Chapter IV: Clumped isotope effects in methane
4.5 Discussion
64
*Figure labels provided for labeling purposes only and are not to be included in final figures.
Figure 4*
-3 -2 -1 0 1 2 3 4 5
0 1 2 3 4 5 6 7
Δ13CH3D(wg)measured (‰)
Δ13CH3D theory-PIMC (‰)
Alumina Nickel 1:1 Line
g-Al2O3
Ni 1:1 line
-10 -5 0 5 10 15 20
0 5 10 15 20 25
Δ12CH2D2 (wg)measured (‰)
Δ12CH2D2theory-PIMC (‰)
a)
b)
Figure 4.3: Comparison of PIMC calculations to experiment1000×ln Δ(𝑤 𝑔)/1000+1 values from experiments (y-axis) vs. 1000 × ln(𝐾13
CH3D) (panel a) and 1000 × ln 8/3×𝐾CH
2D2
(panel b) values from the PIMC calculations obtained at the experimental temperatures. White circles indicate γ-Al2O3 and black circles indicate Ni experiments.
Error bars for replicated experimental data points from this study represent either the±1 s.e.
of replicates or the expected±1 s.e. based on the observed external precision of standards and the number of experimental replicates (i.e., 𝜎𝑒 𝑥 𝑡 𝑒𝑟 𝑛 𝑎𝑙/√
𝑛, where n is the number of experimental replicates), whichever is larger. Error bar on one experimental data point that has not been replicated (theΔCH2D2 value at 350°C) represents±1 s.e. internal precision.
Error bars in the PIMC calculations (x axis error bars,±1 s.e.) are smaller than the symbols.
PIMC RPFRs. The𝛿RPFRBMU-PIMCare as high as 5-6‰for13CH4&13CH3D and 29-95‰for the CH3D & CH2D2over the computed temperature range.
65
*Figure labels provided for labeling purposes only and are not to be included in final figures.
Figure 5*
0 1 2 3 4 5 6 7 8
0 100 200 300 400 500 600
Δ13CH3D (‰)
T(˚C)
Theory (PIMC) Ni
Series3 Theory (PIMC) Ni
g-Al2O3
0 5 10 15 20 25 30
0 100 200 300 400 500 600
Δ12CH2D2(‰)
T(˚C)
a)
b)
Figure 4.4: Temperature dependence of the Δvalues in thermodynamic (absolute) reference frame.Note that experimentalΔvalues are based on convertingΔ𝑤 𝑔(reported in the “working gas reference frame”) intoΔvalues in the “thermodynamic reference frame.”
by the PIMC calculations.
Fig. 4.3 compares the measured vs calculated clumped isotope compositions com- puted with PIMC at the experimental temperatures. We compare experimentally measured values as 1000 × ln Δ(𝑤 𝑔)/1000+1
values vs. computed values of 1000×ln(𝐾13
CH3D)(panel a) and 1000×ln 8/3×𝐾CH
2D2
(panel b) at the exper- imental temperatures. We note that the measured Δ-values (y-axis) are plotted as 1000×ln Δ𝑤 𝑔/1000+1
since the measured values follow the 1000× (𝑅/𝑅∗ −1) notation, while theory appropriately follows the 1000×ln(𝐾) notation91, and by converting the measured 1000× (𝑅/𝑅 ∗ −1) values to 1000 × ln(𝑅/𝑅∗) values we place the measured and computed values on more comparative grounds. Since experimentally measured values are shown in the working gas reference frame, the key aspect of this comparison is the relative difference between theory and experi- ment as a function of temperature. A least squares linear regression through each measured vs. theory (PIMC) dataset yields a slope of 1.02 ±0.04 for the Δ13CH3D
comparison and 0.98±0.05 for theΔCH2D2 comparison (1 s.e.). Thus, both slopes are within 1 s.e. error of 1 over a temperature range of 1-500°C. Given this 1:1 agreement between experiment and theory, lines with slopes of 1 are used to infer the intercept in each plot to obtain estimates of the working gas composition. These yieldΔ13CH3D = 2.59± 0.14‰andΔCH2D2 = 5.86± 0.60‰(1 s.e.). As expected from the 1:1 agreement in Fig. 4.3, the experimental data match the predicted tem-
acomputed from Eqs. (4.5) and (4.6)
bTo facilitate a more direct comparison, these values are reported as 1000×ln(Δ/1000+1).
cIn this instance, error is 1 s.e. (internal precision) because the value represents a single measurement.
