Temperature dependence of isotopic fractionation in the CO₂‐O₂ isotope exchange reaction

Abstract

Rationale

Oxygen isotope exchange between O₂ and CO₂ in the presence of heated platinum (Pt) is an established technique for determining the δ ¹⁷O value of CO₂. However, there is not yet a consensus on the associated fractionation factors at the steady state.

Methods

We determined experimentally the steady‐state α ¹⁷ and α ¹⁸ fractionation factors for Pt‐catalyzed CO₂‐O₂ oxygen isotope exchange at temperatures ranging from 500 to 1200°C. For comparison, the theoretical α ¹⁸ equilibrium exchange values reported by Richet et al. (1997) have been updated using the direct sum method for CO₂ and the corresponding α ¹⁷ values were determined. Finally, we examined whether the steady‐state fractionation factors depend on the isotopic composition of the reactants, by using CO₂ and O₂ differing in δ¹⁸O value from −66 ‰ to +4 ‰.

Results

The experimentally determined steady‐state fractionation factors α ¹⁷ and α ¹⁸ are lower than those obtained from the updated theoretical calculations (of CO₂‐O₂ isotope exchange under equilibrium conditions) by 0.0024 ± 0.0001 and 0.0048 ± 0.0002, respectively. The offset is not due to scale incompatibilities between isotope measurements of O₂ and CO₂ nor to the neglect of non‐Born‐Oppenheimer effects in the calculations. There is a crossover temperature at which enrichment in the minor isotopes switches from CO₂ to O₂. The direct sum evaluation yields a θ value of ~0.54, i.e. higher than the canonical range maximum for a mass‐dependent fractionation process.

Conclusions

Updated theoretical values of α ¹⁸ for equilibrium isotope exchange are lower than those derived from previous work by Richet et al. (1997). The direct sum evaluation for CO₂ yields θ values higher than the canonical range maximum for mass‐dependent fractionation processes. This demonstrates the need to include anharmonic effects in the calculation and definition of mass‐dependent fractionation processes for poly‐atomic molecules. The discrepancy between the theory and the experimental α ¹⁷ and α ¹⁸ values may be due to thermal diffusion associated with the temperature gradient in the reactor.

1. INTRODUCTION

Measurements of Δ′¹⁷O (defined in Equation 1) of oxygen‐containing molecules can provide insight into, for example, the carbon cycle and its link to the hydrological cycle ¹ , ² , ³ , ⁴ , ⁵ , ⁶ , ⁷ , ⁸ , ⁹ , ¹⁰ and are also used to reconstruct paleoclimate. ¹⁰ , ¹¹ , ¹² , ¹³ , ¹⁴ The Δ′¹⁷O of CO₂ is a promising tracer to study the CO₂ exchange flux between the biosphere/hydrosphere and the atmosphere. ³ , ⁵ , ⁶ , ¹⁵ However, due to the mass interference of ¹³C¹⁶O¹⁶O and ¹²C¹⁷O¹⁶O, it is difficult to measure δ ¹⁷O of CO₂ with existing isotope ratio mass spectrometers, because the required mass resolving power of about 55,000 (Δm/m) is a challenge for even the highest resolution instruments. ¹⁶ High precision and accuracy are needed for measurements of both δ ¹⁸O and δ ¹⁷O in order to capture the small, but non‐zero deviations to use Δ′¹⁷O as a tracer. Since the measurement of δ ¹⁷O directly from CO₂ with isotope ratio mass spectrometry (IRMS) is impossible due to the interference of δ ¹³C, except from the fragment ions formed in the ion source, ¹⁶ indirect methods have been developed that enable the measurement of δ ¹⁷O of CO₂. These techniques are either based on isotope exchange or chemical conversion of CO₂ and are described in more detail in Adnew et al ¹⁶ and Hofmann and Pack. ¹⁷

$${\mathrm{\Delta }}^{\prime 17}\mathrm{O}=\ln \left({\delta }^{17}\mathrm{O}+1\right)-0.528\ln \left({\delta }^{18}\mathrm{O}+1\right)$$ (1)

The CO₂‐O₂ exchange method was first developed by Mahata et al ¹⁸ and has been established in several laboratories, including our own. ¹¹ , ¹² , ¹⁶ , ¹⁹ Recently, the method has been extended to measure the Δ′¹⁷O of water. ²⁰ The CO₂‐O₂ exchange method to measure Δ′¹⁷O of CO₂ is less labor‐intensive compared with other conversion/exchange methods and is capable of an impressive precision of < 0.01 ‰. Such precision allows the subtle seasonal Δ′¹⁷O variability of tropospheric CO₂ to be quantified, with reported seasonal amplitudes between 0.02 ‰ and 0.13 ‰. ² , ⁶ , ²¹

The CO₂‐O₂ exchange method is based on facilitating complete oxygen isotope exchange between CO₂ and O₂ at high temperature in the presence of a platinum catalyst. This way, the oxygen isotopic composition of the CO₂ is imprinted on O₂ and the ¹⁷O content of the CO₂ sample can be calculated from measurement of the isotopic composition of O₂ before and after the isotope exchange. Important parameters in the calculation of the Δ′¹⁷O of CO₂ are the molar ratio of CO₂ and O₂ and the isotopic fractionation factors (α).

$${\alpha }^I\left({\mathrm{CO}}_2/{\mathrm{O}}_2\right)=\frac{{\left({{}_{}{}^{\mathrm{I}}\mathrm{R}}_{\mathrm{f}}\right)}_{{\mathrm{CO}}_2}}{{\left({{}_{}{}^{\mathrm{I}}\mathrm{R}}_{\mathrm{f}}\right)}_{{\mathrm{O}}_2}}=\frac{{\left({\delta }^{\mathrm{I}}{\mathrm{O}}_{\mathrm{f}}+1\right)}_{{\mathrm{CO}}_2}}{{\left({\delta }^{\mathrm{I}}{\mathrm{O}}_{\mathrm{f}}+1\right)}_{{\mathrm{O}}_2}}$$ (2)

The index I is a placeholder for either 17 or 18. The fractionation factors α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) determine the distribution of the heavy isotopes between O₂ and CO₂ at isotopic steady state.

The CO₂‐O₂ exchange technique has been used in different laboratories to study the carbon cycle ¹ , ² , ³ , ⁴ , ⁶ , ²² , ²³ and precise measurement of Δ′¹⁷O is used to correct for biases in clumped isotope measurements of CO_2. ²⁴ , ²⁵ , ²⁶ , ²⁷ However, the fractionation factor in CO₂‐O₂ isotope exchange is not well established, and different values have been reported by different laboratories.

