INTRODUCTION

Studies of the isotopic effects of oxygen is of paramount importance in isotope geochemistry. The great majority of isotope geothermometers make use of oxygen isotopes. This paper is devoted to the simultaneous computation of chemical and oxygen isotope equilibria with the GEOCHEQ_Isotope software package. The principal approaches to and methods of calculating oxygen isotopic effects utilized in this study are generally analogous to those we have applied when studying carbon isotopic effects. The first ever simultaneous simulation of chemical and isotope equilibria by minimizing the Gibbs free energy (this method is referred to as the method of isotope–chemical system, and its idea was formulated by B.N. Ryzhenko) was carried out for sulfur isotopes (Bannikova et al., 1987; Grichuk, 1987, 1988, 2000; Grichuk and Lein, 1991). The method enables calculating chemical and isotope equilibria simultaneously, in contrast to the usual approaches (Ohmoto, 1972), in which first chemical and then isotope equilibria are evaluated.

As was pointed out in (Mironenko et al., 2018), one of the main reasons why simultaneous simulations of chemical and isotope equilibria are still used inadequately little is the absence of internally consistent thermodynamic databases on rare isotopologues. One of the tasks of our study is to bridge this gap by developing a database for simulating geochemical processes with regard to oxygen isotopic effects, based on critical analysis of information on isotope equilibria.

APPROACHES AND METHODS

Similar to what has been done in studying carbon isotopic effects (Mironenko et al., 2018), the simulations used herein were conducted with a software package (Mironenko et al., 2020) that utilizes the SUPCRT92 database (Johnson et al., 1992), which has been later updated and corrected. The GEOCHEQ software package minimizes the Gibbs energy by the algorithm of the convex simplex (de Capitani and Brown, 1987). The isotopic effects of an element are calculated within the GEOCHEQ_Isotope extension of the GEOCHEQ software package by using isotopes of the element as independent components (instead of the chemical element as whole), and the list of substances is appended with the corresponding isotopologues (Mironenko et al., 2018). For oxygen, these are 16O and 18O and their isotopologues.

All isotopic effects are computed in GEOCHEQ_Isotope in the approximation of an ideal mixture of isotopes, in which isotope substitutions in different sites do not affect one another, and the isotopic effects are independent of the multiplicity of the isotope substitution. For molecules containing several atoms of a given element, this approximation is commonly formulated as the rule of geometric mean (Galimov, 1973, 1974). The approximation by an ideal mixture is valid for all elements except hydrogen at temperatures above 100 K (Polyakov, 1993). This approximation is deemed to be valid by default in the overwhelming majority of geochemical simulations, if no clumped isotopic effects are computed (Grichuk, 2000; Hill et al., 2014, 2020; Schauble and Young, 2020). Similar to carbon isotopic effects, the calculations in this study were carried out using the formalism of the β-factor. The concept of the β-factor was coined by Ya.M. Varshavskii and S.E. Vaisberg (1957) on the basis of the previously suggested (Bigeleisen and Mayer, 1947) reduced isotopic partition function ratio. However, it was E.M Galimov (1973, 1974, 1982, 2006) who introduced the concept of the β-factor into the practice of geochemical research. It is convenient to express the β-factor of a substance that contains atoms of the chemical element of interest in equivalent settings in a logarithmic form, in terms of the Gibbs energy

$$\ln \beta = \left( {G(P,T) - G{\text{*}}(P,T)} \right)/zRT - 1.5\left( {m{\text{*}}/m} \right),$$
(1)

where β is the β-factor, G is the Gibbs energy, P is the pressure, T is the temperature, R is the absolute gas constant, z is the multiplicity of the isotope substitution, m is the mass of the isotope, and the heavy isotopes are marked with *. According to this, the equilibrium isotope fractionation factor (αA/B) between two substances A and B is equal to the ratio of their β‑factors

$${{\alpha }_{{{\text{A/B}}}}} = {{{{\beta }_{{\text{A}}}}} \mathord{\left/ {\vphantom {{{{\beta }_{{\text{A}}}}} {{{\beta }_{{\text{B}}}}}}} \right. \kern-0em} {{{\beta }_{{\text{B}}}}}},$$
(2а)