PIMC Experiment PIMC Experiment T (°C) Δ𝑒 𝑞13
CH3D
a Δ13CH3Db 1𝜎 residuals ΔCH𝑒 𝑞
2D2
a ΔCH2D2b 1𝜎 residuals
500 0.82 0.76 0.23 -0.06 1.30 -0.06 1.93 -1.36
400 1.18 1.44 0.51 0.27 2.03 3.45 0.40 1.42
350 1.41 1.46 0.17 0.05 2.60 3.43 1.21c 0.83
300 1.70 1.26 0.16 -0.45 3.39 1.86 1.72 -1.52
250 2.07 2.27 0.34 0.20 4.47 4.99 2.49 0.52
165.4 2.95 2.93 0.30 -0.02 7.43 8.84 1.44 1.42
127.8 3.50 3.35 0.11 -0.14 9.46 9.59 0.59 0.14
75.7 4.48 4.47 0.21 -0.01 13.45 13.95 1.98 0.50
50.5 5.09 5.07 0.24 -0.01 16.08 13.99 0.47 -2.09
25 5.81 5.62 0.07 -0.19 19.36 19.45 0.95 0.08
1.2 6.64 7.00 0.09 0.36 23.15 23.23 1.55 0.08
1𝝈 0.22 1.17
Table 4.4: Comparison between theoretical (PIMC) and experimentalΔvalues in ascending order (highest temperature first).
perature dependence from the PIMC calculations over 1-500°C (see Fig. 4.4 and Table 4.4). The computed±1𝜎of the residuals (Table 4.4) are±0.22‰and±1.17‰ for theΔ13CH3D andΔCH2D2 values, respectively, and are comparable to the external precision estimated solely from the experimental replicates at a given temperature (±0.28‰forΔ13CH3D and±1.61‰forΔCH2D2, 1𝜎).
This work yields the important and satisfying result that theoretically calculated Δ𝑒 𝑞13
CH3D andΔ𝑒 𝑞
CH2D2 values using the most rigorous theoretical approach available (PIMC) are in 1:1 (at the 1.s.e. level) agreement with experimental determinations of equilibriumΔ𝑒 𝑞13
CH3DandΔ𝑒 𝑞CH
2D2. This provides confidence in both the theory, ex- periments, and measurement techniques over essentially the full range of formation temperatures of microbial and thermogenic gases on Earth.
Finally, the working gas clumped compositions yield apparent methane-clumped isotope temperatures of 196±13°C forΔ13CH3D and 204±17°C forΔCH2D2 (1 s.e.) as determined by the polynomial fit to the PIMC calibration (Eqs. 4.7 & 4.8). Based on the𝛿DVSMOW(-159.3±2.4‰) and𝛿13CVPDB(-38.37±0.2‰) values of the work- ing gas, the cylinder gas is likely thermogenic in origin.107,108 Such temperatures are reasonable potential gas formation temperatures128 and are consistent with the common observation that apparent methane clumped isotope temperatures of ther- mogenic methane are compatible with expectations of thermogenic gas formation temperatures.113,116,119,120
TheΔ13CH3D andΔCH2D2-based temperatures are within 1 s.e. of each other. Such agreement has been previously seen both in assumed thermogenic gases from com- mercially purchased cylinders125as well as thermogenic gases from natural gas de- posits.113,116Such agreement has been taken as additional evidence that thermogenic gases may form in clumped-isotope equilibrium and that Δ13CH3D andΔCH2D2 may
represent formation temperatures of thermogenic gases (or at least re-equilibration temperatures).