One explanation for these differences may be variations in the effective isotope exchange temperature, as the fractionation factor is expected to be temperature dependent. Furthermore, the mechanism of the CO₂‐O₂ isotope exchange in the presence of a platinum catalyst is unknown. ¹⁶ , ¹⁸ , ¹⁹ , ²⁶ , ²⁸ , ²⁹ In this study we determined α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) for the CO₂‐O₂ exchange method over a temperature range from 500°C to 1200°C. We investigated whether there is a dependency on the initial oxygen isotopic composition of CO₂ and O₂. This also addresses a potential effect of an isotope scale difference between CO₂ and O₂. We revisit and update theoretical calculations of the isotope equilibrium by Richet et al. (1997) ³⁰ and calculated the three‐isotope exponent (θ) of the CO₂‐O₂ equilibrium isotope exchange for a temperature range of 0°C to 1200°C.

2. MATERIALS AND METHODS

2.1. Theoretical calculations of CO₂‐O₂ isotope exchange at equilibrium

Fractionation factors ¹⁷ α and ¹⁸ α for the CO₂‐O₂ isotope exchange can be calculated from statistically corrected equilibrium constants of O‐atom exchange reactions with CO₂ and O₂, respectively. The latter are conveniently designated as β‐factors and were originally introduced as fractionation factors between the fully substituted molecule and O‐atom (^I β _A  = ([^IO]/[¹⁶O]) _A /([^IO]/[¹⁶O])_Atomic−O, where A is either CO₂ or O₂). ³⁰ , ³¹ , ³² , ³³ Since in the measurements of the equilibrium constants multi‐substituted species have been neglected, we define quantities α and β slightly differently:

$$\begin{array}{rcl}{\alpha }^I& =& \frac{Q\left(\mathrm{C}{}_{}{}^{16}\mathrm{O}{}_{}{}^I\mathrm{O}\right)}{Q\left(\mathrm{C}{}_{}{}^{16}{\mathrm{O}}_2\right)}\frac{Q\left({}_{}{}^{16}{\mathrm{O}}_2\right)}{Q\left({}_{}{}^{16}\mathrm{O}{}_{}{}^I\mathrm{O}\right)}\\ & =& \frac{Q\left(\mathrm{C}{}_{}{}^{16}\mathrm{O}{}_{}{}^I\mathrm{O}\right)Q\left({}_{}{}^{16}\mathrm{O}\right)}{Q\left(\mathrm{C}{}_{}{}^{16}{\mathrm{O}}_2\right)Q\left({}_{}{}^I\mathrm{O}\right)}/\frac{Q\left({}_{}{}^{16}\mathrm{O}{}_{}{}^I\mathrm{O}\right)Q\left({}_{}{}^{16}\mathrm{O}\right)}{Q\left({}_{}{}^{16}{\mathrm{O}}_2\right)Q\left({}_{}{}^I\mathrm{O}\right)}=\frac{{}_{}{}^I{\beta }_{{\mathrm{CO}}_2}}{{}_{}{}^I{\beta }_{{\mathrm{O}}_2}},\end{array}$$ (3)

where $Q$ are the total partition functions. For the calculation of these exchange equilibria, molecular partition functions were divided into internal and translational parts. Total internal partition functions and ratios of diatomic molecules were calculated using the formalism of Urey ³⁴ and Bigeleisen and Mayer, ³⁵ including anharmonic corrections and zero‐point energy expressions based on Dunham's ³⁶ expression of energy levels (see Richet et al. (1997) ³⁰ ). Molecular constants of the ¹⁶O₂ species were taken from Huber and Herzberg ³⁷ in combination with recent atomic masses ³⁸ and those of the isotopic variants were calculated using the standard mass dependence of Dunham coefficients. This choice of molecular constants is compatible with a recent recommendation ³⁹ and an agreement to better than five significant digits is obtained across the temperature range. However, our current results are slightly different from a previously published parametrization ⁴⁰ – the less than 0.2 ‰ difference being mainly due to the identification and removal of an error in the calculation of a correction term.

Unfortunately, application of Dunham's approach to larger molecules is not straightforward, because the isotope scaling of anharmonic vibrational constants is neither theoretically well constrained nor experimentally known, and the constant zero‐point energy offset G ₀ is not known either. ³⁰ We have therefore calculated the total internal partition functions of the ¹⁶O‐, ¹⁷O‐, and ¹⁸O‐containing isotopologues of ^IO¹²C¹⁶O as direct sums of overall energy levels obtained from the spectroscopically adjusted ab‐initio line list calculations of Huang et al ⁴¹ , ⁴² (see Prokhorov et al ⁴³ for more details).

Our new calculations were verified by comparing atom‐diatom isotope exchange equilibria (β‐values) with the tables of Richet et al. (1997) ³⁰ using several isotopes and diatomic molecules (¹³C exchange in CO and CS, ¹⁵N exchange in N₂, and ¹⁸O exchange in CO and O₂). These results agreed across the temperature range between 0°C and 1200°C, to better than 0.7 ‰ for all molecules except for O₂, where our values were higher by 4 ‰ at 0°C and 1 ‰ at 1200°C (see Table S1, supporting information). We have discovered a subtle difference in the molecular constants used in this study and that of Richet et al. (1997) ³⁰ but such slight differences cannot account for the observed discrepancy. Interestingly, Richet et al. (1997) ³⁰ report the same 4 ‰ discrepancy when comparing their 25°C results with the calculations of Urey ³² (see Table 16 in Richet et al. (1997) ³⁰ ), while the β‐values for O‐, C‐, or N‐exchange with other diatomic molecules always agree with the work of Urey to within 1 ‰. It needs be pointed out that the discrepancy is not due to our slightly different definition of β and our neglect of excess factors, since it even increases if we calculate β‐values from doubly substituted O₂. This indicates that an error must have occurred in the O₂ calculations of Richet et al. (1997). ³⁰

When comparing our atom‐CO₂ exchange equilibria with Richet et al. (1997) ³⁰ the agreement is better, but still worse compared with diatomic gases when O₂ is excluded from the comparison. For the ¹⁸O exchange, our β‐values are 2 ‰ higher at 0°C and less than 1 ‰ for relevant temperatures above 400°C (see supporting information). For ¹³C exchange, our values at 0°C are 1.3 ‰ higher and the discrepancy becomes already smaller than 1 ‰ at temperatures above 100°C.

The comparison of α ¹⁸(CO₂/O₂) between our calculation and that of Richet et al. (1997) ³⁰ for the temperature range between 0°C and 1200°C is also shown in Table S2 (supporting information). Our α¹⁸(CO₂/O₂) value agrees with that reported by Richet et al. (1997) ³⁰ to within 1 ‰, for temperatures higher than 450°C. However, at low temperature the difference is higher than 1 ‰ (for instance 2.2 ‰ at 0°C; see Table S2, supporting information). This is because our β‐factors for both O₂ and CO₂ are different from the values reported by Richet et al. (1997) ³⁰ the difference being largest in O₂, where we suspect the calculation of Richet et al. (1997) ³⁰ is erroneous (see Table S1, supporting information).

