c1 + c2 = νPCaCO3 = 1. Note that the coefficients c0, c1, and c2 are usually fractions, and so, we call Eq. (17) a fractional reaction equation (as opposed to Eqs. like (3)–(6) which predominantly contain integer stoichiometric coefficients and are thus called integer reaction equations). Effectively, this fractional reaction equation provides an infinite number of possibilities to write the reaction equation of carbonate precipitation. From this infinite set, we can select, however, one set of coefficients ci that has a particular chemical meaning. The ambient pH within our aquatic system induces a specific 2− partitioning of DIC into CO2, HCO− expressed by the 3 , and CO3 ionization fractions (Eqs. (13)–(15)). If we substitute these ionization fractions αci as the coefficients ci into the fractional reaction Eq. (17), we obtain an equation which describes the proton release of calcium carbonate precipitation at ambient pH of the system 2þ

Ca

þ

c α0 CO2

þ

c − α1 HCO3

þ

c 2− α2 CO3

þ

c α0 H2 O→CaCO3

þ

c ð2α0

þ

c þ α1 ÞH

ð18Þ We refer to Eq. (18) as the fractional reaction equation at ambient pH for calcium carbonate precipitation. We now show that the fractional reaction equation at ambient pH forms a key concept in the explicit description of proton cycling in natural waters. Indeed, based on Eq. (18), the proton production rate due to CaCO3 precipitation unambivalently becomes
  c d Hþ P c = SP RP = νHþ RP = 2α0 + α1 RP ð19Þ dt While still not the end of the story, this proton balance does already have a relevant chemical meaning: in a non-buffered system (i.e., without fast acid-base reactions that achieve instantaneous equilibrium), the rate of change of protons due to CaCO3 precipitation would be exactly the proton release as expressed by Eq. (19). However, in a realistic, buffered system, this is no longer the case

because of re-equilibration by the acid-base reactions present. According to the principle of Le Chatelier-Braun, re-equilibration will always counteract the proton release or consumption by kinetic reactions. This implies that the actual rate of change of protons must always be smaller than the nonbuffered rate. We can account for this by introducing the buffer factor β≥1. Using β, the proton balance (Eq. 19) can be modified to
 P d Hþ ν þ 1 c c 2α0 + α1 RP = SP RP = H RP = β dt β

ð20Þ

The remaining challenge is to find an explicit expression for the buffer factor β. 5. The buffer factor To arrive at an explicit expression for the buffer factor, we extend the approach that was followed by Hofmann et al. (2008). It is well known that all quantities in the carbonate system (TA, DIC, [CO2], + 2− [HCO− 3 ], [CO3 ], and [H ]) can be calculated from any two components from this list (Zeebe and Wolf-Gladrow, 2001). As shown above, in the classical approach, one uses TA and DIC as the basal components. Here, however, we follow Hofmann et al. (2008) and take the proton and DIC concentrations as the independent variables. The alkalinity is then expressed as in Eq. (16). Differentiating both the left and right hand side of Eq. (16) with respect to time, and applying the chain rule of differentiation, one immediately arrives at dTA = dt


þ  ∂TA d H ∂TA dDIC + þ dt ∂½H  ∂DIC dt



ð21Þ

where the partial derivatives3 of TA can be analytically calculated. Upon substitution of the rates of change of TA and DIC, which are known from the mass balance Eqs. (11) and (12), and a rearrangement of terms, one also arrives at an expression for the proton balance
 d Hþ = dt

 1 ∂TA  2− RP = SP RP ∂DIC ∂TA − þ

ð22Þ

∂½H 

Using Eq. (16) to explicitly calculate the partial derivatives of TA and knowing that the ionization fractions sum up to one (i.e. αc0 + αc1 + αc2 = 1), one can identify the term4  ∂TA c c P 2− = 2α0 + α1 = νHþ ∂DIC

ð23Þ

A subsequent comparison of the corresponding terms in the proton balances in Eqs. (19) and (22) immediately shows that the buffer factor β must have the form  ∂TA β: =− ∂½Hþ 

ð24Þ

From the above analysis, the partial derivative of the alkalinity with respect to the proton concentration emerges as a central quantity in acid-base theory of natural waters. Moreover, as noted 3 Note that, to reduce notational complexity, subscripts for partial derivatives have been omitted since the exhaustive list of independent variables for our representation of TA given in Eq. (16) implicitly defines the variables being held constant upon partial differentiation. 4 Note that Eq. (23) implies that the stoichiometric coefficient of the proton νHP+ is related to the stoichiometric coefficients of TA ν PT A and DIC ν PD I C by   ∂TA P ∂TA P i νPHþ = − νTA − νDIC which can be generalized to νiHþ = − νTA −∑j νiXj for

∂DIC

∂Xj

process i in a system with total quantities TA and X1, ..., Xn. This relation represents an alternative way of calculating the stoichiometric coefficient of the proton: from the stoichiometric coefficients of the total quantities.

A.F. Hofmann et al. / Marine Chemistry 121 (2010) 246–255

249

Table 1 Values for different terms in Eq. (25), calculated for S = 35, t = 15 °C, DIC = 2000 μmol kg− 1,and TA = 2200 μmol kg− 1. Note that “Numerator” and “Denominator” refers to the fraction given in the second line of Eq. (25). ≈ value

Term in Eq. (25) [H ] 4[H+]K⁎2 K⁎1 K⁎2 Numerator

4.7 × 10− 05 (μmol kg− 1)2 2.2 × 10− 05 (μmol kg− 1)2 8.9 × 10− 04 (μmol kg− 1)2 9.6 × 10− 04 (μmol kg− 1)2

([H+]2)2 ([H+]K⁎1 )2 (K⁎1 K⁎2 )2 Denominator

2.2 × 10− 09 (μmol kg− 1)4 5.9 × 10− 05 (μmol kg− 1)4 8.0 × 10− 07 (μmol kg− 1)4 7.4 × 10− 05 (μmol kg− 1)4