or, in the logarithmic form for the isotope shift (ΔA/B)

$${{\Delta }_{{{{\text{A}} \mathord{\left/ {\vphantom {{\text{A}} {\text{B}}}} \right. \kern-0em} {\text{B}}}}}} \equiv {{\alpha }_{{{{\text{A}} \mathord{\left/ {\vphantom {{\text{A}} {\text{B}}}} \right. \kern-0em} {\text{B}}}}}} - 1 \approx \ln {{\beta }_{{\text{A}}}} - \ln {{\beta }_{{\text{B}}}}.$$
(2b)

In (2b), |β – 1| \( \ll \) 1.

If atoms of an isotopically substituted element occur in nonequivalent settings, equations (1) and (2a) should be refined. Because natural isotope exchange processes involve all of the isotopologues, one should know how the β-factor of a compound at whole relates to the site-specific β-factors for isotope substitutions in certain nonequivalent positions. The theory of intermolecular isotopic effects developed by E.M. Galimov (1973, 1974, 1981, 1082, 2006) for carbon isotopes is universal and can be applied to the fractionation of oxygen isotopes, because geochemical studies often deal with substances containing several oxygen atoms and some of them are in nonequivalent positionsFootnote 1. In particular, E.M. Galimov has demonstrated that, in the case of rare isotope substitutions, the β-factor of the compound is generally related to the β-factors of monosubstituted isotopologues (βi) by the relation (Galimov, 1973, 1981, 1982):

$$\beta \approx \frac{1}{n}\sum\limits_{i = 1}^n {{{\beta }_{i}}} .$$
(3a)

Relation (3a) was referred to by E.M. Galimov as the first additivity law (Galimov, 1981; 1982).Footnote 2 It has been later demonstrated that relation (3a) is valid not only at rare isotope substitutions but also when βi is insignificantly different from unity (Polyakov, 1987; Polyakov and Horita, 2021). Also, the logarithmic form is slightly more precise (Polyakov and Horita, 2021).

$$\ln \beta \approx \frac{1}{n}\sum\limits_{i = 1}^n {\ln {{\beta }_{{i\,}}};\,} \,\,\,\left| {\ln {{\beta }_{{i\,}}}} \right| \approx \,\left| {{{\beta }_{i}} - 1} \right| \ll 1.$$
(3b)

Because the intramolecular effects of oxygen are not computed in this GEOCHEQ_Isotope version, the use of the β-factor of the substance as a whole according to the Eq. (3b) makes it possible to minimize the increase in the database due to introduction of data on the isotopologues of various multiplicity and localization. Hence, in calculating an equilibrium fractionation of oxygen isotopes with the GEOCHEQ_Isotope complex, it is sufficient to consider the 18O isotopologue along with the most abundant 16O one, and a change in the Gibbs energy per one atom of this 18O isotopologue calculated according to (1), analogously to what has been earlier done for the fractionation of 13C/12C carbon isotopes (Mironenko et al., 2018).

The oxygen β-factors insignificantly depend on pressure (Polyakov and Kharlаshina, 1994), and hence, only their temperature dependences are incorporated into the GEOCHEQ_Isotope software. The temperature dependences of the β18O-factors are presented in the form of polynomials in the reciprocal absolute temperatures, as has been done for the carbon β-factors (Mironenko et al., 2018)

$${{10}^{3}}\ln \beta = {{A}_{0}} + {{A}_{1}}{{x}^{1}} + {{A}_{2}}{{x}^{2}} + \,{{A}_{3}}{{x}^{3}} + {{A}_{4}}{{x}^{4}} + {{A}_{5}}{{x}^{5}},$$
(4)

where \(x = {{{{{{{10}}^{6}}} \mathord{\left/ {\vphantom {{{{{10}}^{6}}} T}} \right. \kern-0em} T}}^{2}}\), and Ai is the coefficients of polynomial (4), which are given in Table 1.