We have compared partition function ratios Q(¹³CO₂)/Q(¹²CO₂) of ¹⁶O‐ and ¹⁸O‐containing isotopologues also with results from path integral Monte Carlo calculations of Webb and Miller ⁴⁴ over the 300 to 600 K range and found deviations of less than 0.5 ‰ and 0.3 ‰ at 300 K and 600 K, respectively. This confirms that CO₂‐related uncertainties of ¹⁸ α should be smaller than 0.3 ‰ for most of the temperatures in this study. Higher electronic states of both molecules have been neglected in the calculations. At transition energies of 5.7 eV ($1^1{A}_1^{\prime \prime }$ CO₂ ⁴⁵ ) and 0.98 eV ($a^1{∆}_g$ O₂ ³⁷ ), the population of electronically excited CO₂ is completely negligible and the O₂(¹ $∆$) ground state is only marginally populated (by less than 8 parts in 10⁵), even at the highest temperature. The effect on isotope ratios is much smaller because geometric changes in excited oxygen isotopes are small, and vibrational and rotational constants of the excited state are similar to the ground state values (changing by 7 % at most). ³⁷

2.2. CO₂‐O₂ exchange system

A detailed description of the CO₂‐O₂ exchange method established at Utrecht University is given elsewhere. ¹ , ¹⁶ In brief, equal amounts of CO₂ and O₂ were allowed to equilibrate in a quartz reactor in the presence of platinum sponge as a catalyst (purity ≥ 99.9 %, Sigma Aldrich, USA) at the bottom of the reactor. The CO₂‐O₂ exchange experiments were performed at temperatures between 500 and 1200°C. The temporal evolution of the isotope exchange was determined at 650°C, 750°C, and 850°C by varying the duration of the CO₂‐O₂ exchange. After the exchange reaction, CO₂ and O₂ were separated cryogenically and the O₂ collected at liquid nitrogen temperature on molecular sieve 5 Å pellets (3 pellets, 5 mm long, 1.8 mm o.d., Sigma Aldrich, USA). The isotope measurements were performed on a Delta V Plus isotope ratio mass spectrometer (Thermo Scientific, Germany) in dual inlet mode. A schematic of the exchange reactor is shown in Figure 1. For comparison, the reactor that was used in Adnew et al. (2019) ¹⁶ is shown in Figure S1 (supporting information). The main difference between the current setup and the one used in the previous study ¹⁶ is an electropolished stainless steel extension.

FIGURE 1.

Geometry and dimensions of the CO₂‐O₂ reactor

2.3. Samples

Three commercial O₂ gases with different isotope compositions were used for the measurements (Table 1). The working reference gas (GO₂‐1) has δ ¹⁷O = 9.235 ± 0.011 ‰ and δ ¹⁸O = 18.514 ± 0.011 ‰ (Table 1), determined by Eugeni Barkan, Hebrew University of Jerusalem. For CO₂ we used two gases from high‐pressure cylinders (GCO₂‐3 and GCO₂‐4) and two additional gases (GCO₂‐1 and GCO₂‐2) that were prepared by combusting a graphite rod with two isotopically different O₂ gases ¹⁶ , ⁴⁶ , ⁴⁷ , ⁴⁸ , ⁴⁹ (GO₂‐1 and GO₂‐2, respectively). The isotopic compositions of the CO₂ and O₂ used in this study are shown in Table 1. The experimental procedure for the combustion experiment is described in detail elsewhere. ¹⁶ In summary, the graphite rod (99.9995% purity, 3.05 mm × 20 mm, Alfa Aesar, Part No: 40765, Thermo Fisher Scientific) was wrapped in a sheet of platinum foil and platinum wire and placed inside a quartz reactor. The combustion with O₂ was performed at 800°C. Before starting the experiment, the graphite rod was cleaned for more than 24 h by heating at 1000°C in high vacuum. After each conversion experiment, the graphite rod was again cleaned by heating at 1000°C for at least 1 h.

TABLE 1.

Overview of names, suppliers, and isotopic compositions of the CO₂ and O₂ working standards used in this study. All the CO₂ gases used had a purity of > 99.995 % except for the ones that were produced in the laboratory by combustion of a graphite rod. O₂ gases had a purity of > 99.998 %. All isotopic values are given in ‰, relative to VPDB for δ¹³C and relative to VSMOW for δ¹⁷O and δ¹⁸O. The errors are reported as one standard deviation (σ)

CO2 gases used in this study
Name	Source/supplier	δ 13C	δ 18O	Δ′17O
GCO2‐1	Combustion	−26.160 ± 0.030	17.921 ± 0.042	−0.465 ± 0.018
GCO2‐2	Combustion	−26.154 ± 0.059	−38.205 ± 0.051	−0.518 ± 0.016
GCO2‐3	Linde Gas, The Netherlands	−31.665 ± 0.005	34.655 ± 0.011	−0.242 ± 0.003
GCO2‐4	Air Products, Germany	−10.332 ± 0.005	30.082 ± 0.011	−0.169 ± 0.002

O2 gases used in this study
Name	Source/supplier	δ17O	δ18O	Δ′17O
GO2‐1	Air Products, The Netherlands	9.235 ± 0.011	18.514 ± 0.011	−0.494 ± 0.011
GO2‐2	Air Liquide, The Netherlands	−20.734 ± 0.017	−37.862 ± 0.012	−0.573 ± 0.017
GO2‐3	Linde Gas, The Netherlands	7.040 ± 0.021	14.095 ± 0.011	−0.375 ± 0.020

3. RESULTS

The differences in isotopic composition of the two working standards (GCO₂‐3 and GCO₂‐4) are 4.440 ± 0.015 ‰ for δ ¹⁸O and −21.556 ± 0.007 ‰ for δ ¹³C. respectively (Table 1). In our laboratory, we measured relative differences of 4.479 ± 0.003 ‰ for δ ¹⁸O and −21.448 ± 0.001 ‰ for δ ¹³C, respectively, from six replicates. For each measurement new gas was introduced to the bellows. Similarly, for GO₂‐1 and GO₂‐2, the relative difference was −29.690 (−29.746) ‰ for δ ¹⁷O and −55.456 (−55.432) ‰ for δ ¹⁸O, respectively. The values in parentheses are the assigned differences based on the measurements at Hebrew University (E. Barkan). Overall, the assigned and measured values agree well.

3.1. CO₂‐O₂ isotope steady state

Figure 2 shows the temporal evolution of the CO₂‐O₂ isotope exchange for temperatures of 650°C, 750°C, and 850°C. The exchange proceeds faster at higher temperatures, as expected.

FIGURE 2.