+ 2

above, this partial derivative can be calculated in an analytical form. For our simple example of CaCO3 precipitation, the buffer factor adopts the explicit form  c c dα1 dα2 + 2 DIC = 1 + β = 1− d½Hþ  d½Hþ 

!
þ 2
þ ⁎ ⁎ ⁎ H + 4 H K2 + K1 K2 ⁎ K1 DIC  þ2  2 ½H  + ½Hþ K1⁎ + K1⁎ K2⁎

ð25Þ A closer examination of this expression confirms that β truly describes chemical buffering. In our example, the only chemical buffering is due to the carbonate acid-base system. When there is no carbonate (DIC= 0), the solution has no buffering capacity. In this case, the buffer factor becomes 1, and the protons released or consumed at ambient pH will contribute to proton cycling in an unmodulated fashion. In contrast, when the DIC in the system is very large (DIC↦ ∞), the solution has an infinite buffering capacity (β ↦ ∞). In the latter situation, the solution is buffered to such an extent that it will no longer experience biogeochemical proton cycling and pH changes. Approximations for individual terms (Table 1) allow the conclusion that the numerator of the fraction in the second line of Eq. (25) can, for most marine systems, be approximated by the term K1⁎K2⁎, while the denominator is best approximated by the term ([H+]K1⁎)2. Accordingly, Eq. (25), i.e. the buffer factor β dominated by the carbonate system, can be approximated by β≈1 + 

K1⁎ K2⁎ ½H

þ

2 K1⁎

⁎

K1 DIC = 1 +

K2⁎ DIC ½H þ 2

ð26Þ

This means that, if DIC and pH are treated as independent variables,5 β is inversely correlated with [H+] and directly correlated with DIC. Furthermore, it becomes obvious that, for most marine systems, the second dissociation constant of the carbonate system (K2⁎) is dominating the value of the buffer factor β, rather than the first dissociation constant (K1⁎). If other acid-base systems are taken into consideration besides the carbonate system for the model and thus the definition of TA, the explicit expression for β, i.e. Eq. (25), will feature other total quantities (borate, phosphate, fluoride, silicate, etc.) besides DIC. These other acidbase systems then add to the buffer capacity of the solution. Fig. 1 shows the buffer factor β, calculated with Eq. (25), i.e. considering the carbonate system only (β (CA)), calculated using the approximation in Eq. (26) (β (CA, ap)), as well as calculated including the influence of the borate system, the sulfate system, the fluoride system, and water auto-dissociation (β (full)). Values are shown for standard seawater conditions at present DIC levels and increasing DIC levels. As DIC changes and TA stays constant, pH changes as well (the free scale pH change is overlayed in the plot). It can be seen that β (full) is close to β (CA) which confirms that, in natural marine systems, proton buffering is dominated by the carbonate system. While overestimating at lower and underestimating at higher DIC 5

i.e., TA varies concurrently with either DIC or pH.

Fig. 1. The buffer factor β for global ocean surface waters at current and increasing DIC levels. β is calculated for a constant TA of 2200 μmol kg− 1and DIC between 2000 μmol kg− 1 and 2500 μmol kg− 1. Salinity of S = 35 and temperature of t = 15 °C remain constant. The solid, black line represents β calculated including the contribution of total borate ([∑ B(OH)3]), total sulfate ([∑ H2SO4]), and total fluoride ([∑HF]), all calculated from salinity according to (Dickson et al., 2007), and water auto-dissociation. The dashed, blue line represents β based on carbonate alkalinity (CA) as given in Eq. (25), and the dot-dashed, red line represent the approximation as in Eq. (26). For illustrative purposes, the plot is overlayed by the change in free scale pH (show by the long-dashed, orange line), corresponding to the given change in DIC at constant TA.

concentrations, β (CA, ap) provides an overall reasonable approximation for the buffer factor. The buffer factor β, as defined in Eq. (24), is conceptually equivalent to the buffer capacity defined by Morel and Hering (1993) (see also, e.g., Frankignoulle, 1994; Egleston et al., 2010) and the buffer intensity advanced by Stumm and Morgan (1996). While describing the potential for buffering in the system, the latter quantities were generally only used to calculate the effect of the addition of a fully dissociated strong acid to the system. Here, the buffer factor β is defined as derivative with respect to the proton concentration rather than pH. Together with the concept of fractional stoichiometry at ambient pH introduced above, this allows for a calculation of the effect of any kinetic process on the proton concentration using equations like Eq. (20). The following section will clarify this more. 6. Generalization: proton cycling in natural waters The above analysis applies to a solution with carbonate precipitation being the only biogeochemical process. However, our treatment can be readily generalized to an arbitrarily complex biogeochemical system characterized by multiple chemical reactions and physical transport processes. For each process, the proton-cycling sensitivity can be expressed as Sx =

νxHþ β

ð27Þ

where β is the buffer factor, and νHx + is the stoichiometric coefficient of the proton in the fractional reaction equation at ambient pH for the respective process. It is possible to write a fractional reaction equation at ambient pH for any kinetic process, be it a biogeochemical reaction or a physical transport process. The procedure to do so is detailed in Appendix A for biogeochemical reactions and in (Hofmann et al., 2009b) for physical transport processes.6 6 Note that for a consistent inclusion of advective–dispersive transport, all chemical 2− + species need to be transported (here: CO2, HCO− 3 , CO3 , and H ). Note further that this approach allows for transport with different molecular diffusion constants for each chemical species (cf. porewaters of sediments Boudreau, 1996).

