Systematic review of forsterite dissolution rate data

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Geochimica et Cosmochimica Acta 99 (2012) 159–178 www.elsevier.com/locate/gca

Systematic review of forsterite dissolution rate data J. Donald Rimstidt a,⇑, Susan L. Brantley b, Amanda A. Olsen c a Department of Geosciences, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, United States Earth and Environmental Systems Institute, 2217 EES Building, Pennsylvania State University, Univ Pk, PA 16802, USA c School of Earth and Climate Sciences, 5790 Bryand Global Sciences, Center, University of Maine, Orono, ME 04469, United States b

Received 14 July 2011; accepted in revised form 12 September 2012; available online 19 September 2012

Abstract This paper demonstrates a method for systematic analysis of published mineral dissolution rate data using forsterite dissolution as an example. The steps of the method are: (1) identify the data sources, (2) select the data, (3) tabulate the data, (4) analyze the data to produce a model, and (5) report the results. This method allows for a combination of critical selection of data, based on expert knowledge of theoretical expectations and experimental pitfalls, and meta-analysis of the data using statistical methods. Application of this method to all currently available forsterite dissolution rates (0 < pH < 14, and 0 < T < 150 °C) normalized to geometric surface area produced the following rate equations: For pH < 5.6 and 0° < T < 150 °C, based on 519 data log rgeo ¼ 6:05ð0:22Þ  0:46ð0:02ÞpH  3683:0ð63:6Þ1=T ðR2 ¼ 0:88Þ

For pH > 5.6 and 0° < T < 150 °C, based on 125 data log rgeo ¼ 4:07ð0:38Þ  0:256ð0:023ÞpH  3465ð139Þ1=T ðR2 ¼ 0:92Þ The R2 values show that 10% of the variance in r is not explained by variation in 1/T and pH. Although the experimental error for rate measurements should be ± 30%, the observed error associated with the log r values is 0.5 log units (±300% relative error). The unexplained variance and the large error associated with the reported rates likely arises from the assumption that the rates are directly proportional to the mineral surface area (geometric or BET) when the rate is actually controlled by the concentration and relative reactivity of surface sites, which may be a function of duration of reaction. Related to these surface area terms are other likely sources of error that include composition and preparation of mineral starting material. Similar rate equations were produced from BET surface area normalized rates. Comparison of rate models based on geometric and BET normalized rates offers no support for choosing one normalization method over the other. However, practical considerations support the use of geometric surface area normalization. Comparison of Mg and Si release rates showed that they produced statistically indistinguishable dissolution rates because dissolution was stoichiometric in the experiments over the entire pH range even though the surface concentrations of Mg and Si are known to change with pH. Comparison of rates from experiments with added carbonate, either from CO2 partial pressures greater than atmospheric or added carbonate salts, showed that the existing data set is not sufficient to quantify any effect of dissolved carbonate species on forsterite dissolution rates. Ó 2012 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.

E-mail addresses: [email protected] (J.D. Rimstidt), [email protected] (S.L. Brantley), [email protected] (A.A. Olsen). 0016-7037/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.gca.2012.09.019

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Notation A Ageo ABET D Ea k M mi R Rs

pre-exponential factor in Arrhenius equation (mol/m2 s) specific surface area calculated from grain diameter (m2/g) specific surface area determined by the BET method (m2/g) grain diameter (m) activation energy (kJ/mol) dissolution rate constant (mol/m2 s) mass of solution (g) concentration of species i (mol/kg) gas constant (8.314 J/mol K) grain surface roughness (= ABET/Ageo)

1. INTRODUCTION Thermodynamic models have been a mainstay of geochemistry for more than a century. They are especially effective for high-temperature and high-pressure conditions where equilibrium is rapidly established. However, at low temperatures slow reaction rates often keep geochemical processes from reaching equilibrium and this makes geochemical kinetics an essential tool for modeling these situations. An impressive number of mineral dissolution rate data are already available but they remain widely scattered throughout the literature where they are presented in many different formats. They would be much more useful if they were collected, collated, and correlated into rate equations. In addition, such a synthesis of rate data could point to new mechanistic insights into geochemical kinetics, problems in the datasets, and the needs for new data. Only a few such syntheses have been produced (White, 1995; Palandri and Kharaka, 2004; Marini, 2006; Bandstra et al., 2008; Brantley and Conrad, 2008; Bandstra and Brantley, 2008) and none are up to date. There are many examples of the application of rate equations to understand and model important natural and technological processes. For example, the mathematical forms of rate equations can be used to constrain theories about the chemical nature of the reactions. In addition, there are many scientific and practical uses for the rate equations. For example, the development of rate equations for accurate prediction of weathering reactions in response to acid rain became an important focus of research that started in the 1970s. Reliable rate equations are a necessary part of the success of major new technological and environmental initiatives like subsurface CO2 sequestration (Marini, 2006). All of these activities require highly reliable rate equations and reliability comes from organized and thoughtful methods for collecting, evaluating, and correlating rate information, which is the purpose of a systematic review. Two kinds of activities characterize systematic reviews. In fields such as thermodynamics, very reliable theory is used to select those data that are most consistent with

rgeo(i)

rate of release or consumption of species i per unit geometric surface (mol/m2 s) rBET(i) rate of release or consumption of species i per unit BET surface (mol/m2 s) T temperature (K) n number of data ni reaction order for species i Vm(fo) molar volume of forsterite (4.365  105 m3/ mol) (Robie and Hemingway, 1995) Wm(fo) molar mass of forsterite (140.693 g/mol) Xfo mole fraction of forsterite in the olivine sample n extent of reaction (mol) mi stoichiometric coefficient for species i

the existing body of knowledge. There are many examples of this kind of critical selection of thermodynamic data (Cox et al., 1989; Parker and Khodakovskii, 1995; Hummel et al., 2002). Critical selection requires the efforts of scientists who have extensive experience with experimental methodology along with a thorough understanding of theoretical relationships. Critically selected data are typically documented by narrative and calculations that justify the choices. When theoretical principles are sparse, when information needed for critical selection has been omitted from publications, or when data are contradictory for no apparent reason, methods typically used to compare and evaluate data are strictly statistical. Although the idea of using statistical methods for this task is not new (O’Rourke, 2007; Bandstra and Brantley, 2008), it was Glass (1976) who systematized them under the rubric of metaanalysis, which he defined as “the statistical analysis of a large collection of analysis results from individual studies for the purpose of integrating the findings.” Meta-analysis is now considered to be mostly a statistical exercise in which the findings are not influenced by existing theory and the results are justified using measures of statistical significance. Nonetheless, geochemical kinetics is certainly grounded by some theoretical principles that have been borrowed from chemical kinetics (Lasaga, 1998; Kump et al., 2000; Brantley, 2003, 2008; Brantley and Conrad, 2008). This suggests that some geochemical rate data should be subjected to critical selection. However, the currently used geochemical rate equations lack the kind of rigorous theoretical justification found in geochemical thermodynamics. Thus, the most reasonable strategy for assembling geochemical rate data into a coherent form is a hybrid of critical analysis and meta-analysis. This hybrid methodology is called systematic review (Cook et al., 1997; Akobeng, 2005). Systematic review is the assembly, critical appraisal, and synthesis of all relevant studies on a specific topic using an appropriate combination of critical selection and meta-analysis. The purpose of this paper is to present a strategy for the systematic review of mineral dissolution rates and to illustrate that strategy by considering forsterite dissolution rates.

