Chemical Geology 550 (2020) 119693

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Chemical Geology

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Equilibria, kinetics, and boron isotope partitioning in the aqueous boric T acid–hydrofluoric acid system ⁎ Richard E. Zeebea, , James W.B. Raeb a School of Ocean and Earth Science and Technology, Department of Oceanography, University of Hawaii at Manoa, 1000 Pope Road, MSB 629, Honolulu, HI 96822, USA b School of Earth & Environmental Sciences, The Irvine Building, St. Andrews KY16 9AL, UK

ARTICLE INFO ABSTRACT

Editor: Oleg Pokrovsky The aqueous boric, hydrofluoric, and fluoroboric acid systems are key to a variety of applications, including Keywords: boron measurements in marine carbonates for CO2 system reconstructions, chemical analysis and synthesis, Isotopic and elemental geochemistry polymer science, sandstone acidizing, fluoroborate salt manufacturing, and more. Here we present a compre- Boron isotopes hensive study of chemical equilibria and boron isotope partitioning in the aqueous boric acid–hydrofluoric acid pH proxy system. We work out the chemical speciation of the various dissolved compounds over a wide range of pH, total Analytical and theoretical advances fluorine T(F ), and total boron (BT) concentrations. We show that at low pH (0 ≤ pH ≤ 4) and FT ≫ BT, the Aqueous boric acid–hydrofluoric acid system − dominant aqueous species is BF4 , a result relevant to recent advances in high precision measurements of boron concentration and isotopic composition. Using experimental data on kinetic rate constants, we provide estimates

for the equilibration time of the slowest reaction in the system as a function of pH and [HF], assuming FT ≫ BT. Furthermore, we present the first quantum-chemical (QC) computations to determine boron isotope fractiona- tion in the fluoroboric acid system. Our calculations suggest that the equilibrium boron isotope fractionation − − between BF3 and BF4 is slightly smaller than that calculated between B(OH)3 and B(OH)4 . Based on the QC

− ≃ methods X3LYP/6-311+G(d,p) (X3LYP+) and MP2/aug-cc-pVTZ (MP2TZ), α(BF3−BF4 ) 1.030 and 1.025, re- − 11 − − − ≃ spectively. However, BF4 is enriched in B relative to B(OH)4 , i.e., α(BF4 −B(OH)4 ) 1.010 (X3LYP+) and 1.020 (MP2TZ), respectively. Selection of the QC method (level of theory and basis set) represents the largest uncertainty in the calculations. The effect of hydration on the calculated boron isotope fractionation turned out − to be minor in most cases, except for BF4 and B(OH)3. Finally, we provide suggestions on best practice for boric acid–hydrofluoric acid applications in geochemical boron analyses.

1. Introduction catalyst in organic synthesis and in the electroplating of tin and tin alloys, alongside various applications in polymer science, sandstone Fluroboron compounds have a wide variety of uses. A recent geo- acidizing, manufacturing of fluoroborate salts, and more (e.g., chemical application is in the high precision analysis of boron con- Palaniappan and Devi, 2008; Leong and Ben Mahmud, 2019). centration and isotopic composition (Misra et al., 2014b; Rae et al., While the chemical and isotopic equilibria in the boric acid system

2018; He et al., 2019). These boron measurements have application in in aqueous solution is relatively well understood, including α(B(OH)3−B

− marine carbonates as tracers of the CO2 system (e.g., Branson, 2018; (OH)4 ) (e.g. Zeebe, 2005; Liu and Tossell, 2005; Klochko et al., 2006; Rae, 2018; Hönisch et al., 2019), in silicates as tracers of seawater Nir et al., 2015), the partitioning of boron and its isotopes in the aqu- exchange with oceanic crust (Marschall, 2018) and subduction (De eous boric acid–hydrofluoric acid system has not been investigated in Hoog and Savov, 2018), and various other fundamental and environ- detail. This leads to uncertainties on how best to apply this method in mental uses (e.g., Rosner et al., 2011; Penman et al., 2013; Guinoiseau laboratory procedures and potential pitfalls associated with isotope et al., 2018). Boron trifluoride (BF3) and fluoroboric acid (HBF4) also fractionation between BeF species. Here we work out the equilibria and have a wide range of applications in chemical analysis and synthesis. kinetics of boron species in the presence of hydrofluoric acid, provide

BF3 is used in various organic synthesis reactions, such as the reduction the first quantum-chemical calculations for isotopic fractionation fac- of aldehydes and ketones to alcohols and hydrocarbons (Fry et al., tors in this system, and comment on best practice in the application of

1978), due to its properties as a Lewis acid. HBF4 is also used as a HF in boron analyses.

⁎ Corresponding author. E-mail addresses: [email protected] (R.E. Zeebe), [email protected] (J.W.B. Rae). https://doi.org/10.1016/j.chemgeo.2020.119693 Received 7 January 2020; Received in revised form 19 May 2020; Accepted 22 May 2020 Available online 05 June 2020 0009-2541/ © 2020 Elsevier B.V. All rights reserved. R.E. Zeebe and J.W.B. Rae Chemical Geology 550 (2020) 119693

Fig. 1. Geometries of several key compounds ex- amined in this study. Structures shown were opti- mized using quantum-chemical calculations for iso- lated (“gas-phase”) molecules. Dark-green: boron, light-green: fluorine, red: oxygen, white: hydrogen.

The notations BF3(H2O) and HBF3(OH) will be used synonymously here for structures similar to the

BF3(H2O) geometry shown (for details, see Section 4.1). (I) indicates geometrically unstable (one cal- culated frequency is imaginary). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