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A.F. Hofmann et al. / Marine Chemistry 121 (2010) 246–255

The total rate of change of protons at a particular location and at a particular time is then the sum of the contributions of all individual processes to proton cycling
 d Hþ = ∑ Sx Rx dt x

ð28Þ

decrease by 0.5 units from 8.2 to 7.7. While these time evolutions could also have been obtained with the classical approach, the advantage of the approach presented here is that it provides additional information on the proton release by calcite precipitation and its change over time. The middle panel in Fig. 2 compares the rate of calcite precipitation RP
(solid, black line) to the rate of change of the  proton concentration

The index x runs over all kinetic processes within the system with corresponding rate Rx. A kinetic process is a process that receives a kinetic rate description on the relevant time scale of the model (as opposed to an equilibrium description). In typical model applications of natural waters, biogeochemical reactions (e.g. carbonate precipitation, primary production, and denitrification) as well as physical transport (e.g. CO2 air–water exchange, advective diffusive transport) fall into this category. Processes that operate on much faster timescales than the model timescale are assumed to reside in instantaneous thermodynamic equilibrium. Such equilibrium processes do not receive a kinetic rate description and should not be included in the summation. Their effect is accounted for through the proton-cycling sensitivity Sx for each kinetic process. In typical model descriptions of natural waters, the acidbase reactions fall into this category (see Hofmann et al., 2008, 2009b, for a discussion of kinetic vs. equilibrium processes). Eq. (28) specifies that proton cycling and thus the pH evolution of an aquatic system are determined by the interplay between biogeochemical process rates and proton-cycling sensitivities. This allows to introduce the concept of the proton-cycling intensity of a natural body of water. Some processes will produce protons (Sx N 0 and Rx N 0, or Sx b 0 and Rx b 0), while others will consume protons (Sx b 0 and Rx N 0, or Sx N 0 and Rx b 0). Grouping processes into proton producing (index p) and proton consuming (index c) processes leads to
þ d H = ∑ Sp Rp − ∑ Sc Rc = P−C dt p c

ð29Þ

Typically, the total proton production P and proton consumption C
 d Hþ are much larger than the net rate of change of protons (Hofmann dt et al., 2009a). We define the largest of the two quantities P and C as the proton-cycling intensity. Dividing the ambient proton concentration by the proton-cycling intensity yields the proton turnover time (see also Hofmann et al., 2009a).

7.1. Carbonate precipitation in a closed system To illustrate the above concepts, we simulate the precipitation of calcite under closed conditions (i.e. no exchange of CO2 with the atmosphere). We adopt following kinetic rate formulation for calcite precipitation (simplified from Morse, 1983) 2

ð30Þ

where kP = 1 μmol kg− 1 d− 1 is the rate constant and Ω = [Ca2+][CO2− 3 ]/ Ksp denotes the saturation state. Starting from given initial conditions, the time evolution of the system is simulated over a period of 50 days (see caption Fig. 2 for details) which corresponds to a total calcite precipitation of about 150 μmol kg− 1. It should be noted that, with the classical approach, the TA evolution would be calculated via direct numerical integration and the pH evolution would subsequently be calculated by solving a nonlinear system of equations. In contrast, with the approach advanced here, the proton concentration evolution is calculated via direct numerical integration and the TA evolution is calculated by analytically evaluating Eq. (16) in every timestep. Fig. 2 shows the obtained results. The top panel of Fig. 2 shows that TA decreases by 14% from 2200 μmol kg− 1to 1900 μmol kg− 1and DIC decreases by 8% from 2000 μmol kg− 1to 1850 μmol kg− 1. This entails a free scale pH

= SP RP (dashed, red line). These two

quantities are not simply proportional: the rate of change of protons decreases less strongly than the calcite precipitation rate. Our treatment shows, that this is because the proton-cycling sensitivity SP of CaCO3 precipitation strongly increases over the simulated time interval, i.e., by approximately 450% from 3.1 × 10− 5 to 17 × 10− 5 as shown by the dotted, blue line in the middle panel of Fig. 2. But what mechanisms can explain this increase in SP? Using the introduced fractional reaction equation at ambient pH, we can calculate the stoichiometric coefficient for the proton νHP + linked to calcite precipitation (bottom panel of Fig. 2; solid, black line). This reveals that calcite precipitation produces about 0.9 mol of protons per mole of CaCO3 precipitated with a modest change of 8% from 0.90 to 0.97 over the simulated time interval of 50 days. Plotting β over time (bottom panel of Fig. 2; dashed, red line), shows that it decreases by 80% from around 29,600 initially to around 5800 after 50 days. The increase in SP is thus the combined result of an increase in νHP + and a decrease in β, where the decrease in β is dominant. 7.2. Carbonate precipitation in an open system As a simple illustration of how the interaction of multiple processes induces proton cycling in an aquatic system, we add CO2 air–sea exchange to the above example. CO2 air–sea exchange can be written in the form of a chemical reaction CO2ðatmÞ ⇌CO2

ð31Þ

which leads to a corresponding fractional reaction equation at ambient pH c

c

−

þ

c

2−

þ

CO2ðatmÞ ⇌α0 CO2 þ α1 ðHCO3 þ H −H2 OÞ þ α2 ðCO3 þ 2H −H2 OÞ ð32Þ Using the kinetic rate law   RC = kC ½CO2 sat −½CO2 

7. Example models

RP = kP ð1−ΩÞ

d Hþ dt

ð33Þ

with a rate constant of kC = 0.5 d− 1 and the saturation concentration [CO2]sat being calculated according to Weiss (1970), the total rate of change of protons now has two separate contributions: from calcite precipitation and from CO2 air–sea exchange
 d Hþ = SP RP + SC RC dt

ð34Þ

where the proton-cycling sensitivities SP and SC are calculated as outlined above. Fig. 3 shows the obtained results. The top panel of Fig. 3 shows that TA decreases by 17% from 2200 μmol kg− 1 to 1824 μmol kg− 1and DIC decreases by 13% from 2000 μmol kg− 1 to 1738 μmol kg− 1. This entails a free scale pH decrease by 0.3 units from 8.2 to 7.9. Compared to the closed system simulation, the DIC change is larger and the pH change smaller because of CO2 outgassing. The stoichiometric coefficient for the proton νHC+ associated with CO2 air–sea exchange varies from 1.10 to 1.05, while νHP+ for calcite precipitation varies between 0.90 and 0.95 in this model, and the buffer factor β changes from 29,600 initially to around 9400 after 50 days (Fig. 3, bottom panel). Mainly as a result of the decreasing buffer factor, the proton-cycling sensitivities for both processes, SP and SC, increase over time by roughly 200% (Fig. 3, middle panel).