Table 1.   Coefficients of the polynomial \({{10}^{3}}\ln \beta = {{A}_{0}} + {{A}_{1}}{{x}^{1}} + {{A}_{2}}{{x}^{2}} + {{A}_{3}}{{x}^{3}} + {{A}_{4}}{{x}^{4}} + {{A}_{5}}{{x}^{5}}\) for the calculation of the temperature function of the carbon β-factors
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Another additions to the computation of chemical and isotope equilibria in system involving the 18O isotope is the following modification of the formulas for the molalities of the solute components in view of the occurrence of the H218O isotopologue

$${{m}_{i}} = \frac{{{{n}_{i}} \times 55.51}}{{\left( {{{n}_{{{{{\text{H}}}_{{\text{2}}}}{}^{{16}}{\text{O}}}}} + {{n}_{{{{{\text{H}}}_{{\text{2}}}}{}^{{18}}{\text{O}}}}}} \right)}},$$
(5)

where mi is the molality of component i, and ni is the number of moles of component, and 55.51 is the number of moles corresponding to 1 kg of water.

β-FACTORS IN THE GEOCHEQ_ISOTOPE THERMODYNAMIC DATABASE

Similar to what pertains to the carbon isotopic factors (Mironenko et al., 2018), the accuracy and justification of the calculations of the oxygen isotope equilibria depend, first of all, on the accuracy of data on the β-factors utilized in the GEOCHEQ_Isotope software package. Equilibrium oxygen isotope fractionation factors can be evaluated experimentally and/or theoretically. Equilibrium oxygen isotope fractionation factors are experimentally determined using the Northrop–Clayton partial isotope exchange method (Northrop and Clayton, 1966) and the three-isotope method (Matsuhisa et al., 1978). Oxygen isotope geothermometers are also calibrated using empirical techniques (Clayton et al., 1968; Ustinov et al., 1978; Macey and Harris, 2006). A review of these techniques and their advantages and disadvantages can be found in (Chacko et al., 2001). The theoretical calculations are conducted based on model vibrational spectra (Urey, 1947; Bigeleisen and Mayer, 1947; Kieffer, 1982), the interatomic potential (Patel et al., 1991), and ab initio principles, based on the density functional method (Schauble, 2004; Blanchard et al., 2009; Hill et al., 2014; Krylov et al., 2017; Schauble and Young, 2021). Ab initio computations lately became the most widely used. The oxygen β-factors of many minerals were calculated with the application of the semiempirical method of increments (Schütze H., 1980; Zheng, 1991; 1999), which, in essence, is E.M. Galimov’s method of isotopic bond numbers (Galimov, 1973, 1974) extended to crystalline bodies. The latter results have, however, to be treated with care, because they often come in conflict with both experimental data and theoretical calculations (Horita and Clayton, 2007).

Oxygen Isotope Fractionation Factors in the System H2O–CO2 (gas)–dissolved Inorganic Carbon

The great majority of experimental determinations of the equilibrium oxygen isotope fractionation factors were conducted with the involvement of H2O and CO2 (Chacko et al., 2001), and hence, an accurate choice between the β18O-factors of H2O and CO2 is of principal importance for the calibration of the entire set of data on oxygen isotope equilibria.

β18O-factors of H2O. The oxygen β-factors of water vapor were calculated using various theoretical approaches. The computations on the basis of spectroscopic data and molecular constants were made with regard to the anharmonicity and interaction between oscillations and rotations (Bron et al., 1973; Richet et al., 1977). Recent ab initio computations were carried out based on the density functional (Hill et al., 2014 2020; Guo and Zhou, 2019; Schauble and Young, 2021). These calculations were conducted using diverse techniques, and their results are in reasonably good mutual agreement. The coefficients of approximating polynomial (4) used in the GEOCHEQ_Isotope software are listed in Table 1. The β18O-factors for liquid H2O were computed (Table 1) by adding the oxygen values of lnβ for water vapor and experimentally determined lnα values for equilibrium fractionation of oxygen isotopes between liquid H2O and saturated vapor (Horita and Wesolowski, 1994)