2. Chemical equilibrium HBF2(OH)2, HBF3(OH), and HBF4 need to be considered. For example,

[BF(OH) ][H+ ] The reactions that link the boric acid–hydrofluoric acid system + 3 HBF(OH)3 BF(OH)3 + H ; K1 = , (involving BeF compounds, see Fig. 1) may be written as (Wamser, [HBF(OH)3 ] (11) 1951; Mesmer et al., 1973): and so forth. B(OH)3 + F BF(OH)3 (1) Given the immediate reaction of BF3 with water, gaseous BF3 can + B(OH)3 + 2F + H BF2 (OH)2 + H2 O (2) likely be neglected for dilute solutions at room temperature. For ex- ample, complete absorption of 1.17 mol BF3 per kg H2O has been re- B(OH)+ 3F + 2H+ BF (OH)+ 2H O 3 3 2 (3) ported at 25 °C with no BF3 observed by infrared analysis in the vapor + phase (Scarpiello and Cooper, 1964). Some gas manufacturers state BF3 B(OH)3 + 4F + 3H BF4 + 3H2 O, (4) “solubilities” of ~3.2 g per g H2O at 0 °C (47 mol BF3/kg H2O), yet the with origin and method for obtaining this value is difficult to track down. Also, “solubility” appears misleading here as hydrates are instantly [BF(OH)3 ] [BF2 (OH)2 ] K1 = ; K2 = formed, rather than dissolved BF . Raman spectra of a BF -H O mix [B(OH) ] [F ] [B(OH) ] [F ]2 [H+ ] 3 3 2 3 3 (5) (ratio 1:2) showed three of the four fundamentals of a tetrahedral [BF (OH) ] [BF ] molecule (not of a trigonal molecule like BF3), almost identical to those K = 3 ; K = 4 . 3 3+ 2 4 4+ 3 of an aqueous NaBF (OH) solution (Maya, 1977). These observations [B(OH)3 ][F ] [H ] [B(OH)3 ][F ] [H ] (6) 3 are consistent with Anbar and Guttmann's (1960) estimate of 6 In addition, we take into account the relatively well known dis- [HBF3(OH)]/[BF3] ≃ 5 × 10 based on free energies. sociation reactions: The equilibrium constants Ki have been determined in 1 M NaNO3 + and 1 M NaCl solutions (Grassino and Hume, 1971; Mesmer et al., B(OH)3+ H 2 O B(OH)4 + H (7) 1973); KB, KF, and Kw are relatively well known, also at different ionic + HF F+ H (8) strengths. In this study, we use the set of constants given by Mesmer + et al. (1973) in 1 M NaCl (see Table 1). The acid dissociation constants H2 O H+ OH , (9) Ki′ are less well known. For instance, the acid strength of HBF4 has been with estimated to be similar to HCl and H2SO4 in aqueous solution (Wamser, [B(OH) ][H+ ] [F ][H+ ] 1951; Fărcasiu and Hâncu, 1997). Thus, we assign the value K = 4 ; K = ;K = [H+ ][OH ]. B F w pK4′ = − 3.0 (actual value is of minor importance as long as ≲ − 2.0). [B(OH)3 ] [HF] (10) The acid strength of HBF3(OH) and HBF2(OH)2 was deemed similar to − Furthermore, equilibrium between the four BFi(OH)4−i ions (i=1, CCl3COOH and CHCl2COOH, respectively (Wamser, 1951), broadly …, 4) in reactions (1)–(4) and their protonated forms HBF(OH)3, consistent with estimates based on HBF3(OH) and HBF2(OH)2

2 R.E. Zeebe and J.W.B. Rae Chemical Geology 550 (2020) 119693

Table 1 Equilibrium constants at 25 °C used in this study (pK = −logK).

a a a b a a a b b b b c pK pK1 pK2 pK3 pK4 pKB pKF pKw pK1′ pK2′ pK3′ pK4′ pKh

Value 0.36 −7.06 −13.69 −19.21 8.81 2.89 13.73 2.00 1.35 0.66 −3.00 2.64

a Mesmer et al. (1973), 1 M NaCl. b See text. c Wamser (1948).

10 -5 defaults: pH=1, FT=0.3M, BT=1e-5M concentrations in solution (Mesmer et al., 1973). Hence we use 1 pK3′ = 0.66 and pK2′ = 1.35 (Lide, 2004). For the weakest of the four B(OH) 3 acids, HBF(OH)3, we assign pK1′ = 2.0 (actual value is of minor im- 0.8 B(OH) - portance as long as ≲ 5.0) (Table 1). 4 BF(OH) - To link equilibrium and kinetics, the equilibrium constant for the 0.6 3 BF (OH) - hydrolysis of fluoboric acid will also be required (see reaction (13), 2 2 0.4 BF (OH) - Section 3 below): 3 HBF (OH) 0.2 3 [BF3 (OH) ][HF] Concentration (M) BF - a Kh = , 4 [BF ] 4 (12) 0 0 2 4 6 8 10 −3 which we set to Kh=2.3 × 10 M(pKh = 2.64, Wamser, 1948) at pH 25 °C for consistency with Wamser's kinetic data used in Section 3. 10 -5 Compatibility between Kh and the set of K's from Mesmer et al. (1973) 1 (Table 1), then determines pK4 = − 19.21, because K4 = K3/KF /Kh . 0.8 Note that Mesmer et al. (1973) did not measure K4 but determined its value using the same reasoning as applied here. 0.6 Given equilibrium constants, pH, total boron and total fluorine, the speciation in the boric acid–hydrofluoric acid system can be calculated 0.4

(Appendix A). At low pH (0 ≤ pH ≤ 4) and FT ≫ BT, the dominant − − 0.2 aqueous species is BF4 , followed by BF3(OH) (Fig. 2). Also, under Concentration (M) b most of those conditions, the protonated forms [HBF (OH) ] make up i 4−i 0 -3 -2 -1 0 a small fraction of BT and FT (not shown). One exception is HBF3(OH) 10 10 10 10 with significant concentrations at very low pH(Fig. 2). However, note F total (mol/l) that there is considerable uncertainty in the calculated [HBF3(OH)], as 10 0 only estimated values for pK3′ are available (see above). Our speciation results (Fig. 2) are similar to, but different from, those of Katagiri et al. (2006), who used a different set of constants and did not take into 10 -5 account the protonated forms [HBFi(OH)4−i]. + 0 The rising concentration of HBF3(OH) at [H ] > 10 (negative pH) can be understood considering the relevant equilibria and their pK 10 -10 values (Table 1). At very high [H+], the most abundant species are − − Concentration (M) c BF4 , BF3(OH) , HBF4, and HBF3(OH). As one might expect, at low pH, − − 10 -15 [BF4 ] > [BF3(OH) ]. However, the balance between these species is additionally controlled by the acid strengths of the protonated forms, 10 -8 10 -6 10 -4 10 -2 10 0 B total (mol/l) HBF4 and HBF3(OH), of which HBF4 is the stronger acid (pK's of −3.00 vs. +0.66, see Table 1). It turns out that for the pK set used here, the Fig. 2. Speciation in the boric acid–hydrofluoric acid system at 25 °C asa difference in acid strength between HBF4 and HBF3(OH) dominates that function of (a) pH, (b) FT, and (c) BT. Default parameter values (when not − − −5 between BF4 and BF3(OH) , leading to high [HBF3(OH)] at very low varied) are pH = 1.0, FT = 0.3 M, and BT = 1 × 10 M. Note that [HBF3(OH)] pH. In fact, using equilibrium relations such as Eqs. (6), (10), and (11), is uncertain as only estimated values for pK3′ are available. + ∗ and FT ≫ BT, the hydrogen ion concentration [H ] at which − + ∗ [HBF3(OH)] = [BF4 ] can be estimated as [H ] ≃ K3′K4KFFT/K3, or ∗ should be taken as a basic guide to the speciation in the system. pH ≃ − 1.5, in agreement with the complete speciation calculation However, that speciation only applies to a single ionic strength and may (Fig. 2). also change in the future as new or improved data for equilibrium constants become available. Caution seems also warranted regarding 2.1. Note of caution the available kinetic data for the system, as values reported by two different studies for one particular rate constant differ by a factor of For the present study, we opted for the set of equilibrium constants ~2–3 (see Section 3). (K's) from Mesmer et al. (1973) (Table 1), as the set appears internally consistent. However, these K's apply to 1 M NaCl solutions and not to 3. Kinetics solutions of different ionic strengths in general. Unfortunately, thede- pendence on ionic strength is unknown at present. Mesmer et al. (1973) − Among the reactions in the fluoroboric acid system, only the fol- noted that their equilibrium constant for the hydrolysis of BF3(OH) to − −4 lowing is considered slow (Wamser, 1948; Wamser, 1951; Anbar and produce BF2(OH)2 and undissociated HF (1.8 × 10 ) is considerably Guttmann, 1960): lower than the value 0.011 estimated by Wamser (1951). Furthermore, k1 the acid dissociation constants Ki′ have only been estimated, not mea- BF3 (OH)+ HF BF4 + H2 O, sured, at this point. As a result, our equilibrium calculations (Fig. 2) k2 (13)