A.F. Hofmann et al. / Marine Chemistry 121 (2010) 246–255

251

Fig. 2. Calcite precipitation under closed conditions. The initial condition for DIC is 2000 μmol kg− 1, and for TA 2200 μmol kg− 1, corresponding to a free scale pH of 8.2. Salinity of S = 35 and temperature of t = 15°C remain constant. The saturation state Ω is calculated with a constant calcium ion concentration for seawater of 10282.05 μmol kg− 1 as given in ⁎ (Dickson et al., 2007). The calcite solubility product Ksp is calculated according to (Mucci, 1983), and K⁎ 1 and K2 are calculated according to (Roy et al., 1993). Note that the small contribution of [OH−] formed due to water auto-dissociation to TA (≈ 0.1% for TA = 2200 μmol kg− 1and DIC = 2000 μmol kg− 1) is included in all calculations. The model has been implemented in R (R Development Core Team, 2008) and the numerical integration of the differential equations has been performed using the package deSolve (Soetaert et al., 2010).

Fig. 3. Calcite precipitation in an open system. The model formulation is the same as in the closed system case, only the kinetic process CO2 air–sea exchange has been added as described in the text.

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A.F. Hofmann et al. / Marine Chemistry 121 (2010) 246–255

Fig. 4. Calcite precipitation in an open system: individual
contributions to proton d Hþ cycling, cumulatively plotted. Note that SP RP + SC RC = (thick gray line). dt

Fig. 4 cumulatively plots the contributions of CO2 air–sea exchange and CaCO3 precipitation to proton cycling, i.e., SPRP and SCRC respectively. The gray
line represents their sum, i.e., the total rate of d Hþ

. Note
that change of protons  pH reaches steady state after about dt d Hþ ≈0). Still the proton-cycling intensity 40 days (free scale pH ≈ 7.9, dt

is not zero. The proton production by CaCO3 precipitation balances the proton consumption by CO2 air–sea exchange, and the proton-cycling intensity becomes |SpRP| ≈ |SCRC| ≈ 2 × 10− 4 μmol kg− 1 d− 1. With a proton concentration of 0.013 μmol kg− 1 (pH 7.9), this corresponds to a proton turnover time of about 65 days. Note that this is the proton turnover time after buffering. The proton turnover time before buffering is much smaller, and is arrived at by dividing by the buffer factor β = 9400, providing a turnover time of about 10 min. While the latter helps to understand the model from a rather technical perspective, the quantity relevant for a real-world, buffered system is the proton turnover time after buffering. 8. Proton-cycling sensitivities in an acidifying ocean Combining Eqs. (27) and (28) results in an integrated equation for proton cycling in natural waters
 d Hþ 1 x = ∑ νHþ Rx β x dt

ð35Þ

This equation shows that three principal factors govern proton cycling in an aquatic system: (1) the rates Rx of the kinetic

biogeochemical processes in the system, (2) the stoichiometric coefficients νH+x of protons at ambient pH, and (3) the buffer factor β. This decomposition provides a powerful tool to analyze the proton dynamics of natural waters. This can be illustrated by a simple analysis of the projected future acidification of the global surface ocean. We consider two situations: (1) standard seawater at current global surface ocean conditions with a total scale pH of 8.1 (Dickson et al., 2007) and (2) standard seawater with a total scale pH of 7.7 representing the projected pH decrease of the surface ocean towards the end of the century (IPCC, 2007; Guinotte and Fabry, 2008). For simplicity, we assume that temperature and alkalinity remain constant. Table 2 presents the stoichiometric coefficients for a list of representative biogeochemical processes in these two situations, as well as the corresponding values of the buffer factor. The stoichiometric coefficients reveal exactly how many moles of protons are consumed or produced per mole of reaction. Table 2 shows that these are not integer numbers and that they sometimes differ from the values that intuitively could be expected from the common integer reaction equations. Table 3 shows that, for mineralisation processes and primary production, the stoichiometric coefficients for the proton depend also on the N/C and P/C ratio of the organic matter involved in the reaction. Explicitly defined stoichiometric coefficients for the proton for each process allow for a ranking of processes in terms of their relative impact on proton cycling (i.e. impact per unit of reaction). CO2 air–sea exchange, CaCO3 precipitation, oxic mineralisation and primary production all produce or consume around 1 mol of protons per mole of carbon processed. In contrast, denitrification exhibits only a minor impact on proton cycling, as it only produces about 0.1 mol of protons per mole carbon mineralized. Nitrification, however, produces around 2 mol of protons per mole of ammonium nitrified. Splitting nitrification into its two steps ammonia oxidation and nitrite oxidation reveals that the first step, ammonia oxidation, is responsible for the impact of nitrification on proton cycling, the contribution of the second step, nitrite oxidation, is negligible. Furthermore, one can observe that stoichiometric coefficients are fairly similar for both present-day and acidified conditions. However, it is worth mentioning that, as a result of only slight changes, CaCO3 precipitation/dissolution and oxic mineralisation switch roles in terms of their relative importance for proton cycling. In contrast, the buffer factor β changes significantly: it decreases by a factor of four from around 40,000 at pH = 8.1 to around 10,000 at pH = 7.7. With