Temporal evolution of δ ¹⁷O (O₂) (A) and δ ¹⁸O (O₂) (B) during CO₂‐O₂ exchange on a platinum sponge catalyst in a quartz reactor, for reaction temperatures of 650°C, 750°C, and 850°C. Δδ¹⁷O(O₂) and Δδ¹⁸O (O₂) are the differences in δ¹⁷O and δ¹⁸O, respectively, between the initial and final isotopic compositions of O₂ [Color figure can be viewed at wileyonlinelibrary.com]

The data sets at each of the three reaction temperatures are fitted to an exponential curve of the form y = A + B × e ^{−k × t} , where y is the isotopic composition (δ¹⁷O or δ¹⁸O) of O₂ at time t, A is the isotopic composition at long time scales (Figure 2), A + B is the isotopic composition of O₂ at t = 0, k is the rate constant of the reaction, and t is the reaction time. In our experiments we defined steady state as having been achieved when k determined from the fit for each temperature e ^−kt < 10⁻⁶, i.e. t = 13.816/k. From these fits, the respective steady state values of δ ¹⁷O_f and δ ¹⁸O_f of the O₂ are 2.684 ‰ and 5.032 ‰ at 650°C; 3.018 ‰ and 5.648 ‰ at 750°C; and 3.278 ‰ and 6.100 ‰ at 850°C. The enrichments of ¹⁷O and ¹⁸O in the O₂ at steady state increase with reaction temperature (Figures 2 and 3A).

FIGURE 3.

A, δ ¹⁸O(O₂) and δ ¹⁸O(CO₂) after CO₂‐O₂ exchange at isotope exchange steady state for reaction temperatures ranging from 500°C to 1200°C. B, The corresponding fractionation factors α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂). The vertical dashed line shows the temperature at which α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) cross the value of 1 (horizontal dashed line). In the fit function, T is the temperature in °C [Color figure can be viewed at wileyonlinelibrary.com]

Figure 3A shows the steady state isotopic composition δ ¹⁸O_f (O₂) and δ ¹⁸O_f (CO₂) for reaction temperatures ranging from 500 to 1200°C. In Figure 3B the fractionation factors α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) derived from these steady state enrichments are shown. As can be seen, α ¹⁷ and α ¹⁸ decrease with increasing reaction temperature. The exchange reactions were carried out using gases GCO₂‐4 (δ ¹⁸O = 30.082 ‰) and GO₂‐3 (δ ¹⁸O = 14.095 ‰), see Table 1. Figure 3 shows that below 810°C, δ ¹⁸O_f (CO₂) > δ ¹⁸O_f (O₂) and α ¹⁸(CO₂/O₂) > 1; thus there is a slight preference for the CO₂ to be relatively enriched in ¹⁷O and ¹⁸O. At higher temperatures, this preference is reversed, with α ¹⁸ < 1, i.e. δ ¹⁸O_f (CO₂) < δ ¹⁸O_f (O₂).

3.2. Dependence of α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) on the relative difference between the initial oxygen isotopic compositions of CO₂ and O₂

The isotopic steady state should be a property of the reaction system and independent of the isotopic composition of the initial reagents. To confirm this, we used O₂ and CO₂ with a wide range of isotopic compositions, with a δ ¹⁸O difference of the initial O₂ and CO₂ by up to −65.96 ‰ in exchange experiments at 650°C, 750°C, and 850°C.

Figure 4 demonstrates that the same values for α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) are reached in experiments where the isotopic composition of the starting gases is very different, confirming that α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) values are independent of the initial isotopic composition of CO₂ and O₂.

FIGURE 4.

Dependence of α ¹⁷(CO₂/O₂) (blue symbols) and α ¹⁸(CO₂/O₂) (black symbols) on the initial isotopic composition of CO₂ and O₂ (see legend for the relative difference in δ ¹⁸O between CO₂ and O₂). The rectangle and star symbols are for CO₂ samples produced by the combustion of a graphite rod with O₂. Δδ ¹⁸O is the initial δ ¹⁸O difference between CO₂ and O₂ [Color figure can be viewed at wileyonlinelibrary.com]

3.3. Comparison with theoretical calculations

Figure 5A shows that α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) depend on temperature. This holds for both the experimental steady state results and the theoretically calculated thermodynamic equilibrium values, with the shapes of the temperature dependence being similar between theory and experiment. Interestingly, the calculated results are higher than the experimental data. As shown in Figure 6, the offset between the experimental values and theoretical equilibrium calculation is not constant but has a slight temperature dependence.

FIGURE 5.

A, Dependence of measured α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) on temperature, together with data from theoretical equilibrium calculations. In B) the temperature scale has been adjusted to that of the experiments and the theoretically derived equilibrium values for α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) (solid lines) have been corrected for the respective mean offsets in α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) (dashed lines) [Color figure can be viewed at wileyonlinelibrary.com]

FIGURE 6.

Temperature dependence of the experimental α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) offsets from the theoretical equilibrium values

The average offset between the theoretically calculated and the experimental values is 0.0024 ± 0.001 and 0.0048 ± 0.002 for α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂), respectively. This offset is very close to mass‐dependent fractionation with a triple oxygen isotope fractionation (θ) value of 0.5. For the CO₂‐O₂ exchange experiment, θ is calculated as shown in Equation 4.

$$\theta \left({\mathrm{CO}}_2/{\mathrm{O}}_2\right)=\frac{\ln \left({\mathrm{\alpha }}^{17}\left({\mathrm{CO}}_2/{\mathrm{O}}_2\right)\right)}{\ln \left({\alpha }^{18}\left({\mathrm{CO}}_2/{\mathrm{O}}_2\right)\right)}$$ (4)

In Figure 5B we have shifted the theoretical α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) by the average offset from the experimental value, which results in a very good agreement with the experimentally determined values. The difference between the theoretical and measured α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) values show a slight increase with temperature (Figure 6). This trend diminishes at temperatures above 800°C.

According to the theoretical calculations, the fractionation at higher temperatures should approach unity. However, as mentioned above, experimentally we observe a fractionation of less than unity at higher temperatures, where the δ ¹⁸O of the exchanged O₂ is higher than the δ ¹⁸O of CO₂ (Figures 3 and 4).

3.3.1. Scale difference between O₂ and CO₂

To check if the discrepancy of α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) between the experimental results and theoretical equilibrium estimates is due to the scale difference between the CO₂ and O₂, we used CO₂ formed from combustion of a graphite rod as described in section 2.3. The difference in δ ¹⁸O between GO₂‐3 and GO₂‐1 was −55.335 ‰. After combustion of the graphite rod with both GO₂‐1 and GO₂‐2, the relative difference between the produced GCO₂‐1 and GCO₂‐2 was −55.169 ‰. A potential scale difference is thus of the order 0.2 ‰, but these small differences between δ ¹⁸O of the oxygen used for combustion and the δ ¹⁸O of the resulting CO₂ might also be due to incomplete combustion and/or CO formation. ¹⁶ , ⁴⁶ , ⁴⁷ , ⁴⁸ , ⁴⁹ After assessing the potential scale discrepancies, we performed the same equilibration experiments with these gases at a reaction temperature of 750°C. As shown in Figure 4, the discrepancy between the theoretically estimated equilibrium fractionation in δ ¹⁷O and δ ¹⁸O, and the experimental results after isotopic exchange between CO₂ and O₂ at steady state, is similar for all combinations of CO₂ and O₂, including the gases where scale discrepancies were specifically assessed. This provides evidence that the offset is not due to scale differences between the O₂ and CO₂ isotope scales.