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In this paper we have chosen to review forsterite dissolution rates to illustrate our proposed scheme for conducting a systemic review. The objective in this review was to develop rate equations that connect the observed rates to the most important rate determining variables. These rate equations should be useful for understanding geochemical processes, like chemical weathering, as well as for technological applications, such as CO2 sequestration. Forsterite was chosen because the dissolution reaction is relatively simple, there are many data over a significant range of experimental conditions, and there are several scientific and practical uses for the results. We found 26 different sources that reported the results of 677 rate measurements. This abundance of data reflects the widespread interest in this reaction. This interest arises because the reaction is simple enough that it can be used to test ideas about the silicate mineral dissolution process. Furthermore, the dissolution rate is fast enough to allow easy rate measurements at ambient conditions. With suitable adjustments for the observation that field rates differ from laboratory rates, the rate of forsterite dissolution can be used to investigate interesting geochemical questions such as the chemical basis for Goldich’s “mineral stability series in weathering” (Goldich, 1938; Kowalewski and Rimstidt, 2003) or to estimate the duration of humid conditions on Mars (Olsen and Rimstidt, 2007; Hausrath et al., 2008). Finally, practical applications of the rate data include development of CO2 sequestration technologies (Giammar et al., 2005) and neutralization of acid mine drainage and industrial waste acids (Van Herk et al., 1989). Forsterite is the Mg-rich member of the olivine solid solution series, (MgxFe1X)2SiO4. In this review, only compositions with x > 0.82 are considered. The olivine structure contains no bridging oxygen atoms so forsterite should dissolve congruently. ½6

½4

Mg2 Si½4 O4 þ 4Hþ þ 12H2 O ¼ 2ðMg  6H2 OÞ2þ þ SiðOHÞ4

ð1Þ

Eq. (1) is written to emphasize that mineral dissolution is not simply a bond breaking process but rather involves bond rearrangements. The superscript in square brackets is the coordination number of the atom. During the dissolution process each of the six Mg–O bonds in forsterite (Francis and Ribbe, 1980) is replaced by a Mg–OH2 bond to produce a Mg ion coordinated by six water molecules (Bock et al., 1994). Each of the four SiO–Mg3 bonds becomes a SiO–H bond to form an aqueous silica species. These bond replacements cannot all occur in one reaction step because the probability of the simultaneous correct alignment of all of the reactants in Eq. (1) to form an activated complex is vanishingly small. Bond rearrangements occur as elementary reaction steps, which seldom involve more than a few bonds per step. This means that liberating a Mg atom from forsterite to form a Mg2+ ion might require up to six elementary reaction steps. Likewise, addition of a hydrogen ion to each of the four SiO–Mg3 bonds could require four more elementary steps. Furthermore, even though all forsterite dissolution occurs under very far from equilibrium conditions, in principle all of these steps are

161

reversible, which doubles the number of possible elementary steps. Thus, the observed dissolution reaction is a composite of many elementary steps, each of which is described by a rate law. This means that the fitted values of the A, Ea, and n parameters reported in this paper should be viewed as “apparent” values for some unknown combination of elementary steps rather than values that reflect the nature of a single elementary reaction step. There are several additional sources of information about the forsterite dissolution process. Casey and Westrich (1992) demonstrated that orthosilicate mineral (M2SiO4) dissolution rates correlate with rates of water exchange with their corresponding M cations. This can be interpreted to mean that one or more of the elementary reaction steps that replace Mg–OSi bonds with Mg–OH2 bonds control the overall dissolution rate. This idea is further supported by the observation that the dissolution rate of forsterite is lower in concentrated salt solutions with decreased water activity (Olsen, 2007). Experiments also show that organic ligands, which mount a stronger nucleophilic attack on Mg sites than is mounted by water, increase forsterite dissolution rates. It has been argued that attachment of one of these ligands to a Mg site weakens the adjacent Mg–OSi bonds making them more vulnerable to conversion to Mg–OH2 bonds (Wogelius and Walther, 1991; Olsen and Rimstidt, 2008). The correlation between the rate of forsterite dissolution and the solution’s pH appears to be related to the binding of a hydrogen ion to the bridging oxygen in a Mg–OSi moiety, which would weaken the adjacent Mg–O bonds and increase the rate of replacement of Mg–OSi bonds by Mg–OH2 bonds. Both the ligand effect and the hydrogen ion effect show that the structure of next-nearest-neighbor sites influences the rate of an elementary reaction step. Because Mg atoms can be coordinated by 6, 5, 4, 3, 2, or 1 Mg–OSi bonds and the Si atoms can be coordinated by 4, 3, 2, or 1 SiO–Mg bonds, each of these step-wise changes in coordination can influence the rate of elementary reaction steps at adjacent bonds. The large number of possible configurations for next nearest neighbor atoms further increases the number of possible identifiable elementary steps. In summary, the forsterite dissolution process is a chain reaction with many elementary reaction steps. There is a reasonable possibility for more than one chain reaction path, each with a different initiation and termination step, to occur. Finally, because each elementary step is affected by the geometry of nearby atoms, chain branching is also likely. There is little doubt that the overall reaction process is complex but no doubt some reasonable simplifications are possible. The mathematical methods needed to model such composite reactions (Marin and Yablonsky, 2011) have not yet been applied to silicate mineral dissolution rates. Any model of forsterite dissolution must account for the differences between the structure and composition of the dissolving surface and the unreacted crystal. Numerous studies report that the initial stage of forsterite dissolution, at least at low pH, is incongruent such that Mg is released faster than Si. These observations are confirmed by several studies of reacted forsterite surfaces using XPS and hydrogen depth profiling (Schott and Berner, 1985; Fujimoto

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et al., 1993; Seyama et al., 1996; Pokrovsky and Schott, 1999, 2000; Kobayashi et al., 2001) that show that the surface of forsterite reacted with acidic solutions becomes enriched in Si and H and depleted in Mg and the amount of enrichment varies with the pH of the reacting solution. Some of these studies even suggest that the silica at the surface of reacted forsterite is partly polymerized although forsterite contains no polymerized silica. It is not surprising that an attempt to model forsterite dissolution rate assuming a surface composition equivalent to the bulk solid predicts a rate that is significantly different from observed studies (Morrow et al., 2010). Replacing the connecting oxygen atoms in the crystal with water molecules and OH groups not only alters the bond strengths (Haiber et al., 1997; de Leeuw et al., 2000; Churakov et al., 2003; Shaw and Tse, 2007) but also alters the nearby structure so that some of the bonds will be more exposed to attack by aqueous species while others become shielded (i.e. a steric effect). This effect is unlikely to be as large as observed for polyoxometalate ions (Ohlin et al., 2010; Rustad and Casey, 2012) because the surface species are anchored by the rigid underlying crystal structure but it will nonetheless affect the overall dissolution rate. A model of the forsterite dissolution process that considers all of these factors has yet to be developed but it is useful to keep them in mind when trying to understand the dissolution rate data compiled in this paper. The purpose of a systematic review is not to create a detailed model of a reaction mechanism. Such a model mechanism would certainly consider the rate data reviewed here but also would need to incorporate other existing knowledge about the reaction process. Nonetheless, a model that describes the reaction mechanism must be compared against the best possible rate data – and a systematic review can produce just such a set of data. A systematic review evaluates the consistency among the reported reaction rates, discards unreliable determinations, and summarizes the remaining data into useful rate equations. 2. METHODS Although more detailed steps can be proscribed (Sutton et al., 2000), all systematic reviews that involve critical selection and meta-analysis proceed through five basic stages: (1) identify the data sources, (2) select the data, (3) tabulate the data, (4) analyze the data to produce a model, and (5) report the results (Glass et al., 1981). The only significant difference between them occurs in stage 4 where critical selection is used to choose data that are consistent with the existing theoretical framework whereas meta-analysis uses statistical methods to correlate the data and then deletes outliers for statistical rather than theoretical reasons. Although these stages are discussed separately here, they are often carried out somewhat concurrently. This can create a confusing mass of tasks and documents that are difficult to manage, especially if more than one person is involved. Therefore, it is advisable to develop strict procedures for each stage of the process at the beginning of the project. These include thoughtful selection of standard software and reporting forms. It is

especially important to clearly define the objective of the systematic review at the outset of the project. An extensive explanation of how these methods were applied to the forsterite dissolution rate data is given in the Supporting Online Materials for this paper. That material should be reviewed by anyone trying to implement this approach. The statistical methods used in this paper are explained in most introductory statistics textbooks such as Zar (2010) and are illustrated in Kimball et al. (2010) and Williamson and Rimstidt (1994). 3. RESULTS All the sources that report forsterite dissolution rates are given in Table 1 and all of the rate data are tabulated in the Supplemental Online Material for this paper. This section describes how these data were evaluated and converted into rate equations. 3.1. Properties of the data set Our literature search found 26 independent sources of forsterite dissolution rate data. Four of these sources (Jonckbloedt, 1998; Xiao et al., 1999; Kleiv and Thornhill, 2006, 2008) reported results in ways that were not amenable to conversion into a rate of forsterite dissolution. The remaining 22 documents reported results from 689 experiments from which we were able to compile 661 rate determinations normalized to BET surface area (ABET) and 654 rate determinations normalized to geometric surface area (Ageo) (Fig. 1). The rates based on geometric surface areas are on the average 5.2 times faster than those based on BET surface areas. Thus, the average roughness (Rs = ABET/Ageo) of the forsterite used across all studies is approximately 5. Both sets of rate values span about 8 orders of magnitude. Our data set contains rates determined using either batch reactor (BR) experiments (143) or mixed flow reactor (MFR) experiments (524). The rates were derived using four different methods – see the SOM for further explanation of the methods. Most of the rates were found by direct determination (DD) (432 experiments). The next most common method (128 experiments) calculated rates from the derivative (DV) of concentration versus time data obtained from BR experiments. Some rates (110 experiments) were calculated by fitting the fraction of forsterite reacted away to a shrinking particle (SP) model and a few rates (7 experiments) were calculated from the rate of surface retreat (SR) of forsterite cubes. The experimental conditions spanned a useful range but the experiments were not distributed evenly over that range. Most of the experiments (593) used olivine with Xfo of 0.91 or 0.92 and a few (28) used synthetic forsterite (Xfo = 1.0). Grandstaff (1986) used Hawaiian beach sand with Xfo = 0.82 and Siegel and Pfannkuch (1984) report the results of one experiment using olivine from North Carolina, which was analyzed by wet chemical methods to yield Xfo = 0.85. However, the composition of all other samples from that locality was reported to be near Xfo = 0.92. Based on the calculated values of surface roughness (Rs = ABET/ Ageo) for forsterite grains used in the experiments, there