3 R.E. Zeebe and J.W.B. Rae Chemical Geology 550 (2020) 119693

t (min) [HF] » B which will be assumed the slowest reaction in the system and examined 99% T 10 0 in the following. Reaction (13) leads to the rate law: 1 0.1 10 d[BF ] 4 1 = k1[BF 3 (OH) ][HF]k2 [BF4 ]. dt (14) .1 0 10 100 Wamser (1951) found the forward reaction (13) (left to right) to be acid-catalyzed, with: 10 -1 0 H + k1 = k1 + k1 [H ], (15)

0 −1 −1 H where k1 = 0.064 l mol min and k1 = 7.35 at 25 °C. The mea- 1

sured values in Table II of Wamser (1951) at pH=1.65 and 1.36 (0.244 (mol/l) [HF] −1 −1 and 0.387 l mol min ) are consistent with those in Table IV of 10 -2 100 1000 Wamser (1948) (0.244 and 0.392 l mol−1 min−1) at 25 °C. We are 10 unaware of any other studies that determined k1 experimentally. − Wamser (1948) also studied the BF4 hydrolysis, i.e., the backward 10000 reaction of reaction (13) and provided values for k2 (his Table IV), − which varied with the initial concentration of BF4 . Wamser (1948) -3 + 10 stated that the observed variation of k2 may be in part a result of [H ] -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 changes, which varied simultaneously with initial concentrations. In pH − fact, Anbar and Guttmann (1960) found the BF4 hydrolysis to be first − + + Fig. 3. Calculated time for 99% equilibration of reaction (13) at 25 °C, as- order in [BF4 ] and [H ] and suggested the rate law R = k2′[H ] − − suming [HF] to be much greater than the total boron concentration BT (see [BF4 ]. Assuming also the BF4 hydrolysis to be acid-catalyzed and text). assuming pH values of 1.65 and 1.36 for initial concentrations of 0.0561 M and 0.1105 M in Table IV of Wamser (1948) (cf. Table II, (t99% = − ln (0.01) × τ) is less than ~1 min for [HF] > 0.01 M and Wamser (1951)), we can calculate a catalyzed hydrolysis rate constant + from: [H ] > 1 M due to acid catalysis (Fig. 3). However, t99% increases dramatically to over 10,000 min (167 h) at low [HF] and pH > 2. Thus, 0 H + k2 = k2 + k2 [H ], (16) the equilibration time is very sensitive to [HF] and pH. We note, however, that the specific numbers presented here should be taken with by fitting Eq. (16) to Wamser's k2 values, which yields 0 −4 −1 H −2 −1 −1 caution due to the limited available experimental studies on the subject k2 = 1.47 × 10 min and k2 = 1.69 × 10 l mol min . The −3 −1 −1 and the disagreement between them (discussed above). latter may be compared to k2′ ≃ 7 × 10 l mol min given by H Anbar and Guttmann (1960), which is less than half the k2 value as derived from Wamser (1948). The reason for the discrepancy is unclear 4. Isotopic equilibrium: theory at this point. However, note that the ratio of Wamser's k2/ −3 Isotopic fractionation factors in thermodynamic equilibrium are k1 ≃ 2.3 × 10 M (Table IV, Wamser (1948)), i.e., the equilibrium calculated from first principles based on differences in the vibrational constant (Kh) of reaction (13), was consistent with his value derived −3 energy of molecules. In this study, we determine fundamental fre- from titration data (Kh = 2.04 × 10 M). This lends confidence to the kinetic data of Wamser (1948, 1951), which we use in the following. quencies and molecular forces using quantum-chemical (QC) compu- tations (e.g. Jensen, 2004; Schauble, 2004; Zeebe, 2005; Guo et al., Also, for internal consistency between k1 and k2 as implemented here j 2009; Rustad et al., 2010; Zeebe, 2014). Fractionation factors were (Eqs. (15) and (16)), we made sure that the ratios of individual ki 's that calculated from reduced partition function ratios (Urey, 1947): are dominant at low vs. high pH, respectively, yield the desired Kh 0 0 −4 H H −2 value, i.e., k2 /k1 = 1.47 × 10 /0.064 = k2 /k1 = 1.69 × 10 / Q s ui exp(ui /2) 1 exp(ui ) −3 = , 7.35 = 2.3 × 10 . Q s u exp(u /2) 1 exp(u ) r i i i i (18) Given k1 and k2, the rate law (Eq. (14)) can be solved analytically with some critical assumptions (Appendix B). First, our solution is only with s and s′ being symmetry numbers, ui = hcωi/kT and ui′ = hcωi′/kT − − valid at low pH, where [BF3(OH) ] and [BF4 ] are the dominant BeF where h is Planck's constant, c is the speed of light, k is Boltzmann's − − species and other compounds such as BF2(OH)2 and BF(OH)3 can be constant, T is temperature in Kelvin, and ωi and ωi′ are the frequencies + ignored. Second, [H ] and [HF] are assumed to remain constant during of the isotopic molecules or the solute-water clusters. Note that Eq. (18) + the reaction. If [H ] varies, our acid-catalyzed rates cannot be treated is based on the harmonic approximation and hence requires harmonic using constant k1 and k2 (Eqs. (15) and (16)). Importantly, constant ω's as input (see discussion in Zeebe (2005)), which we calculate here [HF] during the reaction usually requires the hydrofluoric acid con- using QC calculations. In contrast, observed ω's include anharmonicity centration to be much greater than the total boron concentration BT. but will nevertheless be compared to harmonic ω's (see Section 5.1). In The condition “much greater” is critical, as even for initial molar ratios the present case, errors introduced by anharmonicity (e.g., Zeebe, [HF] : BT = 4:1, nearly 4 mol of fluorine per mole of boron maybe 2005) are likely much smaller than those due to different QC methods − consumed, if a large fraction of BF4 is formed during the reaction (see (see below). The theoretical calculations yield β-factors, which, for a discussion in Wamser, 1948, 1951). Hence without a substantial [HF] compound A is given by: excess over BT, the rate may slow down significantly in the course of the 1 Q k reaction as [HF] drops. Note that in such cases, the final equilibrium = A , A Q extent of hydrolysis and product/reactant ratio (Eq. (12)) is given by A r (19) the final, not initial, [HF] (cf. Table V, Wamser, 1948). With the above assumptions, the characteristic (e-folding) time τ for where k is the number of atoms being exchanged (k = 1 for boron in the the reaction may be calculated as (Appendix B): compounds considered here). Finally, the fractionation factor α be- 1 tween two compounds A and B is given by: =(k1 [HF] + k2 ) . (17) A ()AB = . Using k1 and k2 values derived from Wamser (1948, 1951) (Eqs. B (20) (15) and (16)), the calculated time for 99% equilibration