3.4. Dependence of θ (CO₂/O₂) on CO₂‐O₂ equilibration temperature

As shown in Figure 7, θ (CO₂/O₂) is dependent on the CO₂‐O₂ equilibration temperature. θ (CO₂/O₂) takes on extreme values close to the crossover (−∞ to ∞). In addition, for α values close to 1, small experimental errors will lead to large variability in θ (CO₂/O₂). The theoretical calculation does not show any crossover temperature, unlike the experimental values. Due to the crossover in the experiments between 750 and 800°C, θ values are largely outside the canonical range between 0.5 and 0.5305. ³¹ , ⁵⁰ , ⁵¹ , ⁵² , ⁵³ In contrast, the calculated θ (CO₂/O₂) value systematically exceeds the canonical upper limit θ _HTL = 0.5305 and remains close to 0.54 across the temperature range between 0°C and 1200°C. We suspect that the drop in θ (CO₂/O₂) at temperatures >1250°C in Figure 7 is an artefact, likely to be due to not including some high‐energy states in the calculations. ⁴³ , ⁵⁴ , ⁵⁵ This concerns mostly the rarest (¹⁷O containing) isotopologue, on which the least experimental information is available.

FIGURE 7.

Dependence of θ (CO₂/O₂) on reaction temperature and α ¹⁸(CO₂/O₂) value for experimental data (A and C) and theoretical calculation (B and D). In (A) and (C), the dashed line is the θ (CO₂/O₂) value calculated from the fit function of Figure 3: α ¹⁸(CO₂/O₂) = 0.024 exp(−0.002 T) + 0.996; α ¹⁷(CO₂/O₂) = 0.013exp(−0.002 T) + 0.998

4. DISCUSSION

4.1. Reaction rate constant of the CO₂‐O₂ isotope exchange reaction

The rate for the CO₂‐O₂ isotope exchange reaction in the presence of a platinum catalyst depends on the exchange temperature. As shown in Figure 2, isotope steady state between CO₂ and O₂ was reached in less than 1 h at reaction temperatures of 650°C, 750°C, and 850°C. However, at 500°C, the results indicated that the isotopic steady state was not reached within 2 h (α ¹⁷(CO₂/O₂) = 1.0010 and α ¹⁸(CO₂/O₂) = 1.0187), see Table S3 (supporting information). Mahata and co‐authors ²⁸ reported an increase in the δ ¹⁸O value of oxygen with temperature which they attributed to an enhancement in the catalytic activity of the platinum. Our data on the temperature dependence of α ¹⁸(CO₂/O₂) show (after accounting for an offset, discussed later) very good agreement with the theoretical equilibrium calculations over the entire temperature range of 500 to 1200°C. This demonstrates that the increase in the δ ¹⁸O value of the oxygen is not due to the efficiency of the catalyst but reflects the thermodynamically expected behavior. However, we did observe a decrease in the efficiency of the platinum sponge catalyst after it was exposed to a higher temperature. For example, after exposure to 1200°C, it took a long time to reach isotopic steady state at a lower temperature (e.g. at 650°C), which most likely indicates some kind of sintering of the platinum sponge, leading to a reduction in active surfaces and thus a decrease in catalytic efficiency. The efficiency of the catalyst only affects the duration required to attain CO₂‐O₂ isotope exchange, not the final isotopic composition.

4.2. α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) of the CO₂‐O₂ isotope exchange reaction

As described in section 3.3, the experimental values for α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) are lower than the corresponding theoretical equilibrium estimates, and they cross from values above 1 to values below 1 at higher temperatures, in contrast to the theoretical equilibrium calculations. One shortcoming of the current work is that the calculations of α values involve the assumption of the Born‐Oppenheimer approximation in the analysis of spectral data leading to the partition functions in Equation 2. However, this assumption is only approximate and it is possible that neglecting these effects could at least partly explain the difference between experiment and theory. Based on the work of Born and Huang, ⁵⁶ Zhang and Liu ⁵⁷ give an expression for the correction of a partition function ratio for a heavy and light isotope pair, which essentially implies that there is an additional constant shift (ΔE) in the energy levels between the heavy and the light molecules of an isotope pair, leading to an additional term exp(−ΔE/k _B T) in the calculation of the partition function ratios and the equilibrium constant (where k _B is Boltzmann's constant). These effects are strongly temperature dependent and vanish at high temperatures. Such behavior is not compatible with our observation of a temperature‐independent offset and we consequently exclude the notation non‐Born‐Oppenheimer effects could explain the observed discrepancy.

Mahata and co‐authors ¹⁸ also reported α ¹⁸(CO₂/O₂) < 1, ranging from 0.998128 to 0.99463, even at 680°C. Table 2 shows that all recently published α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) values are lower compared with the theoretically calculated equilibrium value for CO₂‐O₂ exchange temperatures of 600°C to 800°C. The presence of the platinum sponge, the geometry of the experimental setups, and the temperature gradient between the cold and hot zones of the reactor were postulated as possible sources of the discrepancy between the measured fractionation value and the theoretically estimated thermodynamic equilibrium value.

TABLE 2.

Theoretically calculated equilibrium values (this study) and experimentally determined values of α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) from this study (*) and from recent publications. All the experimentally determined values are lower than the theoretically calculated equilibrium ones

Temp. [°C]	Theoretical values (this study)		Experimental values			Ref.
	α 17(CO2/O2)	α 18(CO2/O2)	α 17(CO2/O2)	α 18(CO2/O2)	θ (CO2/O2)
600	1.0036	1.0067	‐	1.0031–1.00785		30
670	1.0032	1.0059	0.99977	0.9990	0.229911	29
680	1.0031	1.0058	‐	0.998 to 0.995		19
700	1.0030	1.0056	‐	0.9997–1.00414		30
750	1.0028	1.0052	1.00135	1.00227	0.594987	20
			1.00125	1.00215	0.581657	12
			1.00082	1.00141	0.581732	13
			1.000666	1.000998	0.667445	17
			1.00048	1.00051	0.941191	*
800	1.0026	1.0048	‐	0.9998–1.001269		30
			0.9990	0.9977	0.4345	28
			1.0002065	0.999988	−17.2065	*

Based on the results of the O₂‐CO₂ conversion experiments, the discrepancy between the experimental and theoretically calculated equilibrium values of α ¹⁸(CO₂/O₂) and α ¹⁷(CO₂/O₂) is not due to an isotopic scale difference between O₂ and CO₂. In addition, α ¹⁸(CO₂/O₂) and α ¹⁷(CO₂/O₂) values are independent of the initial oxygen isotope composition of the CO₂ and O₂ and also do not depend on the relative difference in the oxygen isotope composition between the initial CO₂ and O₂. Thus, the discrepancy must have other causes.