Table 1 Summary of publications containing forsterite dissolution rate data along with values of nHþ , A, and Ea (pH < 5) reported by or calculated from each source. Exp.a

Rateb

RPVc

Xfo

D (lm)

Aw00 Awad et al. (2000) Ba76 Bailey (1976) Bl88 Blum and Lasaga (1988) Ch00 Chen and Brantley (2000) Er82 Eriksson (1982) Gi05 Giammar et al. (2005) Go05 Golubev et al. (2005) Gr86 Grandstaff (1986) Ha06 Ha¨nchen et al. (2006) Ha07 Ha¨nchen et al. (2007) Jo98 Jonckbloedt (1998) Kl06 Kleiv and Thornhill (2006) Kl08 Kleiv and Thornhill (2008) Lu72 Luce et al. (1972) Oe01 Oelkers (2001a,b) Ol07 Olsen (2007) Ol08 Olsen and Rimstidt (2008) Po00 Pokrovsky and Schott (2000) Pr09 Prigiobbe et al. (2009) Ro00 Rosso and Rimstidt (2000) Sa72 Sanemasa et al. (1972)

BR

SR

SR

0.91

100

BR

DV

Mg, Si

MFR

DD

Si

0.93

BR

DV

Mg, Si

0.91

BR

DV

Mg

BR

DV

Mg, Si

0.89

MFR

DD

Mg, Si

BR

DV

MFR

105–149

Ageo (m2/g)

0.00974

ABET (m2/g)

NA

75–150

0.0172

0.0340

Mg, Si

1.00 0.91 0.82

<600 <250 125–250 20–50 50–100 50–100 74–149

0.020 0.0267 0.01 0.032 0.026 0.026 0.0173

0.088 15 0.107 0.059 0.930

SP

Mg, Si

0.91

90–180

0.0143

0.0797

MFR

SP

Mg

0.91

0.0143 0.0072

0.1117 0.0457

BR

SP

H

0.94

90–180 180–355 63–150

BR

DV

H

0.93

PFR

DV

H

0.93

100–1000

BR

DV

Mg, Si

0.91

74–149

MFR

DD

Mg, Si

0.89

BR

DV

Si

BR

DV

MFR

nd

T (°C)

pH acid

CO3e

nHþ f

Ea (kJ/mole)

7 T 14 T 12 G 8 T 9 T 5 T 32 T 8 T 44 T 11 T 0

23–90

1–2 H2SO4 4 Phthalate 1.8–11.4

N

/0.42

69.2/70.9

25–65 25 65 6 30–90 25 1–49 90–150 120 60–90

0

25

0.21

0

22

0.0174

0.0445

25

50–100

0.0258

0.0808

0.92

250–350

0.00626

0.0528

Si

0.92

250–350

0.00626

0.0528

DD

Mg, Si

0.91

50–100

0.0251

0.0800

MFR

SP

Mg, Si

0.91

90–180

0.0143

0.0797

MFR

DD

Mg, Si

0.92

250–350

0.00626

0.0356

BR

SP

Mg, Si

0.91

10–20

0.129

5 T 26 T 47 T 37 T 106 T 37 T 253 T 6 G

25–65 25 25 25 120 25–45 35–65

2–5 HCl 3–7 H2CO3 3.1–4.1 H2CO3 2.0–11.8 HCl 3–5 HCl 2–12.5 HCl 2.0–4.8 HCl 0.7–2.0 H2SO4 2 HNO3 1.9–2.4 H2SO4 1.7–9.6 HNO3 2.0 HCl 1.0–4.1 H2SO4 HNO3 1.0–4.1 HNO3 1–11.5 HCl 2–8 H2CO3 1.8–4.1 HNO3 1.0 HClO4/HCl

N

f

57.8/54.3

N

0.56/0.52

N

0.70/0.74

Y

/0.51

Y

/0.39

Y

/0.18

N

1.1/1.1

38.1/37.5

N Y N

0.46/0.46

52.9/51.8

N

0.33/

/10.1

0.46/0.46 66.5/

N N N

/0.33

N

63.8/63.7

N

0.52/0.54

N

0.46/0.43

N Y Y

0.50/0.47

N

0.50/0.50

N

J.D. Rimstidt et al. / Geochimica et Cosmochimica Acta 99 (2012) 159–178

Symbol code Reference

0.52/0.56 42.6/43.4 58.6/63.5 163

(continued on next page)

J.D. Rimstidt et al. / Geochimica et Cosmochimica Acta 99 (2012) 159–178 f

164 Ea (kJ/mole)

180

frequency f

e

c

d

0.0307 0.0598 0.0307 0.0598 63–150 355–630 H

Mg, Si

0.92

0.00568 0.00953 0.00568 0.00953 250–420 149–250 250–420 0.91 1.00 0.91 Mg, Si

BR

DD DV DV MFR BR BR

BET surface area n = 661

120 100 80 60 40 20 0 -12

-11

-10

-9

-8

-7

-6

-5

-4

log r

Fig. 1. Distribution of forsterite dissolution rate values considered in this analysis.

are four cases where the measured ABET value appears to be too high. Giammar et al. (2005) reports that sample D5 has a BET surface area of 15 m2/g but based on the grain size range of 20–50 lm, Ageo = 0.032 m2/g. This would make Rs = 469, which is about two orders of magnitude higher than most of the other values of Rs. (see Fig. 2). Likewise, the calculated surface roughness for grains used by Grandstaff (1986) (Rs = 53), Siegel and Pfannkuch (1984) (Rs = 44), and Van Herk et al. (1989) (Rs = 39) also appear to be too large. It seems that in these cases the measured BET surface area values are too large by one to two orders of magnitude. This could be due to retention of surface fines or simply the result of an instrumental calibration or reading error (Brantley and Mellott, 2000). The remaining values of Rs span the range of 1–10 (Fig. 2) with a mean value of 5.4. Aside from surface area, pH and temperature are the most important rate determining variables. The majority of the experiments (495) were performed at pH < 5 (Fig. 3). About half of the experiments (335) were performed near room temperature, 22° < T < 26 °C. Ten experiments were performed at T < 10 °C and ten experiments were performed at the highest temperature of 150 °C. With the exception of several experiments (68) per-

350 300

number of data

65

25

40–70 0.630 0.0162 105–125 0.92 Mg, Si SP BR

Experiment type: BR = batch reactor; MFR = mixed flow reactor; PFR = plug flow reactor. Rate calculation method: SR = surface retreat method; DD = direct determination of rates from MFR data; DV = derivative method; SP = shrinking particle model. Reaction progress variable. SR = rate of surface retreat; H = rate of hydrogen ion consumption; Mg = rate of Mg release; Si = rate of Si release. Number of data and source: T = table, G = graph. If n = 0 the paper did not report data in a form amenable to conversion to conventional rates in mol/m2 s. N means that PCO2 6 103.5 and no carbonate was added. Y means that PCO2 > 103.5 and/or carbonate or bicarbonate was added to the experiment. Reported in paper/calculated from reported rates. Whenever possible Ea and nHþ (pH < 5) were calculated using multiple linear regression. a

b

N

0.5/0.43 N

0.54/0.48 N

N

N

4 HNO3 1–3 H2SO4, HCl 4.1–12.4 HCl 1.8–6.0 HCl Conc. HCl H2SO4 22

1 T 12 T 16 T 2 T 0 1.3 0.0297 38–42 0.85 Mg, Si DV BR

Si84 Siegel and Pfannkuch (1984) Va89 Van Herk et al. (1989) Wo91 Wogelius and Walther (1991) Wo92 Wogelius and Walther (1992) Xi99 Xiao et al. (1999)

D (lm) Xfo Rateb Exp.a Symbol code Reference

Table 1 (continued)

140

RPVc

Ageo (m2/g)

ABET (m2/g)

nd

T (°C)

pH acid

CO3e

nHþ f

/0.35

30/57.4

160

250 200 150 100 50 0 0

1

2

3

4

5

6

7

8

9

10

Rs

Fig. 2. Distribution of surface roughness (Rs) values for forsterite dissolution experiments, excluding the results of Giammar et al. (2005), Grandstaff (1986), Siegel and Pfannkuch (1984), and Van Herk et al. (1989), who reported ABET surface area values that produced Rs values greater than 10.

