Available online at www.sciencedirect.com Geochimica et Cosmochimica Acta 75 (2011) 2483–2498 www.elsevier.com/locate/gca À On the molecular diffusion coefficients of dissolved CO2; HCO3 , 2À and CO3 and their dependence on isotopic mass Richard E. Zeebe ⇑ School of Ocean and Earth Science and Technology, University of Hawaii at Manoa, 1000 Pope Road, MSB 504, Honolulu, HI 96822, USA Received 21 August 2010; accepted in revised form 7 February 2011; available online 13 February 2011 Abstract À The molecular diffusion coefficients of dissolved carbon dioxide ðCO2Þ, bicarbonate ion ðHCO3 Þ, and carbonate ion 2À ðCO3 Þ are fundamental physico-chemical constants and are of practical significance in various disciplines including geochem- istry, biology, and medicine. Yet, very little experimental data is available, for instance, on the bicarbonate and carbonate ion diffusion coefficient. Furthermore, it appears that no information was hitherto available on the mass-dependence of the dif- fusion coefficients of the ionic carbonate species in water. Here I use molecular dynamics simulations to study the diffusion of the dissolved carbonate species in water, including their dependence on temperature and isotopic mass. Based on the simu- À 2À lations, I provide equations to calculate the diffusion coefficients of dissolved CO2; HCO3 , and CO3 over the temperature range from 0° to 100 °C. The results indicate a mass-dependence of CO2 diffusion that is consistent with the observed 12 13 CO2= CO2 diffusion ratio at 25 °C. No significant isotope fractionation appears to be associated with the diffusion of À 2À the naturally occurring isotopologues of HCO3 and CO3 at 25 °C. Ó 2011 Elsevier Ltd. All rights reserved. 1. INTRODUCTION 1995; Wolf-Gladrow and Riebesell, 1997; Cadours and Bouallou, 1998; Zeebe, 2007b; Kaufmann and Dreybrodt, The molecular diffusion coefficients of the dissolved 2007; Berne et al., 2009; Chhabra et al., 2009). While the À 2À carbonate species (CO2; HCO3 , and CO3 ) in aqueous molecular diffusion coefficient of carbon dioxide in water solution are fundamental physico-chemical parameters. is relatively well known over a range of temperatures, less Knowledge of these parameters at various temperatures is is known about the bicarbonate diffusion coefficient, and of practical value, for instance, in geochemical, biological, little information is available on the diffusion coefficient and medical applications. A few examples include sediment of the carbonate ion. À diagenesis, mineral precipitation and dissolution, fossil fuel As discussed below, diffusion coefficients of HCO3 and 2À carbon sequestration, industrial engineering, carbon uptake CO3 at infinite dilution have been estimated based on and calcification in phytoplankton and zooplankton, stud- conductivity measurements from the 1930s and 1940s ies of duodenal ulceration, O2=CO2 exchange in red blood (Robinson and Stokes, 1959; Li and Gregory, 1974). How- cells, and metabolic models of cornea-contact-lens systems ever, to the best of my knowledge, so far only a single (e.g. Berner, 1980; Uchida et al., 1983; Livingston et al., experimental study has been conducted to directly deter- 2À mine the CO3 diffusion coefficient in water. The few data points were published in a largely unknown short commu- nication by a Japanese group in the 1960s (Kigoshi and ⇑ Address: School of Ocean and Earth Science and Technology, Department of Oceanography, University of Hawaii at Manoa, Hashitani, 1963). Furthermore, it appears that diffusion 1000 Pope Road, MSB 504, Honolulu, HI 96822, USA. Tel.: +1 studies of the ionic carbonate species have as yet been lim- 808 956 6473; fax: +1 808 956 7112. ited to temperatures 6 30 C. While some information is E-mail address: [email protected] available on the mass-dependence of CO2 diffusion in 0016-7037/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.gca.2011.02.010 2484 R.E. Zeebe / Geochimica et Cosmochimica Acta 75 (2011) 2483–2498 12 13 water, e.g. on CO2 vs. CO2 diffusion (O’Leary, 1984; 2. SELF- AND TRACER-DIFFUSION COEFFICIENT Ja¨hne et al., 1987), I am not aware of a theoretical or experimental study that has hitherto tackled the mass- ‘Self-diffusion’ is a process in which the molecules of, for dependence of the diffusion coefficients of the bicarbonate instance, a uniform liquid move randomly over time from and carbonate ion. The mass-dependence associated with one point to another (Robinson and Stokes, 1959). the diffusion of the ionic carbonate species has implications, ‘Tracer-diffusion’ usually refers to a process in which ions for instance, for understanding vital effects in carbonates of a certain kind and of very small concentration diffuse and clumped isotope studies (e.g. Thiagarajan et al., in a large excess of other electrolyte. If both the tracer 2009). and the electrolyte are of the same nature, e.g. 22Naþ in a Given the geochemical, biological, and medical sodium chloride solution, the tracer-diffusion coefficient is significance of