Oxygen isotope exchange between CO₂ or O₂ and the quartz reactor tube has been shown to be negligible in our experiments at 750°C. ¹⁶ , ¹⁹ We did not investigate whether it occurs at higher temperatures, but Barkan and Luz ⁴⁶ reported that isotope exchange between oxygen and quartz at temperatures of up to 950°C is negligible. Similarly, Saeger et al ²⁶ reported no significant exchange between quartz and O₂ nor between quartz and CO₂ at 800°C. The effect of oxygen isotope exchange with quartz will be higher if the δ ¹⁸O value of the quartz is very different from that of the CO₂ or O₂. Quartz glass tubing is mostly made from silica sand having δ ¹⁸O values ranging from 10 ‰ to 20 ‰, ⁵⁸ which is relatively close to the δ ¹⁸O value of the O₂ used for characterizing the CO₂‐O₂ exchange reaction in this study (GO₂‐3). The potential for isotope exchange should depend on the relative difference in oxygen isotopic composition between the quartz, CO₂, and O₂. However, even if there is isotope exchange with quartz, it only affects the δ ¹⁷O and δ ¹⁸O values of CO₂ and O₂ after exchange, not the α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) values.

A possible explanation for the discrepancy between the calculation and the experimental results is isotopic separation due to thermal diffusion. ⁵⁹ , ⁶⁰ , ⁶¹ , ⁶² , ⁶³ , ⁶⁴ In this study, one end of the reactor is positioned inside an oven at a specific temperature (500°C to 1200°C), whereas the other end is at room temperature (see Figure 1). The hot zone of the reactor is only 35 cm³ in volume and approximately half that is outside the oven (Figure 1). Due to thermal diffusion, the heavier isotopes will accumulate in the cold zone of the reactor compared to the hot zone. ⁵⁹ , ⁶⁰ , ⁶¹ , ⁶⁵ , ⁶⁶ The relative separation of the isotopologues is proportional to their mass difference. The thermal diffusion factor of the heavy isotopologue relative to the light isotopologue is (α ₂₁) = α _o(m ₂ − m ₁)/(m ₁ + m ₂) where m ₁ and m ₂ are the molecular masses of the light and heavy isotopologues, respectively, and α _o is the reduced thermal diffusion factor which depends only on temperature and the intermolecular force between isotopes. ⁶⁰ , ⁶⁷ A detailed description of thermal diffusion and the thermal diffusion factor is presented in the literature ⁶⁰ , ⁶⁷ and references cited therein. The isotope separation between the cold and hot zones for O₂ isotopologues is higher than for CO₂ isotopologues. ⁶⁰ , ⁶⁵ Thus, the O₂ in the cold zone of the reactor will be more enriched in heavy isotopes compared with CO₂, resulting in α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂) being lower than as calculated for thermodynamic equilibrium.

The effect of thermal diffusion should depend on the geometry and the temperature gradient between the cold and hot zones of the reactor. Horizontally oriented reactors are less convective and could result in larger isotope fractionation compared to a vertically oriented reactor. Mahata and co‐authors ²⁸ used a horizontal reactor, and in fact the results from their study show a higher discrepancy with respect to the theoretical calculations (3.4 ‰ and 6.9 ‰ for α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂), respectively; Table 2). If thermal diffusion is indeed responsible for the discrepancies between the theoretical equilibrium values and the experimental results, the offset is also expected to depend on the relative volumes of the cold and hot zones of the reactor. Indeed, at 750°C, Adnew et al ¹⁶ reported a slightly higher (thus closer to the theoretical value) value of α ¹⁸(CO₂/O₂), 1.000998 vs. 1.00051 (see Table 1), using a similar experimental setup except that in the previous study the volume of the cold zone of the reactor was smaller and without a stainless‐steel extension (see Figure 1 and Figure S1, supporting information). As shown in Figure 6, the offset between theory and experiment depends on the temperature gradient between the hot and cold zones of the reactor. The cross‐over temperature and the discrepancy between the theoretically calculated equilibrium and experimentally determined steady‐state α values might depend on the geometry of the reactor, the pressure in of the reactor, the volume of the reactor, and the proportion of the cold and hot part of the reactor.

Oxygen three‐isotope phenomena due to thermal diffusion have been investigated previously using pure gases ⁶³ , ⁶⁴ and the experiments indicate that fractionation effects are well established in static reactors exposed to spatial temperature gradients. Due to mass balance, fractionation effects in these studies disappeared when the temperature gradient was removed after the experiment and the system was allowed to equilibrate. This is different to our situation, where cooling the reactor to room temperature implies that the O₂‐CO₂ exchange shuts down.

The fact that the experimental α values differ from theoretically equilibrium calculated data demonstrates that dynamical effects drive the system out of thermodynamic equilibrium. While kinetic isotope fractionation might occur in processes on the surface or at the solid–gas interface, the presence of a catalytic surface alone cannot change the thermodynamic equilibrium in the gas phase, by its very definition. ⁶⁸ This implies that fractionation effects involving the catalytical substance are annihilated in forward and reverse reactions. ⁶⁸ Similar to isotope fractionation in evaporation, surface chemical kinetics can only induce fractionation between O₂ and CO₂ if rapid removal processes in the gas phase prevent equilibrium being established. ⁶⁹ The thermal gradient inducing a disequilibrium in the gas phase could possibly provide the required kinetic removal and thus potentially allow for surface kinetics to impact the O₂‐CO₂ equilibration. However, without further quantitative analysis, such effects cannot be distinguished from the isotope fractionation due to thermal diffusion, which also come into play once a thermal gradient is established. Contrary to previous claims, surface‐induced isotope effects during O₂‐CO₂ equilibration ²⁵ cannot be studied without considering the impact of thermal diffusion.

4.3. θ (CO₂/O₂) of the CO₂‐O₂ isotope exchange

Experimentally determined θ (CO₂/O₂) values are larger than the canonical high temperature limit (θ >0.5305), an observation already made in previous studies. ¹² , ¹⁹ , ⁷⁰ Fosu et al ¹² interpreted this as additional evidence that the CO₂‐O₂ isotope exchange must be kinetically controlled. The experimentally determined θ (CO₂/O₂) takes extreme values when ¹⁸ α (CO₂/O₂) is close to unity (near the crossover temperature where the sign of fractionation between CO₂ and O₂ changes, see Figure 7). These observations are a consequence of a singular behavior of θ close to the crossover, a well described anomaly in isotope equilibria. ³² , ⁷¹ , ⁷² , ⁷³ , ⁷⁴ , ⁷⁵ Nevertheless, the experimentally determined θ values may not represent the CO₂‐O₂ equilibrium exchange due to the possible thermal diffusion effect (see above) and other heterogenous process involving adsorption, desorption, and mixing. ¹² , ¹⁹ , ²⁸ , ²⁹ As a result, the experimentally determined θ values are steady‐state values.