Table 2 Buffer factor β and stoichiometric coefficients for the proton νH+ for some common biogeochemical processes in the surface ocean at present and acidified conditions. Stoichiometric coefficients for the proton are derived from the fractional reaction equations at ambient pH of the respective processes as given in Table A.1 in Appendix A. The middle column defines the stoichiometric coefficients in terms of the ionization fractions α, the two rightmost columns give values for the stoichiometric coefficients for the present-day and acidified conditions. Present-day seawater conditions are S = 35, t = 15 °C, TA = 2400 μmol kg− 1, and a total scale pH of 8.1 (Dickson et al., 2007), corresponding to a DIC concentration of 2136 μmol kg− 1. Acidified conditions have been calculated with the same parameters, but with a total scale pH of 7.7, corresponding to a DIC concentration of 2314 μmol kg− 1. Values for [∑B(OH)3], [∑H2SO4], and [∑ HF] are calculated as functions of salinity of S = 35 as in (Dickson et al., 2007). The contribution of the borate, sulfate and fluoride acid-base systems as well as of water auto-dissociation is added to the definition of total alkalinity and the buffer factor. Note that Redfield stoichiometry for organic matter 16 1 has been used for the calculations (N/C ratio: γN = ; P/C ratio: γP = ). 106

106

νH+:=

Reaction

αc1 + 2αc2 2αc0 + αc1 −(2αc0 + αc1) αc1 + 2αc2 − γNαn0 −(αc1 + 2αc2 − γNαn0) − 0.8 + αc1 + 2αc2 − γNαn0 1 + αn0 2 αn0− αno 0 +1 2 αno 0 αc1 + 2αc2 − γNαn0 + γP(αp1 + 2αp2 + 3αp3)

CO2 air–sea exchange CaCO3 precipitation CaCO3 dissolution Oxic mineralisation Primary production Denitrification Nitrification Ammonia oxidation Nitrite oxidation Oxic min. incl. P Buffer factor (β) =

νH+= (pH = 8.1)

(pH = 7.7)

1.08 0.92 − 0.92 0.94 − 0.94 0.14 1.97 1.97 4 × 10− 6 0.96

1.02 0.98 − 0.98 0.87 − 0.87 0.07 1.99 1.99 1 × 10− 5 0.89

39,069

9554

A.F. Hofmann et al. / Marine Chemistry 121 (2010) 246–255 Table 3 Fractional stoichiometric coefficients for the proton at ambient pH for primary production (νHPP+), oxic mineralisation (νHOM + ), oxic mineralisation including phosphorous release (νHOMP + ), and denitrification (νHDN + ). Definitions in terms of ionization fractions are given in Table 2. Values are calculated using Redfield stoichiometry of organic matter (leftmost column) and using four variations in organic matter x composition (nitrogen rich and poor, phosphorous rich and poor): γN = and 106 y . Note that for x = 106 and y = 1, denitrification even consumes protons γP = 106 instead of producing protons. x

16

106

1

16

16

y

1

1

1

106

0.1

νH+PP νH+OM νH+OMP νH+DN

− 0.94 0.94 0.96 0.14

− 0.11 0.11 0.13 − 0.69

− 1.08 1.08 1.09 0.28

− 0.94 0.94 3.04 0.14

− 0.94 0.94 0.94 0.14

stoichiometric coefficients almost staying constant, this implies that changes in the proton-cycling sensitivities are dominated by changes in the buffer factor which affects all processes concurrently. This means that, with a decrease in β by a factor of four, the surface ocean becomes approximately four times more sensitive to any process: it becomes thus also four times more vulnerable to the addition of acid, be it from future uptake of atmospheric CO2 (e.g., Archer et al., 1998; Archer, 2005; Archer et al., 2009) or from the deposition of other atmospheric acids like HNO3 and H2SO4 (e.g., Doney et al., 2007; Duce et al., 2008), as well as any other proton producing or consuming process.

Acknowledgements This research was supported by the EU (CARBOOCEAN, FP6, 511176-2 and EPOCA, FP7, 2211384) and the Netherlands Organisation for Scientific Research (833.02.2002). This is publication number 4773 of the Netherlands Institute of Ecology (NIOO-KNAW), Centre for Estuarine and Marine Ecology, P.O. Box 140, 4400 AC Yerseke, The Netherlands. Filip Meysman is supported by an Odysseus research grant from F.W.O. (Research Foundation Flanders). Appendix A. A recipe for fractional reaction equations at ambient pH Here we describe a procedure to derive the fractional reaction equation at ambient pH starting from an arbitrary reaction equation. To arrive at the fractional reaction equation at ambient pH, one must substitute all compounds affected by the buffer system, except for H+, in the original (integer) reaction equation using the ionization fractions of the respective acid-base system. The substituted ionized counterparts are accompanied by water molecules and protons to balance masses and charge in the reaction equation. For the carbonate system, the substitution list is given by c

c

We have introduced fractional stoichiometry at ambient pH as a way to unambiguously express the direct proton production or consumption of biogeochemical processes. A buffer factor has been introduced which can be analytically calculated as the partial derivative of the total alkalinity definition equation with respect to the proton concentration. Together, the fractional stoichiometric coefficient for the proton and the buffer factor determine the protoncycling sensitivity of a given biogeochemical process. This allows not only for a direct calculation of pH changes without the detour of alkalinity changes,7 but also for a quantification of influences of processes on the proton concentration. In systems where multiple processes counteract each other in terms of their influences on the proton concentration, proton-cycling intensities and proton turnover times can be calculated. The stoichiometric coefficient of the proton for an arbitrary process expresses the relative importance of this process for proton cycling. Stoichiometric coefficients depend on the current state of the system (pH, total concentrations, etc.), but barely change with predicted ocean acidification over the next century. The buffer factor dominates proton-cycling intensity and expresses the resilience of the pH of an aquatic system to addition of acids or bases. The buffer factor is projected to significantly decrease over the next century due to ongoing ocean acidification. Such drastic changes in the buffer capacity of natural waters, i.e. in the resilience of natural aquatic systems to any proton producing or consuming process, and the associated consequences in terms of chemistry and ecology clearly warrant further research.