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formed at 120 °C there is a notable lack of rate measurements for T > 50 °C and at pH > 6 for temperatures above 25 °C (Fig. 4). Most of the experiments (553) were run in solutions containing little or no carbonate species because they were either equilibrated with air or purged with N2 to remove CO2. However, 124 experiments contained higher amounts of dissolved carbonate either because they were equilibrated with high CO2 pressures or because a carbonate or bicarbonate salt was added to the experiment. Some experiments considered the effects of salinity, Mg2+ concentration, SO42, ionic strength, or activity of water. All of these experiments were included in our compilation with the exception of the experiments with aH2O < 0.90, which were reported to show significantly decreased dissolution rates (Olsen, 2007). Likewise, no experiments containing added organic ligands, such as oxalate, were included in the data set because organic ligands were reported (Olsen and Rimstidt, 2008) to increase the dissolution rate. As a result the main rate determining variables for these experiments appear to be surface area, pH and temperature. Fig. 4 shows that the data span the pH range from 1.0 to 12.4 and the temperature range from 0 to 150 °C. It also shows that the data are very unevenly distributed over this range.

(a)

165

3.2. Data analysis 3.2.1. Choosing a rate equation The compiled rate data alone are of relatively little value unless they are condensed into rate equations that can be used to model geochemical or technological processes. Developing rate equations must be guided by a combination of project objectives and practical considerations. The objective in this study was to develop the simplest possible rate equation that connects the observed rates to the most important rate determining variables. We want this rate equation to be useful for understanding forsterite weathering rates as well as for practical applications, such as modeling CO2 sequestration. Meeting these simple objectives is far more challenging that first appears because the final equation is influenced by numerous decisions that must be made during the data analysis process. These decisions range from selecting the regression variables and the form of the rate equation to filtering the data set to remove inconsistent values. Therefore, it is important to keep a careful record about each of these decisions and to explain the rational for those decisions in any subsequent publication. The data analysis process is non-unique and often not obvious.

120

100

80

60

40

20

0 0

1

2

3

4

5

6

7 pH

8

9

10

11

12

13

14

(b) 350 300 250 200 150 100 50 0 0

20

40

60

80

100

120

140

temperature, º C Fig. 3. (a) Distribution of pH values and (b) distribution of temperatures for forsterite dissolution experiments listed in Table 1.

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J.D. Rimstidt et al. / Geochimica et Cosmochimica Acta 99 (2012) 159–178 150ºC 100ºC 50ºC 25ºC 0ºC

pH

Fig. 4. Reported forsterite dissolution rates as a function of pH and 1/T (K1).

The first step in choosing a rate equation is to decide which independent variables are important enough to include in the model. Several studies (Grandstaff, 1978; Wogelius and Walther, 1991; Olsen and Rimstidt, 2008) have shown that ligands such as oxalate, potassium hydrogen phthalate (KHP), and ethylenediaminetetraacetatic acid (EDTA) increase forsterite dissolution rates; we made no attempt to incorporate this information into the rate equations developed here because of the limited number of data (measurements in the presence of organic ligands were not included in the analysis). Olivine composition also affects the dissolution rate but all the compositions considered here were near fo0.9 so there was not enough compositional variation to consider that effect. Some studies report that dissolved carbonate increases the dissolution rate and some found no effect. That issue is further investigated in this paper. Finally, water activity (>0.9) (Olsen, 2007), Al3+ concentration (<4  104 m) (Chen and Brantley, 2000), SO42 concentration (<3 m) (Olsen, 2007), SiO2 concentration (<30 ppm) (Oelkers, 2001a), Mg2+ concentration (<4 m) (Olsen, 2007), and ionic strength (<12 m) (Olsen, 2007) were reported to not appreciably affect forsterite dissolution rates and were therefore not considered in the development of the rate equations. Thus, we chose pH, temperature, and surface area as master variables in subsequent rate models. We further assumed that rates are related to the hydrogen ion activity and temperature by r ¼ Aeð

E a 1 R ÞðT Þ

anHþ

ð2Þ

The form of the rate equation that we use here is not derived from a physical model. It is a mathematical form that adequately describes the data set compiled here and we make no claims beyond that. There may be other equations that fit the data equally well or even better. Fitting the data to this relationship would require nonlinear regression. This function can be log-transformed to a linear form

 log r ¼ log A 

 Ea 1  n pH 2:303R T

ð3Þ

where pH ¼  log aHþ so that we can use multiple linear regression to find log r = f(1/T, pH). It has been pointed out that regression of log transformed rate data has implications in that it estimates the median of the rate rather than the mean, thus underestimating rates (Bandstra and Brantley, 2008). This issue can be ameliorated by adjusting the weighting of the data to account for the log-transformation; however, when Bandstra and Brantley applied that approach to the forsterite data set that they compiled, they found no difference in their fitting results. Therefore, we created multiple linear regression models using log r and pH and 1/T as regression parameters. This approach is the simplest way to obtain estimates of A, Ea, and n, which could be further refined using nonlinear regression or linear regression with weighting schemes if such refinement was warranted by the quality of the data. pH has a greater effect on rate than temperature over the ranges of interest (Fig. 4). In addition, there is a pronounced curvature in the log r versus pH graph for the 25 °C data. Therefore, it is necessary to decide the best way to deal with this curvature. Linear regression of 317 data for 25 °C over the pH range of 1–12 produces a line with a slope of 0.36 (the same value calculated by an earlier compilation and fit over the same pH range using a purely statistical rate analysis approach (Bandstra and Brantley, 2008)). This slope is considerably lower than the values of near 0.5, determined from fits over the pH range of 1–5 calculated from 12 of the 18 data sets listed in Table 1 (see Fig. 5). Inspection of the residuals from the regression model (shown in Fig. 6b for example for the rates normalized by geometric surface area) shows that slope of the regression line is very much controlled by the leverage of the Po00 and Go05 data on the high pH end of the graph. It is worth noting that regression of the 64 data measured at 120 °C over the pH range of 2–10.5 from three papers (Pr09, Ha07, Ha06) produces a line with slope of 0.49(0.02). This might mean that (scenario 1) there is a systematic problem with the rate measurements at 25 °C and pH > 6 or it might mean that (scenario 2) there is a change in reaction mechanism going from low to high pH at 25 °C that disappears as the temperature approaches 120 °C. Given the scarcity of data between 25 and 120 °C, especially at pH > 6, it is not possible to determine which of these scenarios is most likely. To deal with this problem in a way that best reproduces the data, we split the data set at the break in slope shown on the graph in Fig. 6 and fit the low pH data independently of the high pH data. If scenario 1 is correct, then the low pH rate equation can simply be extrapolated to high pH. If scenario 2 is correct then the low pH rate equation can be used for pH < “break” and the high pH rate equation can be used for pH > “break”. The remaining problem is to determine the best pH to break the data set. This could be done by a visual inspection of residual plot, which shows a minimum at a pH value between 5 and 6. The location of this minimum can be more closely defined by fitting a parabolic function to the residual versus pH data to get residual ¼ 0:4019  0:1904 pHþ

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167

10 8

8

individual studies 6

6

4

4

2

2

8

8

27 data

6

6

4

4

2

2

8

8

106 data

6

6

4

4

2

2

10

10

264 data

8

8

6

6

4

4

2

2

0 0

0. 2

0.4

0.6

0.8

nH+

1

0

20

40

60

Ea, kJ/mol

80

100

Fig. 5. (Top two graphs) Distribution of Ea , and nHþ values for experiments performed at pH < 5 reported by the authors using rates from the studies listed in Table 1. (Six lower graphs) Distribution of Ea , and nHþ values calculated from subsets of 27, 106, and 264 rate data randomly chosen, without replacement, from the pooled dataset for pH < 5.6. These graphs illustrate how the dispersion of the Ea , and nHþ values decreases as the size of the data set increases. The vertical lines show the Ea , and nHþ values calculated from the entire data set (519 data).