Given the agreement in clumped-based temperatures of our working gas inferred for bothΔ13CH3D andΔCH2D2, we could choose to force our working gas to have a ΔCH2D2 composition that corresponds to the temperature derived from theΔ13CH3D
calibration (∼196°C) given that the Δ13CH3D measurements are more precise. This exercise would yield aΔCH2D2value of∼6.17‰for our working gas using Eq. (4.8), which is about 0.31‰higher than, but within 1 s.e. of what we directly infer from our calibration (5.86±0.60‰, 1 s.e.). This may mean that our future measurements ofΔCH2D2 could be biased to ca. 0.3‰higher values based on our calibration, but given our typical external precision (∼1.4‰) we do not expect that any such bias would change any interpretations of environmental or experimental samples.
Comparison of the PIMC calculations to the BMU approximation
PIMC calculations provide a means to compute stable isotope fractionation factors independent of the traditionally employed BMU model and are more rigorous and accurate (assuming a high-quality PES and sufficiently large number of beads𝑛and number of samples for the systematic and statistical error convergence, respectively).
Therefore, comparison of BMU and PIMC calculations can be used to identify errors in BMU calculations.86,93In the current study, all BMU-computed RPFRs exhibit significant departures from the PIMC-computed RPFRs: up to 5-6‰for13CH4/CH4
&13CH3D/CH3D RPFRs and 29-95‰for the CH3D/CH4& CH2D2/CH3D RPFRs over the computed temperature range (-3 to 527°C). Given that both BMU and PIMC computations were performed using the same PES for methane computed using high level coupled cluster theory,133 these are true differences between the BMU and PIMC theoretical treatments. The CH3D/CH4RPFR exhibit 5-20x larger relative errors than 13CH4/CH4 RPFRs, where the larger discrepancy present for D/H exchange is likely due to the long-recognized inadequacies in the simplified treatments of partition functions (PF) in the BMU model to account properly for D/H exchange (harmonic vibrational PF, classical rotational PF).90,92,96,132 The PIMC calculations inherently account for vibrational anharmonicity and quantize the rotational motions, and therefore avoid these well-understood pitfalls of the BMU approach.
Additional insight into the problem may be given by comparisons between the cal- culations of the present study (BMU vs. PIMC) and previous BMU calculations performed with and without anharmonic corrections reported by Liu and Liu130. We first note that such a comparison is ultimately inexact because their130 calcu- lations are based on electronic potential energy surfaces for methane computed at the MP2 level of theory rather than the more accurate couple cluster theories of the present study.133 Nevertheless, the relative difference between uncorrected and corrected CH3D/CH4 RPFRs using the BMU model by Liu and Liu (ca. 112 to 39‰ over -3 to 527°C, respectively, is comparable to what we observe for BMU
and PIMC calculations in this study (95-29‰over -3 to 527°C, respectively). The six corrections applied include those accounting for vibrational anharmonicity and quantum corrections to rotational motions among others.130,132The total correction given by Liu and Liu (a multiplicative factor of∼0.90 to∼0.96 from -3°C to 527°C) is almost entirely driven by the correction for the anharmonic contributions to the zero point energy. This may tentatively suggest that the harmonic treatment of the vibrational partition function may be the source of much of the error in BMU-based D/H-related computations for methane.
Regardless of the precise source of the errors in the BMU model, the contrastingly small (≤0.1-0.4‰) relative differences in the computed equilibrium constants and related Δvalues describing equilibrium clumping in methane from BMU-RPFRs arises from a cancellation of errors in component RPFRs as observed by Webb and Miller.86 One likely reason for this precise cancelation of errors may be due to inherent symmetry preserved in these isotopic clumping reactions. In particular, any errors present in the 13CH4/CH4 RPFR are expected to be similar in nature and magnitude to those present in the 13CH3D/CH3D RPFR, since the RPFRs reflect the same type of isotopic substitution. The same cannot be said for some exchange reactions involving isotopomers (e.g., 14N15NO ⇌ 15N14NO) for which BMU calculations have been shown to only benefit from a partial cancelation of errors.86 Although the RPFR errors are significant on the per mil scale, we find it remarkable that such errors only amount to relative free energy differences of approximately 10−3 and 2×10−2 kcal/mol for the13C/C-related and D/H-related RPFRs, respectively.