4 R.E. Zeebe and J.W.B. Rae Chemical Geology 550 (2020) 119693

Merrick et al., 2007; Zeebe, 2005, 2014). For all molecules and solute- water clusters studied here (see Figs. 1 and 4), geometry optimizations were followed by full Hessian (force-constant matrix) runs to determine frequencies and to ensure that none of the calculated frequencies was imaginary (geometrically unstable, e.g., HBF4 at C2v symmetry, see Fig. 1). As mentioned above, the notations BF3(H2O) and HBF3(OH) are used synonymously here (see Fig. 1). Note that initial geometries in which one hydrogen was positioned near any of the three F atoms − quickly evolved into separate BF2(OH) and HF structures. This was the case for the isolated HBF3(OH) molecule, as well as for a hydrated unit including n = 6 water molecules. The BeO distance, which is large in BF3(H2O) (~1.84 Å, Fig. 1), is substantially smaller in the BF3(H2O) ⋅ (H2O)6 cluster (~1.54 Å) (both at X3LYP/6-311+G(d,p) level).

4.2. Uncertainties in computed α's

The range in the computed fractionation factors from the higher- level DFT and MP2 methods is reported here as an uncertainty estimate, as different QC methods yield significantly different α values (see Fig. 4. Solute-water cluster example. Optimized geometry of a hydrated below). Other approaches have been used in the literature. For instance, − BF3(OH) ion including 22 water molecules (C1 symmetry) based on X3LYP/6- propagated uncertainties in the computed frequencies derived for well- 311+G(d,p) calculated with GAMESS (Gordon and Schmidt, 2005). Dark- studied small molecules have been used as an error estimate for a single green: boron, light-green: fluorine, red: oxygen, white: hydrogen. Dotted lines QC method (e.g., Kowalski et al., 2013). This approach is based on the indicate hydrogen bonds. (For interpretation of the references to colour in this well known tendency of certain methods to systematically under- or figure legend, the reader is referred to the web version of this article.) overestimate frequencies. However, this is a systematic, not random error (commonly corrected for by a scale factor, see Section 4.1), which 4.1. Quantum-chemical computations says little about the QC method's accuracy when applied to a specific system for which experimental frequencies and α's are yet lacking. We used the quantum-chemical software package GAMESS, Sep- (Note that a QC method's precision for a fixed geometry is undefined as 2018-R3 (Gordon and Schmidt, 2005) and different computational one method yields exactly one set of frequencies and one α value for methods (differing in level of theory, LoT, and basis sets) to determine that case.) Furthermore, it is not uncommon that certain QC methods geometries and frequencies of key compounds in the boric acid–hy- give large errors in only a few (but critical) frequencies that deviate drofluoric acid system (Fig. 1). A very basic but fast method (HF/6- substantially from errors accounted for by an average frequency 31G(d), HFb for short) was used for initial guesses and pre-optimization scaling. Also, the computed α value and its error for a given method is (for methods, see e.g., Jensen, 2004; Gordon and Schmidt, 2005). sensitive to the calculated frequency shift upon isotopic substitution, However, α's and β-factors obtained with HFb should be taken with which is not necessarily related to the average error in absolute fre- caution because the method has limited accuracy. The density func- quency for that method. tional theory (DFT) method X3LYP/6-311+G(d,p) (X3LYP+) was Alternatively, by averaging results from different QC methods, a employed for higher level optimizations and large solute-water clusters “theoretical mean” α has been calculated and its ± 1σ standard devia- with up to n = 22 water molecules (Fig. 4). Computations with the most tion been reported as error (e.g., Li et al., 2020). However, it is im- complete basis sets tested here were performed with MP2/aug-cc-pVDZ portant to realize that selected QC methods and their numerical results and MP2/aug-cc-pVTZ (MP2DZ, MP2TZ), which are however compu- do not represent a set whose statistical sample mean approaches “a true tationally expensive and mostly impractical for large solute-water mean value” for large N. Rather, the set of QC methods included in the clusters. analysis is an often arbitrary selection made by the investigator, which We selected the methods HFb, X3LYP+, and MP2TZ because they inevitably leads to bias (some studies, for example, only include density are frequently used in QC computations and similar LoT and basis sets functional theory methods). Furthermore, it is not clear whether results have been applied to boron isotope calculations previously (e.g. Oi, from methods with known limited accuracy such as HF/6-31G(d) 2000; Zeebe, 2005; Liu and Tossell, 2005; Rustad et al., 2010). Also, should be included or not, which may be problematic in some cases, but