0:01523 pH2 . If we set the derivative of this equation to zero we find that the slope goes from positive to negative at pH = 5.50. Another approach is to fit the low pH data (pH < 5) to get log r ¼ 6:09  0:512 pH and the high pH data (pH > 8) to get log r ¼ 7:74  0:233 pH and solve these equations simultaneously to find that their graphs intersect at pH = 5.72. Based on these results, we chose to break the data set at the average of these two values, i.e. pH 5.6. That decision sets the stage for using multiple linear regression to fit the pH < 5.6 data as a function of pH and 1/T. The goal of this first step is to identify and remove outliers in the data set. Regression of all of the 527 data produces log r ¼ 5:76ð0:30Þ  0:49ð0:02ÞpH  3573ð88Þ 1=T ðR2 ¼ 0:78Þ

ð4Þ

The numbers in parentheses are the standard errors of the regression parameters as well as the correlation coefficient, R2. The distribution of residuals from this fit (Fig. 7) shows that the data are slightly skewed to the right. This might be caused by the log transformation of the data; however, the Shapiro–Wilk W-test, which calculates the probability that the data are normally distributed, shows that data are normally distributed. The distribution has a mean of zero and a standard deviation of 0.5. This means that we expect that 68.2% of the data should fall within 0.5 and +0.5 log units (±1r) of the value predicted by Eq. (8), that 95.4% of the data should fall within 1.0 and +1.0 log units (±2r), and that 99.6% of the data should fall within 1.5 and +1.5 log units (±3r). Thus we expect that 359 (68.2%) of the 527 data should fall within ±1r and 451 actually fall within this range. Similarly, 95.4% of the data should fall within ±2r and 4.6% or 24 data should lie beyond ±2r.

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(a)

-7 -8

287

log r

-9

300

173

-10 -11 1 0

0 0 0

2 0

3 4 6

200 33 16

100 0 2

-12

residuals

1

(b)

0 -1 -2 -3 2

4

6

8

10

12

pH Fig. 6. (a) Log r versus pH linear regression model for 25 °C data. (b) The pronounced curvature in the data set is apparent from the plot of the model residuals versus pH. The curved line shows a fit of the residuals to residuals = a + b(pH) + c(pH)2. The legend at the right identifies the source of the data for these and all subsequent figures.

This is reasonably close to the observation that 28 data lie beyond ±2r. Finally, we expect that 99.6% of the data should fall within ±3r so 0.4% or 2 data should lie beyond ±3r. Eight data actually lie beyond ±3r and this suggests that most of those data are not the result of random error. Instead, they are most likely the result of various “blunders”, such as incorrect instrument readings or data recording errors. These 8 data were removed from the data set and the new regression model produced log r ¼ 6:05ð0:22Þ  0:46ð0:02ÞpH  3683ð64Þ1=T ðR2 ¼ 0:88Þ

ð5Þ

Note that filtering the data to remove these relatively few “blunders” increased R2 from 0.78 to 0.88. This equation can be used to predict rates from pH 1.0 to 5.6 from 0 to 150 °C. From the regression model, we find values for A, Ea , and nHþ (see Eq. (6)) that are given in Table 2. Critical selection of data could be used to further improve the rate equation but it was not applied in this study in order to keep the paper short and to the point. The 127 data for pH > 5.6 were likewise fit as a function of pH and 1/T and the distribution of the residuals is shown in Fig. 7b.The residuals are normally distributed and the standard deviation of this distribution is 0.563. Two data fall beyond 3 standard deviations but if the errors were normally distributed only 0.5 data should lie beyond 3 standard deviations. Thus, those two data, shown as open triangles on Fig. 9a, were considered to be outliers and were

Fig. 7. (a) Distribution of residuals for the fit of log r versus pH and 1/T for 527 data for pH < 5.6. The number of data in each bin is shown above the bars. (b) Distribution of residuals for the fit of log r versus pH and 1/T for 127 data for pH > 5.6. The number of data in each bin is shown above the bars.

discarded from the data set. A fit of the remaining data produced log r ¼ 4:07ð0:38Þ  0:256ð0:023ÞpH  3465ð139Þ1=T ðR2 ¼ 0:92Þ

ð6Þ

This equation can be used to predict rates from pH 5.6 to 12 from 0 to 150 °C. Values of A, Ea , and nHþ calculated from the regression parameters are given in Table 2. The foregoing provides a reasonable approach for subsequent analysis of the data set using linear regression models. All subsequent regression models fit log r as a function of 1/T and pH and the pH < 5.6 data are fit separately from the pH > 5.6 data. Furthermore, data that produce residuals falling more than 3 standard deviations away from zero are discarded. Alternately, these data could be examined to discover whether an additional variable may be important with the analysis. This approach allows direct comparison of the values of A, Ea , and nHþ calculated from the regression parameters. Applying this procedure to such a large dataset allows us to consider questions that would be difficult or impossible to address by considering the individual studies alone. In order to illustrate the value of the “systematic review” approach, we will consider three such questions that we pose here and then address in the next section. The first question is whether it is better to report rates as rgeo or as rBET. Studies of quartz dissolution rates by

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169

Table 2 Values of regression parameters from the regression models (top line of each row) and values A, Ea , and nHþ calculated from those regression parameters (bottom line of each row). Dependent variable pH < 5.6 Log rgeo(fo) Log rBET(fo) Log rgeo(Si) Log rgeo(Mg) Log rgeo(fo) No carbonate Log rgeo(fo) Carbonate pH > 5.6 Log rgeo(fo) Log rBET(fo) Log rgeo(Si) Log rgeo(Mg) Log rgeo(fo) No carbonate Log rgeo(fo) Carbonate a b

Log A A (mol/m2 s)a

Ea =R Ea , (kJ/mol)

pH-parameter nHþ

n (used/rejected) R2

6.05(0.22) 1.13(0.57)  106 5.17(0.16) 1.47(0.55)  105 1.20(0.33) 16(11) 0.70(0.34) 0.2(0.2) 5.69(0.20) 4.91(0.23)  105 4.65(0.85) 4.46(0.87)  105

3683.0(63.6) 70.5(1.2) 3675(47.0) 70.4(0.9) 2185(99.3) 41.8(1.9) 1639(104) 31.4(2.0) 3549(60.8) 67.9(1.2) 3278(238) 62.8(4.6)

0.46(0.02) 0.46(0.02) 0.44(0.01) 0.44(0.01) 0.50(0.01) 0.50(0.01) 0.47(0.01) 0.47(0.01) 0.51(0.01) 0.51(0.01) 0.37(0.13) 0.37(0.13)

519/8 0.88 503/15 0.93 378/10 0.88 336/13 0.83 454/15 0.91 57/1 0.79

4.07(0.38) 1.18(0.10)  105 2.34(0.40) 217(201) 8.14(0.22) 7.2(3.6)  109 7.85(0.26) 1.42(0.85)  108 3.40(0.50) 2.52(2.90)  103 3.79(0.57) 6.17(8.10)  103

3465(139) 66.4(2.7) 3179(143) 60.9(2.7)

0.26(0.02) 0.26(0.02) 0.22(0.02) 0.22(0.02) 0.17(0.02) 0.17(0.02) 0.28(0.02) 0.28(0.02) 0.22(0.02) 0.22(0.02) 0.32(0.05) 0.32(0.05)

125/2 0.92 140/3 0.88 77/1 0.42 100/3 0.59 57/2 0.93 66/0 0.92

b

b

3395(153) 65.0(2.9) 3152(257) 60.4(4.9)

The error in A is calculated as dA ¼ jð2:303Þð10log A Þjdðlog AÞ. There are no r(Si) data for T > 25 °C for pH > 5.6 so the rate equation was constrained to 25 °C data only.

Gautier et al. (2001) and Tester et al. (1994) suggest that rgeo values correlate as well as, or perhaps better than rate equations based on rBET values. Furthermore, Brantley and Mellott (2000) argue that BET surface area may not be the appropriate factor for scaling if there is internal surface area (connected porosity). On the other hand they also concluded that BET surface area analysis showed no evidence for internal porosity in olivine. Nonetheless this is an important question, first because BET surface measurements are difficult and expensive as compared to determining grain size by sieving, and second, because determining the BET surface area of grains in natural mineral mixtures is difficult whereas estimating geometric surface areas from sieve size determinations and grain counts is relatively easy. We can address this question in two ways. First we can compare all pairs of rgeo and rBET from our data set to see how well they correlate with each other. Fig. 10 shows that the 628 matched pairs of log rgeo and log rBET values are highly correlated (R2 = 0.978) but the log r values differ by 0.715(0.12). Inspection of Fig. 10 shows that some of the log r values are significantly displaced from the correlation line. This correlation analysis does not allow us to determine whether the problem lies with Ageo or with ABET. In order to address this question the log rgeo and log rBET values were regressed against pH and 1/T using the approach described above in order to see if there is a significant difference in the log rgeo and log rBET regression models.