7 Alkalinity changes and partial derivatives of alkalinity can be analytically calculated at any time using the proton concentration, the concentrations of total quantities other than alkalinity (DIC, total ammonium, etc.), and the definition equation for alkalinity.

−

þ

c

2−

þ

CO2 : α0 CO2 þ α1 ðHCO3 þ H −H2 OÞ þ α2 ðCO3 þ 2H þ −H2 OÞ −

9. Summary

253

c

þ

c

þ

c

−

c

2−

ðA:1Þ þ

ðA:2Þ

2−

ðA:3Þ

HCO3 : α0 ðCO2 −H þ H2 OÞ þ α1 HCO3 þ α2 ðCO3 þ H Þ

2−

CO3

c

−

þ

c

: α0 ðCO2 −2H þ H2 OÞ þ α1 ðHCO3 −H Þ þ α2 CO3

For the ammonia system, the substitution list is given by

þ

n

þ

n

þ

ðA:4Þ

NH3 : α0 ðNH4 −H Þ þ α1 NH3

n

þ

þ

ðA:5Þ

NH4 : α0 NH4 þ α1 ðNH3 þ H Þ

n

The substitution list of other acid-base equilibria that are part of the buffer system can be derived by analogy. If water autodissociation is included in the model, OH− needs to be substituted by H2O − H+ if the ion product of water (KW⁎ = [H+][OH−]) is considered and [H2O] is assumed constant. If the true dissocation constant of water (KH2O⁎ = [H+][OH−] / [H2O]) is considered and [H2O] is modelled dynamically, the substitution list has to be derived analogously to the other acid-base systems considering the ionization fractions for H2O and OH−. Table A.1 summarizes integer reaction schemes and the corresponding fractional reaction equations at ambient pH for a list of common physical and biogeochemical processes. To arrive at the fractional reaction equation at ambient pH for, e.g., oxic mineralisation and primary production, CO2 and NH3, need to be substituted using Eqs. (A.1) and (A.5). Any arbitrary process can be transformed into fractional reaction equations at ambient pH by using the substitution lists given above and, if needed, additional substitution lists for other acid-base systems. Physical transport processes can be treated analogously when they are written as chemical reactions (see Hofmann et al., 2009b, for details).

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A.F. Hofmann et al. / Marine Chemistry 121 (2010) 246–255

Table A.1 Stoichiometric descriptions and according fractional reaction equations at ambient pH for some physical and biogeochemical processes. The stoichiometric coefficients for the proton νHx + are indicated in red. γN is the N/C ratio and γP is the P/C ratio of organic matter. Note that nitric acid has been considered conservative (i.e. always in the dissociated form as no2 − 2 nitrate), while for the nitrous acid system a substitution list similar to the ammonia system has been used, with ionization fractions αno 0 for HNO2 and α1 for NO2 . Since phosphoric acid is a trivalent acid, the substitution list for the phosphorous system is more complex than the substitution lists presented so far. However, it can be derived in completely p 2− p 3− analogous fashion, and is given in (Hofmann et al., 2009b, page 148). Ionization fractions are: αp0 for H3PO4, αp1 for H2PO− 4 , α2 for HPO4 , and α3 for PO4 . Reaction

Reaction equation

Fractional reaction equation at ambient pH

CO2 air–sea exchange

CO2(g)

⇌CO2

CO2(g)

CaCO3 precipitation

Ca2+ + CO2− 3

→CaCO3

CaCO3 dissolution

CaCO3

→Ca2+ + CO2− 3

Ca2+ + αc0H2O + αc0CO2 + αc1HCO− 3 + αc2CO2− 3 CaCO3

Oxic mineralisation

CH2O (NH3)γN + O2

→H2O + CO2 + γNNH3

CH2O (NH3)γN + O2

Primary production

H2O + CO2 + γNNH3

→CH2O (NH3)γN + O2

Denitrification

CH2O (NH3) γN + + 0.8NO− 3 + 0.8H

→γNNH3 + CO2 + 0.4N2↑ + 1.4H2O

(1 − αc1 − αc2)H2O + γN(αn0NH+ 4 + c 2− αn1NH3) + αc0CO2 + αc1HCO− 3 + α2CO3 − CH2O (NH3)γN + 0.8NO3

Nitrification Ammonia oxidation Nitrite oxidation Oxic min. incl. P

NH3 + 2O2 NH3 + 1.5O2 NO− 2 + 0.5O2 CH2O (NH3)γN (H3PO4)γP + O2

+ →NO− 3 + H2O + H + →NO− 2 + H2O + H →NO− 3 →H2O + CO2 + γNNH3 + γPH3PO4

n αn0NH+ 4 + α1NH3 + 2O2 + N αN 0 NH4 + α1 NH3 + 1.5O2 no2 − 2 αno 0 HNO2 + α1 NO2 + 0.5O2 CH2O (NH3)γN (H3PO4)γP + O2