− X3LYP+ and MP2TZ yield values for α(B(OH)3−B(OH)4 ) that cluster inconsequential in others (HF/6-31G(d) fortuitously yields acceptable α around the upper and lower end of the spectrum (Rustad et al., 2010). values in several cases). For the various reasons outlined above, the However, even for isolated (“gas-phase”) molecules (see below), the uncertainty estimate reported here is given as the range in the com- higher-level DFT and MP2 methods differed by up to 5‰ in α's puted α from different methods. We include results from the higher- (Table 2). Given these differences, we refrain from testing further QC level DFT and MP2 methods and exclude results from HF/6-31G(d). methods, which will unlikely narrow down the range of α values. The selected LoT and basis set therefore represents the largest uncertainty in 5. Boron isotope partitioning our calculations, whereas the effect of hydration on the calculated boron isotope fractionation turned out to be less significant in most Given the dominant aqueous species discussed in Section 2, our cases (see below). calculations of boron isotope fractionation factors (α's) will mainly − − − For HFb and MP2DZ frequencies, scale factors of s = 0.92 and 1.03 focus on B(OH)3, B(OH)4 , BF4 , and BF3(OH) . Of those compounds, − were applied, whereas unscaled frequencies were used from X3LYP+ the fractionation between B(OH)3 and B(OH)4 in aqueous solution has and MP2TZ computations. The scale factors applied here are close to been established theoretically and experimentally and is described those obtained from general low-frequency fits to > 1000 observed elsewhere (e.g. Zeebe, 2005; Liu and Tossell, 2005; Klochko et al., frequencies and are consistent with scale factors from our previous 2006; Rustad et al., 2010; Nir et al., 2015). To provide insight into the work on boron, carbon, and oxygen isotopes (Scott and Radom, 1996; systematics of boron fractionation, we also include BF3 for comparison

5 R.E. Zeebe and J.W.B. Rae Chemical Geology 550 (2020) 119693

MP2/aug-cc-pVTZ s=1.00 − + 1600 that resembled BF4 − H3O − H2O was geometrically stable at all 11 BF LoT tested here. 3 The calculated fundamental molecular frequencies (ω's) and their 10 BF 1400 3 shift upon isotopic substitution are key to evaluate Eq. (18). Measured BF- 4 ω's are available, e.g., for BF3 (Vanderryn, 1959; Nakane and Ōyama, − ) 1200 1966) and BF4 from NaBF4 in aqueous solution (Bates et al., 1971) -1 and can be compared to calculated frequencies from our quantum-

(cm chemical computations (Fig. 5). The match is quite good at the highest 1000 1:1 LoT tested here (MP2/aug-cc-pVTZ) — the calculated ω's line up close to the 1:1 line without scaling (scale factor s = 1). At a basic LoT (HF/6- −1 800 31G(d)) the asymmetric stretch in BF3 at ~1450 cm falls slightly − below the 1:1 line, while the asymmetric stretch in BF4 at Calculated ~1100 cm−1 falls above the 1:1 line (s = 0.92, not shown). As a result, 600 − the β-factors of BF3 and BF4 are probably too small and too large, respectively, and hence αBF34 at HF/6-31G(d) is likely underestimated 400 (Fig. 6, Table 2, see Electronic Annex). Despite differences between LoT and basis sets, a few patterns 400 600 800 1000 1200 1400 1600 emerge from our calculations that appear robust. First, all BeF com- Observed (cm -1 ) 11 − pounds considered are enriched in B relative to B(OH)4 (Fig. 6). − − Second, BF3 and BF4 are isotopically heavier than B(OH)3 and B Fig. 5. Calculated vs. observed fundamental frequencies of BF3 and BF4 − (Vanderryn, 1959; Nakane and Ōyama, 1966; Bates et al., 1971). Calculated ω's (OH)4 , respectively, indicating that boron is more strongly bound in obtained at MP2/aug-cc-pVTZ level are unscaled (s = 1.0). the BeF than the B-OH compounds, given the pairwise similar mole- 11 cular geometries (D3h vs. C3h and Td vs. S4 symmetry). The order of B enrichment may have been expected from bond strength and bond − − between α(BF3−BF4 ) and α(B(OH)3−B(OH)4 ) (αBF34 and α34 for short), and length (d) of BeF vs. BeO in these compounds (e.g., dB−F ≃ 1.31 Å in HBF4 and HBF3(OH) to assess the effect of protonation on α's. − BF3, dB−O ≃ 1.37 Å in B(OH)3). Furthermore, the β-factor of BF3(OH) − falls below or close to that of BF4 and the effect of protonation is small − 5.1. Gas phase estimates (compare, e.g., fractionation between BF3(OH) and HBF3(OH), see Fig. 6). It is instructive to consider first quantum-chemical calculations for The details of the calculated β-factors and α's depend, however, on isolated (“gas-phase”) molecules to gain an overview of α's in the bor- the LoT and basis sets used (see Fig. 6 and Table 2). For instance, the − ic–hydrofluoric acid system and to examine differences between levels order of boron isotope enrichment in HBF3(OH) vs. BF3(OH) is re- of theory. It turned out that isolated HBF4 was geometrically unstable at versed for HFb, compared to X3LYP+ and MP2TZ (Fig. 6). Considering − 11 all LoT tested here, i.e., either one calculated frequency was imaginary only the higher-level DFT and MP2 methods, BF4 is enriched in B − (C2v symmetry, Fig. 1), or the geometry optimization led to BF3 + HF, relative to B(OH)4 by ~10‰ for X3LYP+ but by ~20‰ for MP2TZ,

− suggesting that HBF4 should not exist in the gas phase (Fărcasiu and respectively. The two methods differ by 5‰ in α(BF3−BF4 ) (see Table 2). Hâncu, 1997; Otto, 1999). Starting with HBF4 in C2v symmetry, adding Selection of the QC method (level of theory and basis set) thus re- two H2O molecules (HBF4 ⋅ (H2O)2), and then removing all symmetry presents the largest uncertainty in our isotope calculations for most restrictions in the calculation (C1), led to disintegration into a compounds (cf. Section 5.2). BF3 − FH − (H2O)2 configuration, in which3 BF was slightly non- planar (tested with HF/6-31G(d)). An alternative configuration (C1)

HF/6-31G(d) s=0.92 X3LYP-6-311+Gdp s=1.00 MP2/aug-cc-pVTZ s=1.00 1.05 1.05 1.05 a b c BF 1.045 1.045 1.045 3

BF 1.04 1.04 3 1.04 BF 3 B(OH) 1.035 1.035 3 1.035 B(OH) 3 ) ) ) - 4 - 4 1.03 - 4 1.03 1.03

B(OH) 1.025 3 1.025 1.025 BF- 4 -

(X-B(OH) 1.02 (X-B(OH) 1.02 (X-B(OH) 1.02 BF 4 - BF (OH) HBF (OH) 1.015 3 1.015 1.015 3 - HBF (OH) HBF (OH) BF (OH) 3 3 3 - 1.01 1.01 BF 1.01 BF4(OH) - 3 1.005 1.005 1.005 B(OH) - B(OH) - B(OH) - 4 4 4 1 1 1

− Fig. 6. Calculated stable boron isotope fractionation of selected compounds in the boric acid–hydrofluoric acid system at 25 °C relative to B(OH)4 . The level of theory and size of the basis set increase from a to c.