The fitted regression parameters along with derived values of A, Ea, and nHþ are reported in Table 2. The second question is whether Si and Mg perform equally well as reaction progress variables (RPV). In order to produce the regression models, Regression models for rates of forsterite dissolution were calculated from experiments where either Si or Mg or both was the RPV. This approach is justified only if these RPV’s give comparable rates. One way that we can test this assumption is by pair wise comparison of log r(Mg)geo with log r(Si)geo for the same experiments. This comparison of the results from 391 rate measurements shows a mean difference in the log r values of 0.025(0.015). Correlation of these 391 data produces the equation log rgeo(Mg) = 0.004459 + 1.0026672 log rgeo(Si) (R2 = 0.93). Another way to compare the RPV’s is to compare the multiple linear regression models of Si release rates with the models based on Mg release rates. The fitted regression parameters along with derived values of A, Ea, and nHþ are reported in Table 2. Note that because there are no log r(Si) data for T > 25 °C and pH > 5.6, the fits for pH > 5.6 considered only the 25 °C data. The third question is whether aqueous carbonate species affect forsterite dissolution rates as suggested by some studies (Wogelius and Walther, 1991; Pokrovsky and Schott, 2000). We do not have rate data for matching experiments with and without carbonate that would allow us to do a pairwise comparison. However, in the data compilation

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part of this project, we flagged experiments that contained high carbonate concentrations because either CO2 or carbonate salts were added to the experiment. This allows us to separate the data into “carbonate” and “no carbonate” groups and to use our multiple linear regression model to produce rate equations for each group. These equations are listed in Table 2. We will compare the rates predicted by these equations to see if they are different. Use of the statistical approach to address each of these three problems is summarized in the next section. 4. DISCUSSION/CONCLUSIONS 4.1. Data identification, selection, and tabulation Systematic reviews present many challenges and chief among them is compiling a suitable dataset. We found that useful papers are widely scattered throughout the science and engineering literature and sometimes are identified only after considerable detective work because the authors neglected to mention terms like forsterite, olivine, kinetics, or rates in the titles and key word list. In some papers, the experimental methods section is vague or confusing. For example, many experiments were performed using a “flow through reactor”, which usually means that the data were analyzed as if the experiment consisted of a well mixed flow reactor (i.e. continuously stirred tank reactor = CSTR) without recognizing that the term could also be interpreted as meaning a plug flow reactor, which requires a completely different data analysis method. Data are often reported in non-SI units making cumbersome unit transformations necessary. These sorts of data manipulations slow the tabulation process and increase errors in the final data set. Another attribute of metaanalysis such as demonstrated here is creating awareness among authors that data generated to answer narrowly posed questions may eventually be incorporated into larger data sets and used for other purposes. This means that it is in the community’s best interest to present their data in such a way that it is readily available for these other uses. 4.2. Choosing a rate equation Ideally a rate equation has a mathematical form that reflects the underlying nature of a reaction’s chemistry and physics. At the present time there are several competing ideas about how solids dissolve. It would be useful and interesting to fit the rate data compiled here to a variety of rate equations based on these ideas and the dataset accumulated here can now be used for that purpose. In other words, compilation of the data itself is a contribution; nonetheless, our analysis is limited to finding a simple and convenient rate equation that can be used to model natural and technological processes. This choice is in part justified by “Occam’s razor” and by our desire to focus this paper on the topic of critical analysis. Furthermore, such models can be incorporated into reactive transport codes where the use of simple rate equations is the norm.

4.3. Errors and error propagation The regression models developed in Section 3 have two primary uses. First, the regression parameters (log A, Ea/ R, and nHþ ) listed in Table 2 can be used to predict log r as a function of pH and 1/T values and the log r value can be transformed to predict a dissolution rate that could be used to model the behavior of forsterite in geological or technological settings. Second, estimates of A, Ea, and nHþ (Table 2) can be derived from the regression parameters. The magnitudes of A, Ea, and nHþ provide support for theoretical models of the reaction mechanism. However, these regression parameters should not be used for either purpose without considering their associated errors. Furthermore, the error structure of the models contains important information about the data that can be used to improve experimental design, our understanding of the reaction mechanism, and our choice of future experiments. Because it is based on the most extensive data set (n = 519), the model of log rgeo for pH < 5.6 (Eq. (9); first row in Table 2) will be used to illustrate how to evaluate the errors associated with regression models. The R2 value listed in Table 2 is 0.88, which means that 88% of the variance of the data is explained by the fitting variables (pH and 1/T). The 12% of the variance that is not explained by the model shows up as scatter around the line in Fig. 8a and in the residuals of the fit (Fig. 8b–d). The overall distribution of these residuals, shown in Fig. 7, has a standard deviation of 0.5 log units (300% relative error in rate), which is large considering that typical analytical errors associated with the rate measurements leads us to expect an overall error of <30%. This discrepancy between the expected and observed error along with the 12% of variance that is not explained by pH and 1/T might mean that a rate-controlling variable is missing from the model. Considering that water activity (>0.9), Al3+ concentration (<4  104 m), SO42 concentration (<3 m), SiO2 concentration (<8  104 m), Mg2+ concentration (<4 m), and ionic strength (<12 m) (Chen and Brantley, 2000; Olsen, 2007) do not appreciably affect forsterite dissolution rates and that there is no strong support for a rate effect from dissolved carbonate (see later discussion), it seems unlikely that this unaccounted variance involves solution composition. That leaves the method of normalizing for surface area as a most likely source of uncontrolled variance. The term “reactive” surface area (White and Peterson, 1990) was coined to highlight the discrepancy between operationally defined surface area (ABET or Ageo) and the concentration of reactive surface sites that controls mineral reaction rates. When mineral reaction rates are normalized to either ABET or Ageo, the resulting “rate” is actually a measure of the flux of reactants or products through a surface that is circumscribed around the reacting grains. Interpreting that flux as a reaction rate is based on the assumptions that the reacting sites are homogeneously distributed on the mineral’s surface and that all surface sites are equally reactive. For olivine – or perhaps any crystal – we already know that these assumptions are not valid. Awad et al. (2000) showed that the relative magnitudes of the dissolution rates (70 °C,

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(b)

(c)

(d)

171

Fig. 8. (a) Actual versus predicted values of log r for the pH < 5.6 rate data. Data not used in the regression model are shown as open symbols. (b) Distribution of residuals for the fit as a function of the predicted value of log r. (c) Distribution of residuals for the fit as a function of pH. (d) Distribution of residuals for the fit as a function of 1/T. The legend identifying the source of each datum is found on Fig. 6 and in Table 1.

pH 1) along the a-, b-, and c-crystallographic directions of a forsterite crystal vary in the ratio, 1:21:3, and the activation energy for these directions are 114.5, 69.9, and 72.7 kJ/mol respectively. These results clearly show that either the concentration or the reactivity, or both, of exposed sites depends, among other things, upon the crystallographic orientation displayed to the solution. The formation of etch pits on naturally weathered olivine (Velbel and Ranck, 2008) as well as on olivine dissolved in laboratory experiments (Grandstaff, 1978) provide further evidence for the non-uniform distribution of olivine surface reactivity. Surface reactivity not only changes from place to place on the grain surface but it also changes over time. Fig. 1 in Grandstaff (1978) shows that the BET surface area of dissolving forsterite increases as the most reactive parts of the surface dissolve away so that rBET decreases with extent of reaction. In other words, the surface area increased faster than the rate increased during the experiment. Indeed, it is expected that the most reactive sites will dissolve away first, leaving a less reactive surface with time (Bandstra and Brantley, 2008b). Furthermore, preparation of samples by grinding or pulverizing presumably differs from laboratory to laboratory so that strained surface area, grain size distribution, and cracking may differ from one laboratory to the next. Accounting for the variation of site reactivity on dissolving forsterite surfaces with time and location would

likely improve our rate models and our understanding of the reaction mechanism. Given the large residuals associated with the best-fit rate equation, how useful are the model predicted rates? Forward propagation of the errors for each coefficient in Eq. (5) gives the following relationship: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dlog r ¼ ð0:22Þ2 þ ð0:02 pHÞ2 þ ð63:6 1=T Þ2 ð7Þ where dlog r is the error in the predicted value of log r. For the range of pH and T of the data set, dlog r varies from a low of 0.274 log units at pH 1 and 120 °C to a high value of 0.339 at pH 5.6 and 0 °C. These log r errors can be treated with the antilog error transformation dr ¼ 2:303ð10ðlog

rÞ

Þdlog

r

ð8Þ

to find that for pH 1 and 120 °C, r = 1.65(1.04)  104 mol/m2 s (fractional error = 63%) and for pH 5.6 and 0 °C r = 9.22(7.21)  1011 mol/m2 s (fractional error = 78%). Although the error is still large, the uncertainty in the predicted rates is smaller than the scatter displayed by individual experiments because when data are pooled and fit to a model there is a tendency for random errors to cancel each other and make the model predictions more reliable than individual results (Gauch, 1993). Thus, systematic analysis not only condenses the rate information

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(a)

(c)

(b)

(d)

Fig. 9. (a) Actual versus predicted values of log r for the pH > 5.6 rate data. Data not used in the regression model are shown as open symbols. (b) Distribution of residuals for the fit as a function of the predicted value of log r. (c) Distribution of residuals for the fit as a function of pH. (d) Distribution of residuals for the fit as a function of 1/T. The legend identifying the source of each datum is found on Fig. 6 and in Table 1.