β18O-factors of CO2 and dissolved inorganic carbon. The β-factors of CO2 were computed by analogy with what has been done for water: from spectroscopic data and molecular constants (Richet et al., 1977) and ab initio principles (Hill et al., 2014, 2020; Guo and Zhou, 2019; Schauble and Young, 2021). The data were then used to calculate the coefficients of polynomial (4) for O2(gas) (Table 1). Also, experimental data are available on the equilibrium oxygen isotope fractionation between CO2(gas) and H2O(liq) at 25°C, because these values are used in determining water isotope composition. Numerous experiments (see references in Chacko et al., 2001) enabled the IAEA Consultants Group to recommend the value of 1.0412 at 25°C (Hoefs, 2018) as that of equilibrium fractionation coefficient in the system CO2(gas)–H2O(liq). Estimates are also now available for the temperature ranges of 5–100°C (Brenninkmeijer et al., 1983) and 130–350°C (Truesdell, 1974). Comparison of the equilibrium isotope fractionation factors in the system CO2(gas)–H2O(liq) calculated using GEOCHEQ_Isotope data (Table 1) and corresponding experimental values are exhibited in Fig. 1; the experimental and calculation data are obviously in reasonably good agreement.

Fig. 1.
figure 1

Comparison of GEOCHEQ_Isotope data on the equilibrium oxygen isotope fractionation factors in the H2O–CO2–dissolved inorganic carbon system with experimentally determined values. The inaccuracy of the experimental equilibrium oxygen isotope fractionation factors is no larger than the size of the symbols.

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Dissolved inorganic carbon is represented in GEOCHEQ_Isotope by solute carbonate species: CO2, H2CO3, \({\text{HCO}}_{3}^{ - }\), and \({\text{CO}}_{3}^{{2 - }}\). The β-factors of these species were ab initio calculated using the density functional theory approach (Hill et al., 2014, 2020; Guo and Zhou, 2019; Schauble and Young, 2021). The GEOCHEQ_Isotope database (Table 1) incorporates computation results (Guo and Zhou, 2019), which are in the best agreement with experimental data (Beck et al., 2005). Experimental data on solute carbonate species (Beck et al., 2005) are in a good agreement with equilibrium isotope fractionation factors between these solute species and water that were calculated based on data from GEOCHEQ_Isotope database (Fig. 1). Note that the β18O-factors for the dissolved neutral CO2 species were assumed to be equal to those in the gas phase. The calculated and experimental values for the equilibrium fractionation of oxygen isotopes differ by 0.1‰, i.e., their differences is smaller than the inaccuracies of both the measurements and the calculations (Fig. 1)

Oxygen β-Factors of Carbonate Minerals

Fractionation of oxygen isotopes in the system calcite–CO2. From the standpoint of isotope geochemistry, calcite seems to be the most important carbonate mineral. The reason for this is that the fractionation coefficients of oxygen isotopes experimentally determined in the anhydrous system (i.e., without H2O fluid) link to calcite (Chacko et al., 2001; Clayton and Kieffer, 1991). In view of this, it is very important to estimate equilibrium distribution in the system calcite–CO2 because, as is demonstrated above, the β18O-factor of CO2 has been reliably determined. For calcite, the ab initio calculations (Hill et al., 2014, 2020; Guo and Zhou, 2019; Schauble and Young, 2021) and those on the basis of the model oscillation spectra of the Kieffer type (Clayton and Kieffer, 1991; Chacko and Deines, 2008; Polyakov, 2008) agree well to each other. In GEOCHEQ_Isotope, the values of the β18O-factor are assumed equal to the ab initio computation results. At temperatures above 400 K, these data coincide with estimates in (Clayton and Kieffer, 1991). The difference between the values assumed for the β18O-factor of calcite in GEOCHEQ_Isotope and calculated based on the Kieffer spectrum model in a quasiharmonic approximation (Polyakov, 2008) are also much smaller than the scatter of the experimental data (Fig. 2). A somewhat greater difference was found between the CaCO3/CO2 fractionation curve (Chacko and Deines, 2008) and the fractionation curve calculated using the GEOCHEQ_Isotope database. This difference is explained by the underestimation of the Einstein oscillator frequency ν3 = 1460 cm–1 (Mironenko et al., 2018), which leads to underestimated values of the calcite β-factor. This effect is smaller for the β18O-factor of calcite than for the β13C-factor. Equilibrium fractionation in the system calcite–CO2 that was calculated using the values of the β18O-factor for calcite from the GEOCHEQ_Isotope database is generally in a good agreement with the experimentally measured equilibrium oxygen isotope fractionation factor for this system (Fig. 2).