6 R.E. Zeebe and J.W.B. Rae Chemical Geology 550 (2020) 119693

BF- X3LYP+ Table 2 4 a BF- MP2DZ Calculated gas phase β-factors and α's at 25 °C (see Electronic Annex). 1.24 4 - BF3(OH) X3LYP+ - a HFb X3LYP+ MP2TZ BF3(OH) MP2DZ 1.23 HBF3(OH) X3LYP+ B(OH) X3LYP+ β 3 B(OH) - X3LYP+ B(OH)3 1.2313 1.2340 1.2363 1.22 4 − B(OH)4 1.2011 1.1913 1.1964

BF3 1.2467 1.2401 1.2507 − 1.21 BF4 1.2273 1.2035 1.2197 − BF3(OH) 1.2199 1.2013 1.2125 b HBF3(OH) 1.2190 1.2043 1.2145 1.2 c HBF4 I I I − BF(OH)3 1.2002 1.19 − BF2(OH)2 1.2057 α d α34 1.0252 1.0358 1.0333 d αBF34 1.0158 1.0304 1.0254 1.01 a HFb = HF/6-31G(d), X3LYP+ = X3LYP/6-311+G(d,p), MP2TZ = MP2/ X3LYP+ b MP2DZ aug-cc-pVTZ with scale factors s = [0.921.001.00]. 1.008 )

b - HBF3(OH) = BF3(H2O). c

I = one ω imaginary (geometrically unstable, see text). (OH) 1.006 d α34 = α(BF3−BF4−), αBF3BF4 = α(B(OH)3−B(OH)4−). 3

- BF 1.004 - 4 5.2. Solute-water clusters (BF 1.002 We also performed geometry optimizations and Hessian (force- constant matrix) runs for large solute-water clusters with up to n = 22 1 water molecules using the density functional theory (DFT) method 0 5 10 15 20 n(H O) X3LYP/6-311+G(d,p) (X3LYP+) (cf. Fig. 4). MP2/aug-cc-pVDZ and 2

MP2/aug-cc-pVTZ (MP2DZ and MP2TZ) are computationally too ex- − − Fig. 7. Selected β-factors and α(BF4 −BF3(OH) ) from computations including pensive and mostly impractical for large solute-water clusters (we solute-water clusters with up to n = 22 water molecules. tested MP2DZ for n ≤ 6). It turned out that the effect of hydration on the calculated β-factors for boron isotope exchange is minor in most − were evaluated at 25 °C using MP2TZ-calculated values for most α's cases, except for BF4 and B(OH)3 (Fig. 7). For instance, the hydration 11 (Table 2), the experimental α34 (Klochko et al., 2006), and δ BT = 0‰ effect reduces α(B(OH) −B(OH) −) by ~6‰ at 25 °C as n increases from 0 3 4 (Fig. 8). The β-factors of BF(OH) − and BF (OH) − at MP2TZ level to 20 for X3LYP+ (included in our calculations on temperature de- 3 2 2 were calculated as 1.2002 and 1.2057 (Table 2). The concentrations of pendence, see Section 5.4). However, the effect of hydration is much the protonated forms HBF(OH) , etc., are very small and make virtually smaller in most other cases and, importantly, less significant for MP2DZ 3 no difference, except for HBF (OH) at very low pH. The β-factors of the than for X3LYP+ (Fig. 7), suggesting that our “gas-phase” estimates 3 protonated forms were taken equal to the corresponding non-proto- from MP2TZ are reasonable approximations to boron isotope fractio- − nated forms. As expected, over the pH range where BF4 dominates, nation in aqueous solution in most cases (Table 2). Clearly, the overall 11 − 11 the δ B of BF4 is close to δ BT = 0‰. The boron isotope partitioning uncertainties introduced by different QC methods (Fig. 6) are sub- − − only shifts above pH ≃ 4, first towards3 BF (OH) , then B(OH)3, and stantially larger than those resulting from hydration, except for BF4 − 11 11 finally B(OH)4 , having respective δ B values close to δ BT (Fig. 8). and B(OH)3 at X3LYP+ (Fig. 7).

5.3. Boron isotope partitioning vs. pH 5.4. Effect of temperature

Given the calculated fractionation factors (α's) between the various Given that the effect of hydration on boron fractionation for most compounds (e.g., Table 2) and the speciation vs. pH (Fig. 2), the boron BeF compounds is small relative to the effect of different LoT(Fig. 7), isotope partitioning in the boric acid–hydrofluoric acid system asa we calculated the temperature dependence of α(BF −BF −) based on function of pH can be calculated. The correct mass balance calculation 3 4 isolated molecules at the highest LoT tested here (MP2/aug-cc-pVTZ, uses fractional abundances rather than isotope ratios, R's (e.g., Hayes, Fig. 9). For α(B(OH) −B(OH) −) = α34 we use results from our X3LYP/6- 1982). However, the difference is at most 0.15‰ in the present case 3 4 311+G(d,p) calculations of solute-water clusters with n = 20. We also (see Appendix C). Using R's, a general isotope mass balance at any given ∗ include an α34 , which uses the measured α34 seawater value at 25 °C pH may be written as: (Klochko et al., 2006) and an estimated temperature dependence based ∗ RTT X= Ri c i, (21) on the slope of our calculated α34 scaled by the ratio α34 /α34 at 25 °C (Fig. 7). From 0 to 40 °C, the calculated α's are very nearly linear vs. where index T refers to ‘Total’, and XT and ci are the total inventory and temperature for which we provide a fit of the form: individual concentrations of compounds containing element X, re- 3 spectively. If we express all α's relative to a single compound A, we can = ( 1)10=25 + (T C 25), (24) write Ri = αiRA. Then, where TC is temperature in °C. From 0 to 300 °C, we use a fit of the RXR= c , T T A i i (22) form: which can be solved for R : 2 A =a + b//, T + c T (25) RRXA= T T/. ic i (23) where T is temperature in Kelvin (for fit coefficients, see Table 3). The All remaining R's are calculated from Ri = αiRA. These expressions maximum errors of our fits are less than ~0.15‰.