Fig. 10. Although log rBET are highly correlated to log rgeo values (R2 = 0.957, n = 628), some log rBET values are significantly displaced from the correlation line. The legend identifying the source of each datum is found on Fig. 6 and in Table 1.

into a convenient predictive equation but it also yields more reliable rate equations. In addition to using the fitted rate equation to predict rates, the parameters in the rate equation are typically used to estimate values of log A, Ea , and nHþ (see Table 2). The large spread of Ea and nHþ values reported by individual

studies (Fig. 5) suggests that deriving these parameters from individual studies is problematic even if the R2 value and parameter errors for each study appear to be reasonable. Furthermore, although the nHþ reported by individual studies cluster near the value found from the pooled data, most of the Ea values are much smaller. Because Ea and nHþ values are often used as a justification for models of reaction mechanisms, this discrepancy is a cause for serious concern. Two factors seem to be at play. First, most of the individual studies of forsterite dissolution rates yield relatively small data sets, typically fewer than 50 data. Second, the data are not randomly (homogeneously) distributed over the pH and 1/T ranges. This is especially true for 1/T (Fig. 4) where most of the data cluster either near to 25 or 120 °C. Furthermore, this range of temperature means that 1/T values only span the interval of 3.35  103 to 2.54  103, which represents only 0.25% of the range of possible values (0–1) whereas the values of log r span eight log units. As a result the slope of the line on the Arrhenius graph is very sensitive to the error in log r. The relatively small data sets and large amount of scatter characteristic of individual studies produce Ea values that range from 10 to 71 kJ/mol (Table 1). In addition, parameters estimated from small data sets tend to show more scatter for several other reasons (Gauch, 1993). This phenomenon is illustrated by the graphs at the bottom of Fig. 5, which show that values of Ea and nHþ tend to cluster more and

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7.5

7

6.5

6

5.5

5 65

70

75

80

Ea, kJ/mol Fig. 11. Correlation between log A and Ea values from 20 regression models of randomly chosen (without replacement) subsamples (n = 264) from the pooled data set (see Fig. 5). The Ea values span a range of 10 kJ/mol causing the values of A to span a range of about 2 orders of magnitude. The log A and Ea values are highly correlated (R2 = 0.98).

more closely to the value determined from the entire data set as the number of data, randomly selected from the pooled data set, increases from 27 (5%) to 106 (20%) to 264 (50%). A further consequence of the scatter of Ea values is that the intercept (log A) values are very scattered and more importantly are highly correlated to the Ea values (Fig. 11). This suggests that values of log A and Ea generated from small data sets should be viewed with considerable skepticism. A good example of this problem is illustrated by the conclusion of Chen and Brantley (2000), calculated correctly but based on only a few studies, that the Ea of olivine dissolution equals 30 kcal/mol: the value based on Fig. 5 is considerably higher. Several steps are needed to improve estimates of log A and Ea. First, the scatter in log r values must be reduced and the log r values should be more evenly spaced over the range of 1/T. Second, the 1/T range needs to be expanded as much as possible. Performing experiments at temperatures above 150 °C is possible but difficult and eventually will lead to a different reaction process. An alternative strategy would be to perform more experiments at temperatures below 25 °C. Experiments performed in an ice bath at 0 °C would have rates displaced from the 25 °C rates by the same amounts as experiments performed at 55 °C. Third, most of the pH values reported for the high temperature studies were measured in quenched samples so the pH values used in the regression model are not the actual pH of the solution that reacted with the mineral. Because strong acids or bases were used to adjust pH for most of these experiments the difference between the quench pH and the pH at the temperature of the experiment is not large but it does contribute to the error associated with the regression model.

One of the reasons to compile and organize rate data from many different experiments it to explore questions about the reaction process that cannot be addressed by limited data from individual studies. The compiled data set not only explores for correlations over a larger range of conditions but it produces more statistically robust conclusions. In order to illustrate situations where this approach is successful and situations where it is not successful, we now address the three questions posed previously. The first question is: how do models based on log rgeo compare to those based on log rBET? The discussion of errors above concluded that neither Ageo nor ABET give an accurate measure of the concentration of reactive surface sites because they both assume that the mineral surface is homogeneously reactive even though we know that site concentration and/or reactivity differs along different crystallographic directions and varies as a function of time. The regression models resulting from Ageo and ABET normalization are given in the first two rows of Table 2. The R2 values are nearly the same and t-tests show that the Ea values are not significantly different from each other for both the pH < 5.6 and pH > 5.6 data sets. However, t-tests show that both nHþ values are different at the 95% confidence level. Matched pair analysis finds that the geometric and BET rates are highly correlated with R2 = 0.957. Using critical selection to exclude samples with roughness factors greater than 10 (see Fig. 2) (Grandstaff, 1978; Van Herk et al., 1989; Giammar et al., 2005) produces a fit with an R2 of 0.989. Thus, these results do not provide a compelling statistical reason to choose one surface area normalization scheme over the other. Furthermore, as suggested by Brantley and Mellott (2000), ground olivine shows no evidence for internal surface area due to connected porosity; therefore, BET and geometric surface area should simply be interchangeable in predicting rates, as long as an appropriate value of roughness is chosen. However, there are practical reasons to use geometric surface area based rates rather than BET surface area based rates for natural systems. First, particle size determination is relatively reliable compared to determining BET surface area and BET surface areas vary depending on the gas used. As a result, calculated geometric surface areas are much less likely to contain large errors in accuracy that are found all too frequently in BET surface area determinations. Furthermore, determining particle size is simpler, quicker, and cheaper than BET surface area measurements, especially for field systems. Finally, using rate equations based on geometric surface area to model natural systems is relatively simple because it is easy to estimate or even measure grain sizes for the natural samples, whereas it is difficult to separate mineral grains for BET surface area measurements. It is of interest to note that the question of BET versus geometric surface area has been addressed theoretically by some workers (Jeschke and Dreybrodt, 2003). Those authors argued that when dissolution rates are fast compared to transport, that geometric surface area should be used. When the reverse is true, BET surface area should be used. The fast dissolution rates of olivine may mean that

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geometric surface area is the better choice for normalization in this case. However, Jeschke and Dreybrodt point out that the use of geometric surface area is especially appropriate when ink-bottle shaped pores are present in the mineral: no evidence for such pores have been identified in olivine (Brantley and Mellott, 2000). The second question is whether rates based on experiments with Mg as the reaction progress variable are comparable to those with Si as the RPV. When freshly broken forsterite is first introduced into aqueous solution, it dissolves incongruently releasing Mg faster than Si at low pH and vice versa at high pH. This produces a Si-rich surface at low pH and a Mg-rich surface at high pH e.g., (Pokrovsky and Schott, 2000). Many of the studies listed in Table 1 recognize this effect and most designed their experiments so that they measured rates under conditions of congruent dissolution. We tested this claim by pair wise comparison of Si and Mg rates. A matched pair comparison of 391 rates shows that they are highly correlated (R2 = 0.933), have a mean difference of only 0.025 log units, and a t-test does not reject the null hypothesis that they are the same at the 95% confidence level. In a way, this is a remarkable observation because it means that even though the surface composition of the dissolving forsterite is highly variable and only stoichiometric near pH 6, close to the pH where the low temperature rates show a break in slope (Fig. 6), the “steady state” dissolution rate appears to be stoichiometric across the entire pH range. Note that this justifies the use of either Mg or Si release rates, or both, as a basis for calculating the rate of forsterite destruction using Eq. (2). Consistent with these conclusions, many researchers have suggested that olivine dissolution, due to its orthosilicate structure, after a brief period of incongruent dissolution, dissolves congruently (Oelkers, 2001b). The third question is whether dissolved carbonate species affect forsterite dissolution rates. Wogelius and Walther (1991) report that exposing the forsterite dissolution experiments to atmospheric levels of carbon dioxide at pH 11 reduces the dissolution rate by about 1 order of magnitude but there is no significant effect at lower pH. Pokrovsky and Schott (2000) showed that at pH > 7, forsterite dissolution rate slows in proportion to the CO32- concentration but Prigiobbe et al. (2009) found no evidence for an effect of dissolved carbon dioxide on forsterite dissolution rate for pH < 8 at 120 °C. Because forsterite has been proposed as a source of Mg for mineral trapping of CO2 (Giammar et al., 2005; Be´arat et al., 2006; Andreani et al., 2009; Jarvis et al., 2009), the effect of dissolved carbonate species on its dissolution rate is an important issue. Testing for this effect using the data compiled here is difficult because the data set does not contain data from matched pairs of experiments, with and without added carbonate (which might result in the formation of magnesite), that can be compared directly. However, we can compare predicted rates from the carbonate absent and carbonate present rate equations in Table 2. Fig. 12 shows the difference in log r between rates predicted from rate equations based on experiments with no added carbonate, beyond that dissolved from the atmosphere, versus rate equations based on experiments that contained additional dissolved carbonate either because the partial