It is interesting to investigate further the calculated θ (CO₂/O₂). Theoretically calculated values for ¹⁷ α (CO₂/O₂) and ¹⁸ α (CO₂/O₂) lead to values of θ = ln(¹⁷ α)/ln(¹⁸ α) which are outside the “canonical” range, i.e. θ > 0.5305 (see Figure 7). The upper bound of this conventional range arises from the high temperature limit (HTL) of the harmonic oscillator approximation (e.g. Young et al, ⁵⁰ Cao and Liu, ³¹ Dauphas and Schauble, ⁵³ Matsuhisa et al, ⁵¹ Kaiser et al ⁷⁶ ).

$${\theta }_{\mathrm{HTL}}=\frac{m_{16}^{-1}-m_{17}^{-1}}{m_{16}^{-1}-m_{18}^{-1}}=0.5305$$ (5)

Examples in support of the approximation usually come from considering diatomic molecules, such as CO or O₂, for example (Cao and Liu, ³¹ Wang et al ⁵⁴ ). Here, we provide evidence that this empirical rule may not hold for more complex molecules like CO₂. The calculated temperature dependence of the three‐isotope exponents for O + O₂ (${\theta }_{\mathrm{O}-{\mathrm{O}}_2}$) and O + CO₂ (${\theta }_{\mathrm{O}-{\mathrm{CO}}_2}$) isotope exchange reactions is shown in Figure S2 (supporting information). The three‐isotope exponent for atom‐molecule exchange (κ) is calculated as shown in Equation 6) following Cao and Liu. ³¹

$${\theta }_{\mathrm{O}-\mathrm{XO}}={\kappa }_{\mathrm{XO}}=\frac{\ln \left({}_{}{}^{17}{\beta }_{\mathrm{XO}}\right)}{\ln \left({}_{}{}^{18}{\mathrm{\beta }}_{\mathrm{XO}}\right)}$$ (6)

where ${\kappa }_{\mathrm{XO}}$ or $\left({\theta }_{\mathrm{O}-\mathrm{XO}}\right)$ is the three‐isotope exponent for atom‐molecule exchange, which should also be restricted to the parameter range < ${\theta }_{\mathrm{HTL}}$. This is indeed the case for O₂, but ${\kappa }_{{\mathrm{CO}}_2}$ clearly exceeds the expected range. According to our calculations, it reaches a HTL of around 0.533. We suppose that some structure in the results above 800°C is likely to be due to the smallness of the isotopic signatures, as well as to the precision and potential incompleteness of excited states in the line lists. While our ${\kappa }_{O_2}$ value further agrees with the calculation of Cao and Liu, ³¹ to the fourth decimal place (or even better), our ${\kappa }_{{\mathrm{CO}}_2}$ value is higher by 0.0027 to 0.0035 over the full T range and it falls well above the canonical range, even at room temperature, where ${\kappa }_{{\mathrm{CO}}_2}$(25°C) = 0.5317 (see Figure S2, supporting information).

Cao and Liu ³¹ have shown that the three‐isotope exponent (θ _a−b ) for isotope exchange between two molecules a and b can also be calculated as a linear combination of the three‐isotope exponents ${\kappa }_a$ and ${\kappa }_b$ of the atom–molecule exchange equilibria for molecules a and b (see Equation 6), respectively:

$${\theta }_{a-b}=\left(1-c\right){\kappa }_a+c{\kappa }_b,\text{where}\mathrm{c}=-\frac{\ln \left({}_{}{}^{18}{\beta }_b\right)}{\ln \left({}_{}{}^{18}{\alpha }_{a-b}\right)}$$ (7)

Interestingly, the coefficient c in Equation 7 does not depend on ¹⁷O, it only depends on ¹⁸O via the ¹⁸ β‐value of molecule b and the (statistically corrected) equilibrium constant ¹⁸ α _a−b between molecules a and b.

We can use this relationship and the well‐studied H₂O‐CO₂ system for an independent estimate of ${\kappa }_{{\mathrm{CO}}_2}$. The three‐isotope relationship between liquid water and gaseous CO₂ (${\theta }_{{\mathrm{CO}}_2-{\mathrm{H}}_2\mathrm{O}\left(\mathrm{l}\right)}$) was experimentally determined to be 0.5229 at 25°C (${{\alpha }^{18}}_{{\mathrm{CO}}_2-{\mathrm{H}}_2\mathrm{O}\left(\mathrm{l}\right)}$ = 1.041036). ⁷⁷ Using ${\kappa }_{{\mathrm{H}}_2\mathrm{O}}$ = 0.5300 calculated for water vapor by Cao and Liu ³¹ (a similar ${\kappa }_{{\mathrm{H}}_2\mathrm{O}}$ value was derived by the same authors from literature data on the gas vapor equilibrium) and ${}_{}{}^{18}{\beta }_{{\mathrm{CO}}_2}$ = 1.1194 (this study), ${\kappa }_{{\mathrm{CO}}_2}$ can be calculated using Equation (7), yielding 0.5324 at 25°C. This also is clearly larger than the canonical HTL. The result is only slightly different (+0.0001) when ${}_{}{}^{18}{\beta }_{{\mathrm{CO}}_2}$ = 1.1171 (from Richet et al ³⁰ ) is used, in contrast to ${\kappa }_{{\mathrm{CO}}_2}$ = 0.5280 as obtained by Cao and Liu. ³¹ As a cautionary note, it needs to be pointed out that the ${\kappa }_{{\mathrm{H}}_2\mathrm{O}}$ value of Cao and Liu ³¹ was obtained without taking into account non‐Born‐Oppenheimer effects that seem to be important for water at room temperature. ⁵⁷ When we use our value of ${\kappa }_{{\mathrm{CO}}_2}$(25°C) = 0.5317 to determine ${\kappa }_{{\mathrm{H}}_2\mathrm{O}\left(\mathrm{l}\right)}$ via Equation 7, from the measurements of ${{\alpha }^{18}}_{{\mathrm{CO}}_2-{\mathrm{H}}_2\mathrm{O}\left(\mathrm{l}\right)}$ = 1.041036 ⁷⁷ and ${\theta }_{{\mathrm{CO}}_2-{\mathrm{H}}_2\mathrm{O}\left(\mathrm{l}\right)}=$ 0.5229 ⁷⁷ and ${}_{}{}^{18}{\beta }_{{\mathrm{CO}}_2}$ = 1.1194, we obtain ${\kappa }_{{\mathrm{H}}_2\mathrm{O}\left(\mathrm{l}\right)}=$ 0.5294. Interestingly, Barkan and Luz ⁷⁸ have also determined the equilibrium fractionation factors between the gaseous and liquid phases of H₂O, from which the three‐isotope coefficient ${\theta }_{{\mathrm{CO}}_2-{\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)}=$ 0.5241 for the equilibrium fractionation of water and CO₂ in the gaseous phase can be derived. The corresponding ${{\alpha }^{18}}_{{\mathrm{CO}}_2-{\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)}$ = 1.05082 implies that ${\kappa }_{{\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)}=$ 0.5377 using our ${\kappa }_{{\mathrm{CO}}_2}=0.5317,$ or ${\kappa }_{{\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)}=$ 0.5311 when using the lower ${\kappa }_{{\mathrm{CO}}_2}$ from Cao and Liu. ³¹ Independently of the way of calculating ${\kappa }_{{\mathrm{CO}}_2}$, available data on the CO₂‐H₂O system seem to indicate that ${\kappa }_{{\mathrm{H}}_2\mathrm{O}\left(\mathrm{g}\right)}$ also exceeds the canonical range limit. This implies that $\kappa$ values of a chemical compound in different states might not be identical.