Fig. 2.
figure 2

Comparison of various estimates of the equilibrium oxygen isotope fractionation factor in the calcite–CO2 system.

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Effects of oxygen isotope fractionation between carbonate minerals. The method applied to calculating the coefficients of polynomial (4) for carbonate minerals was analogous to the method used for their β13C-factors (Mironenko et al., 2018). Considering that the fractionation between carbonate minerals is well described by the model (Chacko and Deines, 2008) but the β18O-factors of calcite are underestimated compared to the values assumed in GEOCHEQ_Isotope (Fig. 2), the difference between the values of 103lnβ of calcite accepted in the GEOCHEQ_Isotope database and those in the model (Chacko and Deines, 2008) was added to the values of 103lnβ for carbonate minerals

$$\begin{gathered} \ln {{\beta }_{{{\text{carb}}}}} = \ln {{\beta }_{{{\text{calc}}}}} + \left( {\ln \beta _{{{\text{carb}}}}^{{{\text{CD}}}} - \ln \beta _{{{\text{calc}}}}^{{{\text{CD}}}}} \right) \\ = \ln \beta _{{{\text{carb}}}}^{{{\text{CD}}}} + (\ln{{\beta }_{{{\text{calc}}}}} - \ln \beta _{{{\text{calc}}}}^{{{\text{CD}}}}), \\ \end{gathered} $$
(6)

where βcarb and βcalc are the β18O-factors for the carbonate mineral and calcite in the GEOCHEQ_Isotope database, and \(\ln \beta _{{{\text{carb}}}}^{{{\text{CD}}}}\) and \(\ln \beta _{{{\text{calc}}}}^{{{\text{CD}}}}\) are the same factors calculated by the model (Chacko and Deines, 2008). The coefficients of polynomial (4) for the β18O-factors of carbonates calculated according to (6) are given in Table 1.

Figure 3 makes it possible to compare the fractionation of oxygen isotopes in the system dolomite–water according to data in GEOCHEQ_Isotope and experimental calibrations of other authors. The experimental values of the oxygen equilibrium fractionation factors between dolomite and water and the calculation data found using the GEOCHEQ_Isotope database are obviously in a good agreement.

Fig. 3.
figure 3

Fractionation of oxygen isotopes in the system dolomite–water.

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Oxygen β-Factors of Silicates and Oxides in the GEOCHEQ_Isotope Software Package

Many applications of the geochemistry of stable isotopes are related to the fractionation of oxygen isotopes in silicates and oxides. The coefficients of polynomial (4) for the calculation of the β18O-factors of these minerals, which were incorporated into GEOCHEQ_Isotope, are presented in Table 1. Most of these data were obtained by means of ab initio calculations. On the one hand, these results agree well with experimental and empirical data, and on the other hand, they enable extrapolations of the results to conditions inaccessible in the experiments. Some results obtained by various researchers with the use of different methods must be reconciled. An example of this is the calibrations of the β18O-factors of TiO2 polymorphs: rutile, anatase, and brookite. The β18O-factors of these polymorphs computed ab initio with the application of the supercell method (Krylov and Kuznetsov, 2019) are in very good agreement with the empirical calibration of the geothermometers (Zack and Kooijman, 2017). However, the β18O-factors of rutile are in notably poorer agreement with experimental and theoretical estimates of the oxygen isotope fractionation factors between the rutile–quartz and rutile–calcite (Fig. 4) than the calibration for rutile calculated based on a Kieffer model spectrum (Clayton and Kieffer, 1991). We have reconciled the various calibrations using an approach analogous to that applied for carbonates. We estimated the difference between the β18O-factors obtained for rutile by these two methods and then corrected the β18O-factors of the TiO2 polymorphs for this value and thus retained the calibration of the geothermometers and ensured reasonably good agreement with experimental and empirical estimates for the rutile–quartz and rutile–calcite isotope fractionation coefficients.