150 0.2 120

90

0

0 -0.2

-0.2

60 -0.4

-0.4

-0.6

30

0 2

4

6

8

10

pH Fig. 12. Difference between the log r values predicted by rate equations based on experiments with added carbonate versus those without added carbonate contoured over the pH and temperature range of the data set. This figure shows that over much of the range the rate equations developed from the experiments with added carbonate predict rates that are slower by as much as 0.7 log units (5) near pH 5.5 at low temperatures.

pressure of CO2 exceeded atmospheric levels or because a carbonate salt was added. Fig. 12 shows that at pH 5.5 at low temperatures the predicted forsterite dissolution rate for carbonate-added conditions is nearly 5 times slower than for carbonate-absent conditions. This result is inconsistent with the published studies that found carbonate effects occur at pH > 7. Furthermore, we would expect that CO32 ions, which are predominant at high pH, to have the greatest effect on the rates because they are likely to react most strongly with the Mg2+. In addition, the maximum difference in predicted rates shown on Fig. 12 are no larger than two times the forward-propagated, standard error for the predicted values of log r as discussed above. Thus, the differences in log r shown in Fig. 12 are likely not statistically significant and are most likely the result of distortion of the rate equations caused by breaking the data set at this pH. This should not be interpreted to mean that dissolved carbonate does not affect the rates but rather it means that the current data set is too imprecise and too incomplete to identify such an effect. This comparison provides an excellent case for why careful evaluation of statistical results is needed to insure that conclusions are consistent with other knowledge about the reaction. The data analysis presented in this paper is mostly statistical in nature. Such meta-analysis is a reasonable first step toward extracting useful knowledge from the published rate data. Even with a largely statistical approach, decisions are made in the meta-analysis that lead to differences in proposed rate equations. For example, Bandstra and Brantley (2008a,b) used a statistical approach for a similar dataset but assumed that there was no difference in mechanism above and below pH 5.6. To distinguish between the utility of different rate equations and parameters requires (i) further experimental work; (ii) approaches that can provide theoretically based rate equations; (iii) modeling efforts that use rate equations to predict field systems. The latter

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approach can allow testing of rate models as they have been manifested in natural systems over much longer time periods than attained in the laboratory. An interesting corollary of our analysis is that we need further laboratory experiments and theoretical modeling to determine whether there is a break in slope of the rate – pH data around pH 5.6. Furthermore, it is possible that field analyses might be modeled using reactive transport codes incorporating either the model equation proposed here or that proposed by Bandstra and Brantley to determine if the model with a break in slope at pH 5.6 or the simpler model is warranted. Considerable effort has been expended toward understanding the mechanism of silicate mineral dissolution. In the strictest sense the reaction mechanism should map out all of the elementary reaction steps that lead to the observed overall reaction. The discussion in the Introduction explains that the forsterite dissolution process involves chain reactions with an unknown but possibly large number elementary reaction steps. The fractional reaction order, n = 0.5, for the hydrogen ion activity term found here and many previous studies is a characteristic of branching chain reactions (Laider, 1987). Although the fitted values of A, Ea, and n do constrain any proposed model of the forsterite dissolution process, they should be interpreted as apparent values that are most suitable for modeling macroscopic rates. A reasonable next step for the olivine meta-analysis would also be to further subject the data to critical selection, which would use expert knowledge to filter out data that are not consistent with our understanding of the dissolution process. Although statisticians might consider this to be a source of unjustifiable bias, we consider this as a fully justified way to make the data consistent with existing theory and experience. Both approaches are necessary steps toward producing a useful model. Critical selection/rejection of data might consider several factors. For example, because we are interested in the rate of congruent dissolution, we might filter out all results where the rate based on Mg release differs from the rate based on Si release by more than three times (0.5 log units). Furthermore, we might choose to filter out rate data for experiments where the surface roughness of the grains exceeds 10. This is justified for two reasons. First, researchers have found that typical surface roughness for crushed silicate mineral grains is near 7 (White and Peterson, 1990). But even more pertinent to this study is the conclusion of Brantley and Mellott (2000) that log ABET (cm2/g) for mineral grains = b + m log D where D is grain diameter in microns. As pointed out by Brantley and Mellott for both olivine and quartz, m = 1, a determination which is consistent with insignificant surface area from connected porosity within the grains. For both minerals, they also concluded that b = 5.2. Using Eq. (3) from Brantley and Mellott and simple geometric arguments for grains with insignificant porosity, then roughness = 10bq/ a. Here q is grain density, i.e. 3.27 g/cm3 for olivine. Since we have assumed spherical grains in calculating olivine geometric surface area, a  6. To ensure that roughness is dimensionless, we include a term of 104 to account for the cm to micron conversion: the average roughness of grains

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inferred by Brantley and Mellott (2000) for two sets of olivine ground in the laboratory therefore equals (105.2  3.27)/ (104  6)  8.6. It is interesting to note that the rates based on geometric surface areas compiled in this paper are on the average 5.2 times faster than those based on BET surface areas. Given that both sets of rate values span 8 orders of magnitude, it is remarkable that the roughness inferred in this paper from rate measurements (5.2) is so close to the value (8.6) inferred based on two laboratory-ground samples in Brantley and Mellott (2000). For all of these reasons, we would suggest filtering out rate data with high roughness values. This study has identified some significant problems with our current approach to determining, interpreting, and reporting mineral dissolution rates. Chief among these is our present lack of understanding of the relationship between mineral grain surface areas, whether calculated from grain diameters (Ageo) or determined by gas adsorption (ABET) and the concentration of reactive sites. This lack of understanding appears to be responsible for increasing the relative error associated with measured rates by about one order of magnitude from 30% to 300%. This large amount of scatter found in the current data leads to very unreliable estimates of regression parameters for 1/T and pH when data sets are small so that rate equations derived from small data sets often show significant disagreement (Fig. 13) with each other. This also means that values of activation energy and reaction order based on relatively few data are very scattered (see Fig. 5) and unreliable. Additional scatter and uncertainty comes from the lack of standardized experimental and reporting for dissolution

-6

-8

-10

-12 0

2

4

6

pH Fig. 13. The log rBET values predicted by rate equations reported in various papers (see Table 1 for paper codes) often differs by >1 log unit for a given pH. The solid lines represent the range of pH of the study and the dashed lines show how these equations extrapolate beyond the range of the fitted data. The Ro00 and Ol08 equations agree closely with the results reported in this paper because their equations are also based on pooled data sets that included many of the same data considered in this paper.

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rate experiments. These problems raise the application of the systematic review process described in this paper from a luxury to a necessity because it is the only way to pool and process the data to increase the reliability of the rate equations to a level that they can be used for scientific and technological modeling. In order for systematic reviews of dissolution rate data to become standard and routine, we need to develop a standardized way of collecting, compiling, and sharing data so that it is conveniently and widely available to the community of scientists and engineers who are interested in mineral dissolution rates. A key feature of such a database is that it should be easy to access, to add to or to extract from, its contents. Furthermore, all of the data should be fully documented and sourced from peer-reviewed, published papers. The supplemental materials provide an example of some of the features that might be useful in such a database. An effort toward this end is ongoing whereby datasets can be uploaded and registered for DOI values in an organized fashion to enable easy access for kinetic analysis such as demonstrated in this paper. ACKNOWLEDGMENTS This research drew out of ongoing kinetic syntheses at VPI and Penn State. S.L.B. and A.A.O. acknowledge funding from the Penn State Center for Environmental Kinetics Analysis, an Environmental Molecular Sciences Institute funded by the National Science Foundation under NSF Grant No. CHE-0431328. We thank three anonymous reviewers for their helpful suggestions.

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