References Archer, D., 2005. Fate of fossil fuel CO2 in geologic time. Journal of Geophysical Research-Oceans 110 (C9). Archer, D., Eby, M., Brovkin, V., Ridgwell, A., Cao, L., Mikolajewicz, U., Caldeira, K., Matsumoto, K., Munhoven, G., Montenegro, A., Tokos, K., 2009. Atmospheric lifetime of fossil fuel carbon dioxide. Annual Review of Earth and Planetary Sciences 37 (1), 117–134. Archer, D., Kheshgi, H., Maier-Reimer, E., 1998. Dynamics of fossil fuel CO2 neutralization by marine CaCO3. Global Biogeochemical Cycles 12 (2), 259–276. Aris, R., Mah, R.H.S., 1963. Independence of chemical reactions. Industrial & Engineering Chemistry Fundamentals 2 (2) 90–94&.3. Ben-Yaakov, S., 1970. A method for calculating the in situ pH of seawater. Limnology and Oceanography 15 (2), 326–328. Boudreau, B.P., 1987. A steady-state diagenetic model for dissolved carbonate species and pH in the porewaters of oxic and suboxic sediments. Geochimica et Cosmochimica Acta 51 (7), 1985–1996. Boudreau, B.P., 1996. Diagenetic models and their implementation. Springer. Culberson, C.H., 1980. Calculation of the in-situ pH of seawater. Limnology and Oceanography 25 (1), 150–152. Dickson, A.G., 1981. An exact definition of total alkalinity and a procedure for the estimation of alkalinity and total inorganic carbon from titration data. Deep-Sea Research Part A: Oceanographic Research Papers 28 (6), 609–623. Dickson, A.G., Sabine, C., Christian, J.R., 2007. Guide to best practices for ocean CO2 measurements. PICES Special Publications 3, 1–191. DiToro, D.M., 1976. Combining chemical equilibrium and phytoplankton models — a general methodology. In: Canale, R.P. (Ed.), Modelling Biochemical Processes on Aquatic Ecosystems. Science, Ann Arbor, pp. 233–255. Doney, S.C., Mahowald, N., Lima, I., Feely, R.A., Mackenzie, F.T., Lamarque, J.-F., Rasch, P.J., 2007. Impact of anthropogenic atmospheric nitrogen and sulfur deposition on ocean acidification and the inorganic carbon system. Proceedings of the National Academy of Sciences 104 (37), 14580–14585. Dore, J.E., Lukas, R., Sadler, D.W., Church, M.J., Karl, D.M., 2009. Physical and biogeochemical modulation of ocean acidification in the central North Pacific. Proceedings of the National Academy of Sciences of the United States of America 106 (30), 12235–12240. Duce, R.A., LaRoche, J., Altieri, K., Arrigo, K.R., Baker, A.R., Capone, D.G., Cornell, S., Dentener, F., Galloway, J., Ganeshram, R.S., Geider, R.J., Jickells, T., Kuypers, M.M., Langlois, R., Liss, P.S., Liu, S.M., Middelburg, J.J., Moore, C.M., Nickovic, S., Oschlies, A., Pedersen, T., Prospero, J., Schlitzer, R., Seitzinger, S., Sorensen, L.L., Uematsu, M., Ulloa, O., Voss, M., Ward, B., Zamora, L., 2008. Impacts of atmospheric anthropogenic nitrogen on the open ocean. Science 320 (5878), 893–897. Egleston, E.S., Sabine, C.L., Morel, F.M.M., 2010. Revelle revisited: buffer factors that quantify the response of ocean chemistry to changes in DIC and alkalinity. Global Biogeochemical Cycles 24. doi:10.1029/2008GB003407 GB1002. Follows, M.J., Ito, T., Dutkiewicz, S., 2006. On the solution of the carbonate chemistry system in ocean biogeochemistry models. Ocean Modelling 12 (3–4), 290–301. Frankignoulle, M., 1994. A complete set of buffer factors for acid-base CO2 system in seawater. Journal of Marine Systems 5, 111–118. Gazeau, F., Quiblier, C., Jansen, J.M., Gattuso, J.P., Middelburg, J.J., Heip, C.H.R., 2007. Impact of elevated CO2 on shellfish calcification. Geophysical Research Letters 34. Guinotte, J.M., Fabry, V.J., 2008. Ocean acidification and its potential effects on marine ecosystems. Year in Ecology and Conservation Biology 1134, 320–342.

c 2− c c ⇌αc0CO2 + αc1HCO− 3 + α2CO3 − (α1 + α2)H2O + (αc1 + 2αc2)H+ →CaCO3 + (2αc0 + αc1)H+

→Ca2+ + αc0H2O − (2αc0 + αc1)H+ + αc0CO2 + c 2− αc1HCO− 3 + α2CO3 →(1 − αc1 − αc2)H2O + (αc1 + 2αc2 − γNαn0)H+ + n c c − c 2− γN(αn0NH+ 4 + α1NH3) + (α0CO2 + α1HCO3 + α3CO3 ) →CH2O (NH3)γN + O2 + (γNαn0 − αc1− 2αc2) H+ →0.4N2 ↑ + γNαn0NH4+ + γNαn1NH3 + αc0CO2 + c 2− c c αc1HCO− 3 + α2CO3 + (1.4 − α1 − α2)H2O + (− 0.8 + αc1 + 2αc2 − γNαn0)H+ + n →NO− 3 + H2O + (1 + α0)H + no2 − n no2 2 →αno 0 HNO2 + α1 NO2 + H2O + (α0− α0 + 1)H no2 + →NO− 3 + α0 H n →(1 − αc1 − αc2)H2O + γN(αn0NH+ 4 + α1NH3) + c 2− c c n (αc0CO2 + αc1HCO− 3 + α3CO3 ) + (α1 + 2α2 − γNα0 + γP(αp1 + 2αp2 + 3αp3))H+