Fig. 4.
figure 4

On the problem of reconciling the β18O-factors of rutile.

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When selecting the calibrations of the β18O-factors for the GEOCHEQ_Isotope database, we have paid much attention to the consistency of the data. It is the consistency of the estimates of the β18O-factors obtained by different techniques (or the possibility of making these data mutually consistent) that has been applied as the criterion of the reliability of the data.

The β18O-factors of oxides were determined, along with conventional techniques mentioned above, also by some new approaches. In this situation, it is particularly important to verify the results obtained by the new techniques by using conventional techniques, which have already proven to be reliable. For oxides whose cations possess a “Mössbauer” isotope, β18O-factors can be estimated from experimental data on the gamma-resonant scattering and heat capacity. This method was applied to measurements of the β18O-factor for hematite (Polyakov et al., 2001) and cassiterite (Polyakov et al., 2005). For cassiterite, this test has been done experimentally, using the Norton–Clayton method (Hu et al., 2005) and by the method of empirical calibration with the use of other isotope geothermometers and temperature estimation from data on gas–liquid inclusions (Macey and Harris, 2006). Similar to what has been done for hematite, the test can be conducted by comparing the β18O-factors of hematite calculated ab initio and derived from experimental data on the temperature shift in the Mössbauer spectra and heat capacity. These estimates for the β18O-factor of hematite practically exactly coincide: even at room temperature, the difference in 103lnβ is ~0.3 (0.4% of the 103lnβ value), and this difference further diminishes with increasing temperature.

Analogously, data can be made consistent for grossular, for which two calculations are available (Krylov and Glebovitsky, 2017; Schauble and Young, 2020), both based on the density of states functional approach but obtained using different methods. The consistency of these data is slightly poorer (~3.3% of the 103ln β value) but is still acceptable (Fig. 5). Both computations are in a good agreement with the experimental fractionation coefficients for grossular–calcite (Rosenbaum and Mattey, 1995, 1996) and grossular–aqueous supercritical fluid. The values of the β18O-factor of grossular accepted in GEOCHEQ_Isotope obtained by averaging the two calibrations (Table 1).

Fig. 5.
figure 5

Comparison of the β18O-factor of grossular calculated by different techniques with experimental data.

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The usage of β-factors in databases requires the values of these factors computed for broad temperature ranges. This is most commonly done in theoretical calculation studies, but sometimes no such calculation data are available. In such situations, data obtained by various authors can be made consistent. This situation occurs with the β18O-factors for magnetite. Reliable experimental data are available for the equilibrium fractionation of oxygen isotopes in the system magnetite–H2O at temperatures higher than 300oC (Cole et al., 2004), and there are unsystematic scarce data on isotope fractionation in the system magnetite–H2O at low temperatures (Blattner et al., 1999; Zhang et al., 1995; Mandernack et al., 1999). We used these experimental data to plot an interpolation polynomial of form (4) for the magnetite–H2O fractionation curve (Fig. 6). To obtain a polynomial for the β18O-factor of magnetite, the coefficients of the polynomial for water (liquid or supercritical fluid, T > 647 K) from Table 1 were added to the coefficients of the polynomial for the fractionation curve. The polynomials thus obtained were used to plot the temperature dependence of the β18O-factor of magnetite. This curve was smoothed by approximating it again using a polynomial of form (4). The results are summarized in Table 4.

Fig. 6.
figure 6

Equilibrium fractionation of oxygen isotopes (18O/16O) in the magnetite–H2O system.

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Above, we described the main principles, methods, and techniques used to reconcile the data and compile a base of natural isotopic data in GEOCHEQ_Isotope. Detailed discussions of particulars of some related issues is beyond the scope of this paper and shall be rather referred to a manual for the user of this software package.

