Knowledge of the dissolved inorganic carbon (DIC) system is central to a range of research fields and applications, among which are chemical oceanography, environmental geochemistry, carbon sequestration, and experimental geochemistry (e.g., Stumm and Morgan, 1996; Zeebe et al., 1999; Rollion-Bard et al., 2003; Lal, 2008; Hansen et al., 2013; Daëron et al., 2019; Christensen et al., 2021). An understanding of the DIC kinetics factors into such fields, for example, when planning experiments, when deciding on dynamic versus equilibrium representation of reactions in models of aqueous environments, when evaluating the likelihood of isotopic equilibrium for applications in paleoclimate and paleohydrology, and when engineering applications of descaling or carbon sequestration.

The chemical reactions in the DIC system progress at different rates (e.g., Zeebe and Wolf-Gladrow, 2001; Sade and Halevy, 2017; Sade and Halevy, 2018). Dissociation of carbonic acid and water are rapid reactions, whose reactants may be considered to be in equilibrium on a time scale of seconds: H 2 C O 3 ⇌ H C O 3 – + H + , (1) H C O 3 – ⇌ C O 3 2 – + H + , (2) H 2 O ⇌ H + + O H – . (3)

In contrast, CO₂ hydration and hydroxylation are slower reactions, which determine the equilibration times of the sum of H₂CO₃, HCO₃ ⁻, and CO₃ ²⁻ with aqueous CO₂ (henceforth CO₂(aq)), and therefore, the equilibration time of the entire DIC reservoir: C O 2 ( a q ) + H 2 O k + 4 ⇄ k – 4 H 2 C O 3 , (4) C O 2 ( a q ) + O H – k + 5 ⇄ k – 5 H C O 3 – , (5) where k _+i and k _−i are the forward and reverse rate constants of reaction i. In freshwater at 25°C, k ₊₄ and k ₋₄ are 0.026 s⁻¹ and 27 s⁻¹, whereas k ₊₅ and k ⁻ ₅ are 8,416 L mol⁻¹ s⁻¹, and 0.00019 s⁻¹, respectively (Roughton, 1941; Pinsent et al., 1956).

Usdowski (1982) derived equations to describe the time required to achieve chemical equilibrium in a DIC system that is closed to gas exchange with the atmosphere during pH transitions in buffered solutions. The equations of Usdowski (1982) specifically regard an initial low-pH solution composed exclusively of CO₂(aq), which re-equilibrates at an arbitrary, new, fixed pH. The mathematical derivation is based on a schematic reaction between two distinct DIC pools, CO₂(aq) and S, [ C O 2 ( a q ) ] k ⇄   k ′ S , (6) where S is the sum of [H₂CO₃], [HCO₃ ⁻] and [CO₃ ²⁻], which are internally equilibrated with each other {DIC = [CO₂(aq)] + S}. In reaction 6, k, and k’ are rate constants of composite, rather than elementary reactions. Specifically, k and k’ are related to the CO₂ hydration and hydroxylation reactions, and their values depend on the solution pH, k = k + 4 + k + 5 [ O H – ] , (7) k ′ = k – 4 [ H 2 C O 3 ] S   + k – 5 [ H C O 3 –   ] S   = k – 4 1 + K H 2 C O 3 [ H + ] + K H 2 C O 3 ⋅ K H C O 3 – [ H + ] 2 + k – 5 1 + [ H + ] K H 2 C O 3 + K H C O 3 – [ H + ] . (8)

In Eq. 8, K H 2 C O 3 and K H C O 3 – are the first and second carbonic acid dissociation constants. Note that K H 2 C O 3 is distinct from Ka1, the first acid dissociation constant for the sum of CO₂(aq) and H₂CO₃ (Supplementary Information). Then, the equilibration time following a pH transition from an initially CO₂(aq)-dominated DIC solution to a new buffered solution is given by (cf. Eq. I-2–22 in Usdowski, 1982): t = l n ( 1 − S S e q ) – ( k + k ′ ) , (9) where the subscript “eq” denotes the equilibrium concentration at the new pH, and the ratio S/S _eq, the equilibration degree of the S reservoir, approximates the equilibration degree of the entire DIC reservoir at a given time t.

The derivation of Usdowski (1982) closely approximates the chemical equilibration time only under certain conditions. A general derivation should accurately represent a DIC solution at any initial pH and associated DIC speciation. Furthermore, a conservative estimate of the equilibration degree of the DIC reservoir should consider the smaller of the CO₂(aq) and S pools at the new pH, which is always farther from equilibrium. For example, consider an initial solution with 100 units of DIC, all as CO₂(aq), which re-equilibrates to a new buffered pH, at which CO₂(aq):S is 60:40. As equilibrium is approached, for example, when CO₂(aq):S is 65:35, the CO₂(aq) pool is 8.3% away from equilibrium ([CO₂(aq)]/[CO₂(aq)]_eq = 65/60 = 1.083), whereas the S reservoir is 12.5% away from equilibrium (S/S _eq = 35/40 = 0.875). Conversely, during equilibration to a pH at which CO₂(aq):S is 40:60, at 5 units away from equilibrium, the CO₂(aq) pool is 12.5% away from equilibrium and the S pool is 8.3% away from equilibrium. In both cases, the smaller pool at the new pH is farther from equilibrium during the chemical evolution.

In this study, we expanded the equations of Usdowski (1982) to derive general analytical expressions for the chemical equilibration time of DIC solutions of any initial speciation, at any new, buffered pH. We additionally used a numerical model to explore the equilibration times of the DIC in unbuffered solutions. Lastly, we compared the equilibration times of the DIC during pH transitions in buffered and unbuffered solutions with equilibration times calculated by perturbation theory at a given pH (Zeebe and Wolf-Gladrow, 2001), and discussed the sensitivity of these equilibration times to the ionic strength (salinity) of the solution.