Hofmann, A.F., Meysman, F.J.R., Soetaert, K., Middelburg, J.J., 2008. A step-by-step procedure for pH model construction in aquatic systems. Biogeosciences 5 (1), 227–251. Hofmann, A., Soetaert, K., Middelburg, J.J., Meysman, F.J.R., 2009a. pH modelling in aquatic systems with time-variable acid-base dissociation constants applied to the turbid, tidal Scheldt estuary. Biogeosciences 6, 1539–1561. Hofmann, A. F., Middelburg, J. J., Soetaert, K., Wolf-Gladrow, D. A., Meysman, F. J. R., 2009b. Buffering, stoichiometry and the sensitivity of pH to biogeochemical and physical processes: a proton-based model perspective. In: Quantifying the influences of biogeochemical processes on pH of natural waters, A.F. Hofmann, PhD thesis, URL http://igitur-archive.library.uu.nl/dissertations/2009-0513200343/UUindex.html, 129–151. Hofmann, A.F., Soetaert, K., Middelburg, J.J., Meysman, F.J.R., 2010. AquaEnv — an Aquatic acid-base modelling Environment in R. Aquatic Geochemistry. doi:10.1007/ s10498-009-9084-1. Software: http://cran.r-project.org/package=AquaEnv7. IPCC, 2007. Climate Change 2007: Synthesis Report. Contributions of Working Groups I, II, and III to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change [Core Writing team, Pachauri, R.K and Reisinger, A. (eds.)]. IPCC, Geneva, Switzerland. Kleypas, J.A., Feely, R.A., Fabry, V.J., Langdon, C., Sabine, C.L., Robbins, L.L., 2006. Impacts of Ocean Acidification on Coral Reefs and Other Marine Calcifiers: A Guide for Future Research, report of a workshop held 18–20 April 2005, St. Petersburg, FL, sponsored by NSF, NOAA, and the U.S. Geological Survey. Tech. rep.1. Luff, R., Haeckel, M., Wallmann, K., 2001. Robust and fast FORTRAN and MATLAB® libraries to calculate pH distributions in marine systems. Computers & Geosciences 27 (2), 157–169. Meysman, F., 2001. Modelling the influence of ecological interactions on reactive transport processes in sediments. Ph.D. thesis, Netherlands Institute of Ecology. Morel, F.M., Hering, J.G., 1993. Principles and applications of aquatic chemistry. John Wiley & sons. Morse, J.W., 1983. The kinetics of calcium carbonate dissolution and precipitation. In: Reeder, R.J. (Ed.), Carbonates: Mineralogy and Chemistry: Mineral. Soc. Am. , pp. 227–264. Mucci, A., 1983. The solubility of calcite and aragonite in seawater at various salinities, temperatures, and one atmosphere total pressure. American Journal of Science 283 (7), 780–799. Olander, D.R., 1960. Simultaneous mass transfer and equilibrium chemical reaction. Aiche Journal 6 (2), 233–239. Orr, J.C., Fabry, V.J., Aumont, O., Bopp, L., Doney, S.C., Feely, R.A., Gnanadesikan, A., Gruber, N., Ishida, A., Joos, F., Key, R.M., Lindsay, K., Maier-Reimer, E., Matear, R., Monfray, P., Mouchet, A., Najjar, R.G., Plattner, G.K., Rodgers, K.B., Sabine, C.L., Sarmiento, J.L., Schlitzer, R., Slater, R.D., Totterdell, I.J., Weirig, M.F., Yamanaka, Y., Yool, A., 2005. Anthropogenic ocean acidification over the twenty-first century and its impact on calcifying organisms. Nature 437 (7059), 681–686. R Development Core Team, 2008. R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria3-900051-07-0. URL http://www.r-project.org. Roy, R.N., Roy, L.N., Vogel, K.M., PorterMoore, C., Pearson, T., Good, C.E., Millero, F.J., Campbell, D.M., 1993. The dissociation constants of carbonic acid in seawater at salinities 5 to 45 and temperatures 0 to 45 degrees C (Erratum: Mar. Chem., vol. 44, p. 183, 1996). Marine Chemistry 44, 249–267. Saaltink, M.W., Ayora, C., Carrera, J., 1998. A mathematical formulation for reactive transport that eliminates mineral concentrations. Water Resources Research 34 (7), 1649–1656. Santana-Casiano, J.M., Gonzalez-Davila, M., Rueda, M.J., Llinas, O., Gonzalez-Davila, E.F., 2007. The interannual variability of oceanic CO2 parameters in the northeast Atlantic subtropical gyre at the ESTOC site. Global Biogeochemical Cycles 21 (1).

A.F. Hofmann et al. / Marine Chemistry 121 (2010) 246–255 Skoog, D.A., West, D.M., 1982. Fundamentals of analytical chemistry. Holt-Saunders International Editions, Holt-Saunders. Soetaert, K., Petzold, T., Setzer, R.W., 2010. Solving differential equations in R: package deSolve. Journal of Statistical Software 33 (9), 1–25. Stumm, W., Morgan, J.J., 1996. Aquatic chemistry: chemical equilibria and rates in natural waters. Wiley Interscience, New York. Weiss, R.F., 1970. Solubility of nitrogen, oxygen and argon in water and seawater. DeepSea Research 17 (4), 721–735. Wolf-Gladrow, D.A., Zeebe, R.E., Klaas, C., Koertzinger, A., Dickson, A.G., 2007. Total alkalinity: the explicit conservative expression and its application to biogeochemical processes. Marine Chemistry.

255

Wootton, J.T., Pfister, C.A., Forester, J.D., 2008. Dynamic patterns and ecological impacts of declining ocean pH in a high resolution multi-year dataset. Proceedings of the National Academy of Sciences 105 (48), 18848–18853. Zeebe, R.E., Wolf-Gladrow, D., 2001. CO2 in seawater: equilibrium, kinetics, isotopes, 1st Edition. Elsevier Oceanography Series, No. 65. Elsevier. Zeebe, R.E., Zachos, J.C., Caldeira, K., Tyrrell, T., 2008. Oceans — carbon emissions and acidification. Science 321 (5885), 51–52.