SIMULTANEOUS COMPUTATIONS OF EQUILIBRIUM OXYGEN AND CARBON ISOTOPE EFFECTS AND CHEMICAL COMPOSITION IN A HYDROTHERMAL SYSTEM

The GEOCHEQ_Isotope software now enables one to simulate chemical processes with regard to oxygen and carbon isotope effects. As an illustrative example of this simulation, we chose a system, which is analogous to that described in (Mironenko et al., 2018). The calculation method of chemical equilibria with regard to both oxygen and carbon isotope equilibria is generally analogous to those applied to models that involved estimates only of equilibrium carbon isotopic effects (Mironenko et al., 2018, 2021; Sidkina et al., 2019). A natural change was the addition of the 18O isotope as an independent component and 18O isotopologues to the list of substances subject to Gibbs energy minimization. The molality of the components of aqueous solutions was calculated by formula (5).

As an example of the application of GEOCHEQ_Isotope, we have computed chemical and isotope equilibria in CO2-bearing hydrothermal system of the bulk composition 1 mole CO2 + 1 kg H2O + 1.4 moles CO2 at 15, 25, and 40oC and a pressure of 40 bar, depending on pH. The 18O/16O and 13C/12C isotope ratios were specified in the system according to the natural abundances of these isotopes. We considered 16O-, 18O-, 12C-, and 13C-isotopologues of H2O, CO2(aq), \({\text{HC}}{{{\text{O}}}_{{\text{3}}}}^{ - }\), \({\text{C}}{{{\text{O}}}_{{\text{3}}}}{{^{{\text{2}}}}^{{\text{ - }}}}\),CaCO3(aq), \({\text{CaHC}}{{{\text{O}}}_{{\text{3}}}}^{{\text{ + }}}\), CO2(gas), CH4(gas), and H2O(gas). For this system, the dependence of the composition of dissolved inorganic (carbonate) carbon on the equilibrium fractionation of oxygen isotopes between solute carbonate species and water depending on pH have been studied experimentally and theoretically (Beck et al., 2005). These experimental results only insignificantly different from the carbonate system discussed herein. In the GEOCHEQ_Isotope simulations, we analyzed the possibility of formation of the solute CaCO3(aq) and \({\text{CaHC}}{{{\text{O}}}_{3}}^{ - }\) species, whereas these species were ignored in (Beck et al., 2005). At the same time, theoretical modeling in (Beck et al., 2005) involved the solute H2CO3 species. It should be stressed that the H2CO3 concentration was, according to the modeling results, close to 0.2–0.3% of the total concentration of dissolved carbonate carbon at pH < 5 and drastically decreased at increasing pH. The results obtained by GEOCHEQ_Isotope simulation of the composition of the carbonate species are shown in Fig. 7. Our simulation results agree well with earlier results in (Beck et al., 2005).

Fig. 7.
figure 7

Dependence of the composition of dissolved carbon on pH: data of GEOCHEQ_Isotope simulations. Various species are dominant at different pH values. Calcium-bearing species only insignificantly contribute to the overall budget of carbonate carbon. The \({\text{CaHC}}{{{\text{O}}}_{{\text{3}}}}^{{\text{ + }}}\) fraction reaches 7% only at low pH.