Equilibration times of DIC during pH transitions are determined by 1) the initial distance from the equilibrium speciation at the new pH and 2) the rates of DIC conversion during the chemical evolution (Figure 2). Given a fixed DIC concentration, the DIC speciation depends upon the solution pH. Therefore, the difference between the initial and new (buffered or unbuffered) pH values of the solution determines how much of the DIC needs to be converted during the chemical evolution at the new pH. Specifically, if the new pH is higher than the initial pH, CO₂(aq) is converted to S by CO₂(aq) hydration and hydroxylation reactions (forward reaction 6). In contrast, if the new pH is lower than the initial pH, CO₂(aq) is produced by H₂CO₃ dehydration and HCO₃ ⁻ dehydroxylation reactions (reverse reaction 6). However, the solution pH affects not only the amount of DIC that must be converted, but also the conversion rate. The rate constants of CO₂(aq) hydration and hydroxylation, H₂CO₃ dehydration, and HCO₃ ⁻ dehydroxylation, alongside pH-dependent fractions of H₂CO₃ and HCO₃ ⁻ out of S, result in a pH-dependence of the rate of DIC conversion. Specifically, with increasing pH, rates of CO₂(aq) consumption increase whereas rates of CO₂(aq) production decrease (Figure 2B).

Overall, equilibration times are shorter when the converted DIC fractions are smaller, and when the CO₂(aq)-S conversion rates are higher. For instance, the maximal equilibration time in buffered solutions is achieved during a transition of a CO₂(aq)-dominated solution to a new pH of 7.8 (245 s; Figure 1A). During this transition, 97% of the DIC is converted from CO₂(aq) to S (Figure 2A, blue) at rates that are close to the minimum rates of CO₂(aq) consumption under these conditions (Figure 2B, blue). In comparison, the equilibration of an S-dominated solution at the same buffered pH of 7.8 is faster (142 s; Figure 1A), despite CO₂(aq) production rates in this case that are lower by an order of magnitude than the CO₂(aq) consumption rates in the CO₂(aq)-dominated solution described above (Figure 2B, red). The equilibration time of the S-dominated solution is shorter because only 3% of the DIC needs to be converted from S to CO₂(aq) (Figure 2A, red).

Notably, equilibration times of DIC in unbuffered solutions are shorter relative to buffered solutions (Figure 1). In unbuffered solutions, introduction of Ca²⁺ or Cl⁻ ions (as simulated in our numerical model) immediately alters the solution alkalinity, resulting in a transient pH excursion to higher and lower values, respectively, than the final equilibrium pH value. The reason for this transient excursion is that it takes time for DIC speciation [through CO₂(aq) production/consumption] to compensate for the instantaneous change in the solution’s alkalinity. For example, introduction of 5.1 mM Ca²⁺ into a CO₂(aq)-dominated 10 mM DIC solution (pH₀ = 0) instantaneously increases the simulated pH to 12, before decreasing back to the new equilibrium pH of 8.6 (Supplementary Figure S3). Since CO₂(aq) production/consumption rates are higher at lower/higher pH values, these overshoots to higher-than-equilibrium and lower-than-equilibrium pH values decrease the overall equilibration times relative to conditions of a fixed, buffered pH (Figure 2).

Our numerical equilibration times for a pH decrease in unbuffered solutions are comparable to equilibration times calculated by perturbation theory, which are commonly cited in the literature (Zeebe and Wolf-Gladrow, 2001). This similarity arises from the decay of a perturbation to DIC speciation being the driver of the chemical evolution in both cases. In the perturbation theory, DIC is perturbed directly, whereas in our numerical simulations of unbuffered solutions the alkalinity is changed, and the DIC speciation is “perturbed” relative to what it would be at the final new pH. Differences between our numerical equilibration times for both pH decrease and increase and the perturbation theory stem from different amounts and rates of CO₂(aq) consumption/production during the relaxation from the perturbed to the equilibrium pH (Supplementary Information).

Salinity affects the DIC equilibration times in buffered and unbuffered solutions differently (Figure 3). Applying equilibrium and rate constants for seawater (Supplementary Information) in Eq. 13 and in our numerical model results in shorter and longer equilibration times in buffered and unbuffered solutions, respectively, relative to freshwater. For example, the maximal equilibration time of initially CO₂(aq)-dominated seawater buffered solution (Figure 3A), is 25% shorter than in freshwater. In both cases (buffered and unbuffered solutions), the seawater curves shift towards lower pH values relative to freshwater, in accordance with the shift of pKa1 (in seawater, pKa1 = 5.86; Zeebe and Wolf-Gladrow, 2001).

Our analytical expressions for the chemical equilibration times of DIC in (strongly) buffered solutions are applicable to a range of artificial settings, such as laboratory experiments of mineral precipitation. The numerical model for equilibration times in unbuffered solutions expands the applicability to a wide range of natural and artificial settings. Examples include rapid productivity-driven changes in DIC concentration and speciation in the photic zone of lakes and oceans, exhalation of hydrothermal fluids into seawater, injection of CO₂-bearing fluids into marine sediments, or pH variations driven by oxidation of sulfide and/or ferrous iron in initially anoxic spring water or acid mine drainage. As the relaxation to equilibrium is governed by the rates of reaction among the DIC species and with water (H₂O in the case of hydration, OH⁻ in the case of hydroxylation), the chemical equilibration times presented here bear also on the time to carbon isotopic equilibrium within the DIC system and on the time to DIC-water oxygen isotopic equilibrium.