7 R.E. Zeebe and J.W.B. Rae Chemical Geology 550 (2020) 119693

10 -5 FT=0.3M, BT=1e-5M 1 Table 3 Coefficients for temperature fits Eqs. (24) and (25). B(OH) a 0.8 3 −3 −6 - ε25 λ a b × 10 c × 10 Notes B(OH) 4 - 0.6 BF(OH) − 3 α(BF3−BF4 ) 25.4 −0.0627 −7.3143 14.3867 −1.3806 ‡

- − BF (OH) α(B(OH)3−B(OH)4 ) 29.4 −0.0921 −8.9720 15.0475 −1.0757 # 2 2 ∗ − 0.4 - α(B(OH)3−B(OH)4 ) 27.2 −0.0852 & BF3(OH) HBF (OH) 0.2 3 ‡, MP2/aug-cc-pVTZ. Concentration (M) BF - 4 #, X3LYP/6-311+G(d,p), n = 20. 0 &, see text.

borate ion at elevated pH. Notably, as samples, standards, and blanks 11 BT = 0 o/oo −1 30 are introduced in 0.5 M HNO3, and as the ~3 ml min NH3 gas flux is −1 b diluted in the spray chamber by ~1 l min Ar, the bulk solution in the 20 spray chamber remains acidic (pH < 1 when tested). We therefore

10 suggest that the suppression of boron volatility caused by NH3 addition

/oo) results from pronounced elevation of pH (to greater than the boric acid o 0 pK ≃ 9) on the surface layer (and perhaps in the aqueous diffusion B (

11 -10 boundary layer) of otherwise acidic droplets. Addition of HF offers an alternative method of improving boron -20 washout. This was first noted by Makishima et al. (1997) and studied in -30 more detail by Misra et al. (2014b), Misra et al. (2014a), Rae et al. 0 2 4 6 8 10 (2018), and He et al. (2019). The examination of boron partitioning pH here allows us to provide further insights into best practice in boron Fig. 8. Speciation and boron isotope partitioning in the boric acid–hydrofluoric analysis in an HF matrix. Avoiding the presence of volatile B(OH)3 re- acid system at 25 °C. (a) Note that [HBF3(OH)] is uncertain as only estimated quires that all boric acid is converted to fluoroboric species. This re- − values for pK3′ are available. (b) The boron isotope composition of total boron quires an excess of F ions over total boron, which is found with a 11 (BT) in the systen was set to δ BT = 0‰. combination of low pH, high total fluorine, and low total boron (Fig. 2). Boron is typically analyzed at sub-ppm concentrations in geochemistry, 6. Best practice in boron analyses with HF addition so overly high boron concentrations are unlikely to present an issue (cf. Fig. 2c). However, relatively high total concentrations of HF are needed Recently, it has been shown that addition of hydrofluoric acid can to ensure complete consumption of B(OH)3 (Fig. 2b). This is a function improve washout times of boron in the introduction systems of (MC-) of HF's pK (~3), which means that at pH = 1, only ~1% of total − ICPMS instruments (e.g., Wei et al., 2013; Misra et al., 2014b; Rae et al., fluorine is present as free F (Eq. (10)). Low pH (< 5) is also required 2018; Li et al., 2019; He et al., 2019). Slow washout of boron results (Fig. 2a), which is typical in geochemical analyses, and cautions against attempting to use HF and NH3 in combination to reduce boron washout. from the volatility of B(OH)3 (Brenner and Cheatham, 1998), which can be entrained from droplets coating the walls of ICPMS spray chambers The kinetics of B(OH)3 reaction with HF are generally fast, except − (Al-Ammar et al., 1999), leading to persistence of up to 50% of the for the formation of the final product,4 BF (reaction (13)). As a result, initial signal after ~5 min of wash, and associated memory effects be- volatile B(OH)3 may still be present in solutions to which HF has re- tween samples and standards. cently been added. As well as reducing the efficiency of the washout, Previously, addition of ammonia gas to the spray chamber has been this has the potential to impart isotopic fractionation, via preferential used to help combat this issue, leading to improved signal memory of loss of the isotopically heavy trigonal B(OH)3. To avoid this, solutions 2% after ~3 min wash (Al-Ammar et al., 2000; Foster, 2008). This re- should have HF added in advance of analysis depending on pH and − sult can be explained by conversion of volatile boric acid to non-volatile [HF] (see Fig. 3). Slow conversion to BF4 may also limit the efficiency

Temperature ( oC) 50250 100 200 300 34 35

BF34 BF34 30 32 34 34 * 34 25 BF4x 3

/oo) 30 /oo) o o ( ( 20 3 28 3 10 10 15

-1) 26 -1) 10 ( ( 24 5 a b 22 0 0 10 20 30 40 4 3.5 3 2.5 2 1.5 o Temperature ( C) 10 3/T (K -1 )

− − Fig. 9. Temperature dependence of α's. αBF34 = α(BF3−BF4 ) at MP2/aug-cc-pVTZ level; α34 = α(B(OH)3−B(OH)4 ) from X3LYP/6-311 + G(d,p) calculations of solute- ∗ water clusters with n = 20. (a) α34 uses the measured α34 seawater value at 25 °C (Klochko et al., 2006) and the slope of the calculated α34 (dashed line) scaled by the ∗ − − ratio α34 /α34 at 25 °C. (b) αBF4x3 = α(BF4 −BF3(OH) ).

8 R.E. Zeebe and J.W.B. Rae Chemical Geology 550 (2020) 119693

− of washout if HF is used only as a rinse solution rather than being added than total boron (BT), the dominant aqueous species is BF4 . Our es- to the analytes. timated time for 99% equilibration (t99%) of the slowest reaction in the − The current procedure in the STAiG laboratory at the University of system (forming BF4 ) is less than ~1 min at constant [HF] > 0.01 M + St Andrews is to run MC-ICPMS boron isotope analyses in 0.5 M and [H ] > 1 M, assuming FT ≫ BT. However, t99% increases dramati- HNO3 + 0.3 M HF (see Rae et al., 2018). This matrix is used for all cally to over 167 h at low [HF] and pH > 2, suggesting that the equi- solutions — standards, samples, and blanks — and results in washout to libration time is very sensitive to [HF] and pH. Our quantum-chemical ~0.5% in 3 min (cf. ~3% when using NH3 gas). Samples, initially in (QC) computations suggest that the equilibrium boron isotope fractio- − 0.5 M HNO3 following chemical purification, are “spiked” with a small nation between BF3 and BF4 is slightly smaller than that calculated − − 11 volume of concentrated HF (to avoid dilution) about 1/2 h prior to between B(OH)3 and B(OH)4 . Yet, BF4 is enriched in B relative to B − − − analysis (the estimated 99% equilibration time at this pH and HF (OH)4 in all our calculations (α(BF4 −B(OH)4 ) > 1.0), regardless of the content is ~10 min, see Fig. 3). Maintaining a constant matrix for all QC method tested. Unfortunately, even considering only the higher- solutions in the run is generally desirable to avoid differences in mass level QC methods tested, the calculated α values differ by ~10‰ in