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The dominance of one or another solute species of carbonate carbon depending on pH, with regard to the difference in the values of the β18O- and β13C-factors of the solute carbonate species (Table 1, Fig. 1), predetermines the dependence of the equilibrium fractionation coefficient of oxygen and carbon isotopes between total dissolved inorganic carbon and coexisting phases. Figure 8 shows examples of such dependences simulated with GEOCHEQ_Isotope. For oxygen isotopes, reliable experimental data (Beck et al., 2005) are currently available on the equilibrium fractionation of dissolved carbonate carbon and water at temperatures of 15–40°C. Our simulation results are in good agreement with these experimental data (Fig. 8). The decrease in dissolved carbonates in the heavy oxygen isotope relative to water at increasing pH (Fig. 8) is explained by the successive changes in the dominant carbonate species CO2(aq) → \({\text{HCO}}_{3}^{ - }\) → \({\text{CO}}_{3}^{{2 - }}\), along with that the equilibrium fractionation factors between these carbonate species and water (and hence, their β18O-factors) decrease in the same sequence (Fig. 1). The situation with carbon isotopes is opposite: the β13C-factors of the aforementioned carbonate species occur in the opposite order (Mironenko et al., 2021), and hence, the depletion of dissolved carbonate carbon in the 13C heavy isotope is reduced as pH is increased. The isotope shift for carbon between dissolved carbonate carbon and calcite at high pH shows an inversion, because the β13C-factor for the dominant carbonate species \({\text{C}}{{{\text{O}}}_{{\text{3}}}}^{{{\text{2}} - }}\) is greater than that of calcite (Fig. 8).

Fig. 8.
figure 8

Dependence of the equilibrium fractionation factors of oxygen isotopes (in the system dissolved inorganic carbon–water) and carbon isotopes (in the system dissolved inorganic carbon–calcite) on the pH of the solution.

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CONCLUSIONS

We present an extension for the database of the GEOCHEQ_Isotope software package. This extension was designed to enable simultaneous simulations of chemical and isotope equilibria by means of minimizing the Gibbs free energy for the calculation of oxygen isotope equilibria. Similar to what has been done with carbon isotopes, this software version is underlain by the β-factor formalism in an approximation of an ideal mixture of isotopes.

The presented version of GEOCHEQ_Isotope uses the GEOCHEQ database (Mironenko et al., 2000) to describes nonisotope thermodynamic properties of substances. The Gibbs energy of the isotopologues is calculated using known values of the β-factors according to Eq. (1), with regard to additivity relations between the lnβ of a compound as a whole and lnβi of the monosubstituted isotopologues.

In this version, the computational algorithms are adapted to situations when oxygen isotopes are analyzed, and the accuracy of the computations is improved. In particular, the current equation for calculating the molality takes into account the presence of the H218O isotopologue.

Available information on equilibrium fractionation factors of oxygen isotopes has been critically analyzed, and the data have been made mutually consistent. The temperature dependences of the oxygen β-factors are unified and written in the form of polynomials (4) in reciprocal absolute temperature.

The application of the GEOCHEQ_Isotope software package for simultaneous calculations of chemical, oxygen and carbon isotope equilibria was tested by simulating the dependence of the composition of solute carbonate species and the 18O/16O equilibrium fractionation coefficient between solute carbon and water, depending on the pH of the solution. The data are in a good agreement with experimental results (Beck et al., 2005). The enrichment of dissolved carbonate carbon in the 18O heavy isotope decreases with increasing pH because of a systematic change of the dominant carbonate specie in the sequence CO2(aq) → \({\text{HCO}}_{3}^{ - }\) → \({\text{CO}}_{3}^{{2 - }}\), which is characterized by a decrease in the values of the β18O-factor (\({{\beta }^{{{\text{18}}}}}{{{\text{O}}}_{{{\text{C}}{{{\text{O}}}_{2}}}}} > {{\beta }^{{18}}}{{{\text{O}}}_{{{\text{HCO}}_{3}^{ - }}}} > ~~{{\beta }^{{18}}}{{{\text{O}}}_{{{\text{CO}}_{{\text{3}}}^{{{\text{2}} - }}}}}\)). For carbon, a pH increase leads to an increase in the fractionation coefficient for dissolved carbonate carbon–calcite because \({{\beta }^{{{\text{13}}}}}{{{\text{C}}}_{{{\text{CO}}}}}_{{_{{\text{2}}}}} < {{\beta }^{{13}}}{{{\text{C}}}_{{{\text{HCO}}_{3}^{ - }}}} < ~~{{\beta }^{{13}}}{{{\text{C}}}_{{{\text{CO}}_{{\text{3}}}^{{{\text{2}} - }}}}}\). At high pH (~11), the equilibrium isotope shift exhibits an inversion, and the dissolved inorganic carbon enriches in the 13C isotope relative to calcite.