The independently derived ${\kappa }_{{\mathrm{CO}}_2}$ value of 0.5324 is somewhat higher than the value calculated directly from the experimentally improved ab‐initio data from Huang et al, ⁴² ${\kappa }_{{\mathrm{CO}}_2}$= $\ln \left({}_{}{}^{17}{\beta }_{{\mathrm{CO}}_2}\right)/\ln ({}_{}{}^{18}{\beta }_{{\mathrm{CO}}_2})$ = 0.5317, and supports our finding that ${\kappa }_{{\mathrm{CO}}_2}$(slightly) exceeds the value of 0.5305 even at room temperature. At high temperatures (~1000°C), ${\kappa }_{{\mathrm{CO}}_2}$ should be even higher, by 0.01 to 0.02. This is in striking contrast to the ${\kappa }_{{\mathrm{CO}}_2}$ values estimated by Cao and Liu ³¹ which never exceed 0.5305, but their underlying calculations of partition function ratios seem to include only a limited number of anharmonic or higher order corrections to the simple harmonic oscillator approximation. Such corrections seem to be mostly negligible in the calculation of $\kappa$ for diatomic molecules, such as O₂, for which the ${\kappa }_{{\mathrm{O}}_2}$ values calculated in this study (${\kappa }_{{\mathrm{O}}_2}$ at 25°C = 0.5282, for example) are in perfect agreement with the values estimated by Cao and Liu. ³¹ Nevertheless, they appear to be important for triatomic molecules (H₂O, CO₂, etc.) and probably for more complex molecules too.

Finally, Equation 7 can be used to determine ${\theta }_{{\mathrm{CO}}_2-{\mathrm{O}}_2}$ independent of our evaluation of ¹⁷ α (if it is suspected that the ¹⁷O data are less reliable than the other data, for example) or even independent of ¹⁸ α. Using ${\kappa }_{{\mathrm{CO}}_2}$(25°C) = 0.5324 (see previous paragraph), ${\kappa }_{{\mathrm{O}}_2}$(25°C) = 0.5282, ¹⁸ α = 1.0352 (1.0372) and ¹⁸ β _O2 = 1.0813 (1.0773), results in ${\theta }_{{\mathrm{CO}}_2-{\mathrm{O}}_2}=$ 0.5419 (0.5410). The values in parentheses are the corresponding values of Richet et al. ³⁰ For comparison, the calculation based on the direct sum method for CO₂ used in this study yields ${\theta }_{{\mathrm{CO}}_2-{\mathrm{O}}_2}$ = 0.5396 (0.5388) at 25°C, with the value in parentheses again being from the compilation of Richet et al. ³⁰ Both routes of calculation yield values around 0.540, well beyond the “canonical range”.

According to Equation 7, the combination of both the unusually large ${\kappa }_{{\mathrm{CO}}_2}$> 0.5305 (exceeding ${\kappa }_{{\mathrm{O}}_2})$ and the relatively large and opposite‐sign coefficients of c ≃ −2 and (1 − c) ≃ 3 leads to the exceptionally high value of ${\theta }_{{\mathrm{CO}}_2-{\mathrm{O}}_2}$ = 0.5396. Nevertheless, the O + CO₂ exchange alone with ${\kappa }_{{\mathrm{CO}}_2}$≃ 0.533 at temperatures above 300°C must be regarded as the origin of this exception since standard values of ${\kappa }_{{\mathrm{CO}}_2}$ close to ${\kappa }_{{\mathrm{O}}_2}$ would not lead to such departures, regardless of the coefficient c in Equation (7). The solid theoretical and experimental evidence for ${\kappa }_{{\mathrm{CO}}_2}$> 0.5305 and θ (CO₂/O₂) ~ 0.54 implies that analysis of three‐isotope effects in CO₂ needs to go beyond the simple rigid‐rotor harmonic oscillator analysis. This might also apply to other non‐diatomic molecules and clumped isotope effects. ⁷⁹ The above analysis of the CO₂‐H₂O system seems to support this hypothesis concerning the oxygen isotopic fractionation of the H₂O molecule.

5. CONCLUSIONS

The steady‐state fractionation factors, α ¹⁷(CO₂/O₂) and α ¹⁸(CO₂/O₂), for the CO₂‐O₂ isotope exchange reaction depend on temperature, largely explaining differences between previously reported values. Nevertheless, all our experimentally determined values are lower than those obtained from the updated theoretical calculations of equilibrium fractionation, and the offset is strictly mass‐dependent. The discrepancy between the theoretical and experimental fractionation factors might be caused by thermal diffusion effects. While this requires further investigation, it provides an explanation for the inconsistent values reported in the literature.

We have shown that the CO₂‐O₂ isotopic exchange under equilibrium conditions is characterized by unusual oxygen three‐isotope behavior. Calculated θ (CO₂‐O₂) values based on the direct sum method for CO₂, which includes anharmonic corrections, are larger than the canonical range limit for mass‐dependent fractionation processes (0.5305), which possibly needs to be extended to 0.54. The result can be derived independently using the experimentally determined θ (CO₂/H₂O) and α ¹⁸(CO₂/H₂O) values, together with the theoretically calculated fractionation factor (β) for oxygen exchange between CO₂ and atomic O. Our analysis shows that the unusual behavior is linked to the energy levels of CO₂ isotopologues. This could imply the need for more accurate representation of energy terms (beyond the simple Bigeleisen‐Mayer model) for poly‐atomic molecules.

6.

PEER REVIEW

The peer review history for this article is available at https://publons.com/publon/10.1002/rcm.9301.

Supporting information

Figure S1. Geometry and dimensions of the CO₂‐O₂ exchange reactor used by Adnew et al (2019) ¹.

Figure S2. Three isotope exponents κ of O + O₂ and O + CO₂ isotope exchange equilibria as a function of temperature.

Table S1. β factors, κ and θ values calculated in this study for CO₂ and O₂. β factors (i.e. the ¹⁸O fractionation between the compound [in this case oxygen and carbon dioxide] and a dissociated and non‐interacting atom O ²⁻⁷) are also compared to the previously reported values by Richet et al,(1977) ². Reldev stands for the relative deviation of β factors for O₂ and CO₂ calculated in this study and Richet et al,(1977)² in per mill (‰) (β (Richet et al)/β (This study) – 1).

Table S2. Comparison of α¹⁸(CO₂/O₂) values calculated in this study to the previously reported values by Richet et al ².

Table S3. The CO₂−O₂ exchange time to reach an isotopic steady state at 500°C.

Adnew GA, Workman E, Janssen C, Röckmann T. Temperature dependence of isotopic fractionation in the CO₂‐O₂ isotope exchange reaction. Rapid Commun Mass Spectrom. 2022;36(12):e9301. doi: 10.1002/rcm.9301

DATA AVAILABILITY STATEMENT

All the data used in this study are reported in the form of Figures and Tables.