− − − bias or background contamination, though we note that solutions of the α(BF4 −B(OH)4 ) and by ~5‰ in α(BF3−BF4 ). Selection of the QC method standard NIST 951 run as test samples with HF concentrations ranging (level of theory and basis set) thus represents the largest uncertainty in from 0 to 0.5 M show no systematic differences. For analysis of trace our isotope calculations for most compounds. The effect of hydration on boron concentrations, however, we use HF only in the wash solution. the calculated boron isotope fractionation is much smaller in most − We do not add HF to our trace element samples, despite the potential cases, except for BF4 and B(OH)3 computed with the density func- further improvement in washout, due to the lower overall boron con- tional theory method X3LYP/6-311+G(d,p). centrations, higher throughput of samples, risk of contamination of The results of our study should be helpful for implementing and other elements, and reduced contribution of boron washout to preci- advancing geochemical applications in high precision analyses of boron sion. Also note that HF should not be added to samples that are to be concentration and isotopic composition. One specific application is the separated by ion exchange chromatography with Amberlite 743, as addition of hydrofluoric acid to boron samples, which has recently been − BF4 does not interact well with this resin. shown to improve washout times of boron in the introduction systems Alongside the improved washout, use of HF also has advantages of (MC-)ICPMS instruments. However, beyond geochemical applica- over NH3 in terms of machine sensitivity, avoiding a signal decrease of tions, our study should serve as a general resource for a variety of ~10–20% with NH3. Stability and run time can also be increased, as studies dealing with equilibria, kinetics, and boron isotope fractiona- there is no risk of ammonium nitrate salt deposition in the injector. The tion in the aqueous boric acid–hydrofluoric acid system, including ap- main drawback of HF use is the safety hazard, requiring careful oper- plications in physical chemistry, polymer science, sandstone acidizing, ating procedures (for instance use of neoprene rather than nitrile fluoroborate salt manufacturing, and more. gloves), though note that a 0.3 M solution is equivalent to a ~1% di- lution, considerably less hazardous than fully concentrated HF (29 M). Hydrofluoric acid use also requires an “inert” sample introduction kit, Declaration of competing interest including self-aspirating Teflon nebulizer, Teflon spray chamber, and sapphire injector. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influ- 7. Summary and conclusions ence the work reported in this paper.

In the present study, we have examined the equilibria and kinetics in the aqueous boric acid–hydrofluoric acid system using available Acknowledgments experimental data. We have presented the first quantum-chemical computations to determine boron isotope fractionation in the fluor- REZ is grateful to Lance Agavulin and Glen Morangie for their oboric acid system and have provided suggestions on best practice in spiritual support. JWBR was supported by the European Research the application of HF in experimental boron analyses. Our results show Council (ERC Grant 805246) and the Natural Environment Research that at low pH (0 ≤ pH ≤ 4) and for total fluorine (FT) much greater Council (NERC grant NE/N011716/1).

Appendix A. Equilibrium speciation

− − To simplify the calculations, we use in the following xi = [BFi(OH)4−i ], yi = [HBFi(OH)4−i](i = 1, …, 4), b3 = [B(OH)3], b4 = [B(OH)4 ], − + f = [F ], g = [HF], h = [H ], and Li = 1/Ki′. Using equilibrium relationships (see Section 2), we can thus write: yi = xihLi, g = fhLF, and b4 = b3KB/ h. Given pH, total boron (BT) and total fluorine (FT), the mass balance equations read:

BT = b3 (1 + KB / h ) + xi (1 + hLi ) (A1)

FT = f(1 + hLF ) + ixi (1 + hLi ), (A2) in which we substitute xi's using Ki's (Eqs. (5) and (6)): i BT = b3 [1 + KB / h + ai (1 + hLi ) f ] (A3)

i FT = f(1 + hLF ) + b3 iai (1+ hLi ) f , (A4)

2 3 with a1 = K1, a2 = K2h, a3 = K3h , a4 = K4h , and eliminate b3: i i BT ia i (1+ hLfi ) = [ FfT (1+ hLF )][1 + KhB / + ai (1 + hLfi )]. (A5)

This expression can be solved numerically for f at given pH. b3 can now be obtained from Eq. (A3), all xi determined from Eqs. (5) and (6), and all yi from yi = xihLi.

9 R.E. Zeebe and J.W.B. Rae Chemical Geology 550 (2020) 119693

Appendix B. Analytical solution of kinetic rate equation

The kinetic rate Eq. (14): d[BF4 ] = k1[BF 3 (OH) ][HF]k2 [BF4 ] dt (B6)

− − can be solved analytically with some critical assumptions. We assume low pH where [BF3(OH) ] and [BF4 ] are the dominant BeF species, hence − − + we set BT = [BF3(OH) ] + [BF4 ]. Furthermore, we assume constant [H ] and [HF] during the reaction, which should hold approximately for FT ≫ BT. Then: d[BF4 ] = k1 (BT [BF4 ])[HF]k2 [BF4 ] dt (B7) d[BF4 ] = (k1 [HF]+ k2 )[BF4 ] + k1 BT [HF] = k [BF4 ]+ , dt (B8)

−1 where k = (k1[HF] + k2) = τ is the overall rate constant (inverse characteristic time scale) and γ = k1BT[HF] is a constant. The solution is:

[BF]()4 t = ([BF]4 0 /)exp(k kt )+ /,k (B9)

− − where index ‘0’ indicates initial [BF4 ] at t = 0 and γ/k = (k1BT[HF])/(k1[HF] + k2) equals [BF4 ] in equilibrium, which can be shown using k2/ − − k1 = Kh = [BF3(OH) ]eq[HF]/[BF4 ]eq, as it should be. Our solution (Eq. (B9)) is similar to Eq. (4) of Fucskó et al. (1993), except that our solution − allows for explicitly specifying [BF4 ]0. The time evolution is given by the exponential term and hence the characteristic (e-folding) time −1 τ = (k1[HF] + k2) .

Appendix C. Isotope mass balance using fractional abundance

The correct isotope mass balance using fractional abundances (r's) reads (e.g., Hayes, 1982): rT X T= r i c i, (C10) where index T refers to ‘Total’, and XT and ci are the total inventory and individual concentrations of compounds containing element X, respectively. Using ri = Ri/(1 + Ri) and expressing all α's relative to a single compound A, we can write Ri = αiRA and thus: rT X T= c i R i/(1 + R i ) = ci iR A/(1+ iR A ), (C11) which can be solved numerically for RA. All remaining R's are calculated from Ri = αiRA. In the present case, the result only differs from a mass balance using R's (Eq. (23)) by at most 0.15‰.

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