the molecular diffusion coefficients of assumed to be equal to the self-diffusion coefficient. À 2À dissolved CO2; HCO3 , and CO3 , a comprehensive study of these important parameters appears warranted. From a 2.1. Limiting conductivity geochemical point of view, such an effort also appears timely, given the growing number of studies dealing with Limiting tracer- or self-diffusion coefficients of ions have the chemistry of dissolved CO2 in seawater and the cou- been estimated based on measurements of the limiting con- pling of diffusion and reaction within the system (for funda- ductivity using the Nernst–Einstein equation (Robinson mentals, see e.g. Wolf-Gladrow and Riebesell, 1997; Zeebe and Stokes, 1959; Li and Gregory, 1974): et al., 1999; Zeebe and Wolf-Gladrow, 2001). In the present RTk0 study, I have used molecular dynamics (MD) simulations to D0 ¼ ð1Þ 2 2 examine the diffusion coefficients of the dissolved carbonate z F species in water, their temperature-dependence, and their where D0 is the limiting tracer- or self-diffusion coefficient, dependence on isotopic mass. Among other results, I will R ¼ 8:3145 J KÀ1 molÀ1 is the gas constant, T is temperature provide equations to calculate the diffusion coefficients of in Kelvin, k0 is the limiting conductivity (per mole), z is the dissolved carbonate species over the temperature range charge, and F ¼ 9:6485 Â 104 C molÀ1 is the Faraday 0 0 2 from 0° to 100 °C. constant. Using kHCOÀ and k 2À of 4.45 and 13:86 m 3 CO3 Progress has recently been made in understanding diffu- mS molÀ1 (Robinson and Stokes, 1959; Li and Gregory, À 2À sion processes using experimental as well as theoretical 1974), the self-diffusion coefficients of HCO3 and CO3 at methods. This includes, for instance, laboratory experi- 25 °C and infinite dilution may be estimated as 1.19 and ments and molecular dynamics studies to elucidate the fun- 0:92 Â 10À9 m2 sÀ1, respectively. damentals of diffusion and the nature of ionic diffusion in Considering ion mobility, a relation between conductiv- aqueous solution and the influence of isotopic mass ity and diffusion is to be expected. However, several (Koneshan et al., 2001; Richter et al., 2006; Bourg and fundamental differences exist. For instance, in conduction Sposito, 2007; Li et al., 2010). Several of these studies positive and negative ions move in opposite directions, highlight the critical role of hydration for diffusion in liquid whereas in diffusion they move in the same direction. Also, water, which is also fundamental to understanding in conduction ions move independently at very low concen- hydrogen-bonding environments, solvation motifs, calcite tration, whereas in diffusion they have to move at equal growth, and carbon and oxygen isotope fractionation be- speeds to ensure electroneutrality of the solution. Uncer- tween dissolved compounds and water in thermodynamic tainties in estimating self-diffusion coefficients based on equilibrium (e.g. Zeebe, 1999; Zeebe, 2007a; Rustad et al., conductivity data may arise from various issues, including 2008; Zeebe, 2009; Kumar et al., 2009; Garand et al., the fact that conductivity measurements at finite concentra- 2010; Raiteri et al., 2010; Zeebe, 2010). tion have to be extrapolated to zero concentration (for a de- Advances in computational power and numerical meth- tailed discussion of uncertainties, see Robinson and Stokes, ods including molecular dynamics now allow accurate 1959). Nevertheless, for a number of ions, measurement of calculation of diffusion coefficients in many systems (see the limiting conductivity ðk0Þ provide quite accurate num- e.g. Section 5; Bourg and Sposito, 2007; Bourg and Sposito, bers for the self-diffusion coefficients. À 2À 0 2008; Kerisit and Liu, 2010). The system of dissolved CO2 In the case of HCO3 and CO3 , the k values used in the in water is the focus of the present work. The manuscript past to estimate their diffusion coefficients (e.g. Li and is organized as follows. A few basics on diffusion and earlier Gregory, 1974) actually originate from conductivity mea- estimates of ionic diffusion coefficients will be reviewed in surements in the 1930s and 1940s (Shedlovsky and Section 2. The methods employed in the present study MacInnes, 1935; Monk, 1949). The conductivity measure- 0 and system-size effects on calculated diffusion coefficients ments to derive k 2À (Monk, 1949) showed drifts over time CO3 will be described in Sections 3 and 4. Several tests allowing and required several corrections, including conductivity evaluation of the accuracy of MD-calculated diffusion coef- corrections for NaOH and NaHCO3. Three different values 0 ficients will be provided in Section 5, while results for the of kCO2À at 25 °C are listed in Landolt-Bo¨rnstein (1960). 3 0 carbonate species’ diffusion coefficients and their mass- Experimental values for kCO2À are available at 0°,18°, and 3 0 dependence will be presented and discussed in Sections 6 25 °C, while measurement of k À appears to be limited to HCO3 and 7.