. (9)
Similarly, the stochastic mole fraction for12CH4 isx 12CH4 =x H(1)2x H(2)2x 12C. In our example, the true stochastic value for x 12CH2D2 is 1.11885 ×10-7. This is lower by 42‰ than 1.16545×10-7, the value obtained from equation (6). Using the stochastic estimate from equation (9), the example gas has a true Δ12CH2D2 of +25‰.
In practice, one would not have any prior knowledge of the actual distribution of the D/H ratios and would have to apply equation (6) rather than equation (9) for the calculation of Δ12CH2D2. The elevated x12CH2D2 from equation (6) is the cause of the negative Δ12CH2D2 value. The reason for the discrepancy is a mathematical truism (Yeung et al., 2016, Rockmann et al., 2016, Taenzer et al., 2020). It is straightforward to show that the actual stochastic isotopologue ratio12CH2D2/12CH4is proportional to the square of the geometric mean for the two D/H ratios (to see this, divide Equation 9 byx 12CH4 =x H(1)2x H(2)2x 12C) while the measured values, by necessity, yield the square of the arithmetic mean of the two ratios, i.e., the measured bulk D/H ratio (divide Equation 6 byx 12CH4 =x H4 x 12C, mindful of Equation 8). The geometric mean is always less than the arithmetic mean, unless R1 = R2. Thus in general, where molecular positions with distinct isotope ratios are indistinguishable, and the average isotope ratios rather than the site-specific isotope ratios are all that is accessible by the data (the usual case), the calculated stochastic ratio of the multiply-substituted molecule to the major isotopologue (x12CH2D2) is overestimated. This in turn means Δ values derived from these overestimated stochastic ratios are underestimated. This is the combinatorial effect.
The building of CH4 molecules from CO (or CO2) is thought to involve a sequence of steps of carbon reduction followed by additions of hydrogen molecules (Wang et al., 2011). The set of reactions can be represented by these steps:
CO \(\rightarrow\) CO* \(\rightarrow\) HCO* \(\rightarrow\) HCOH*\(\rightarrow\) CH* \(\rightarrow\) CH2* \(\rightarrow\)CH3* \(\rightarrow\) CH4*\(\rightarrow\) CH4 (10)
where *corresponds to a surface-adsorbed species. Here H2 is the source of electrons for reduction. This reaction network allows for the various hydrogens added to methane during sequential steps to have different D/H ratios. A 500‰ difference between two of the hydrogens relative to the other two can account for our observed Δ12CH2D2deficits of approximately −40‰. The particular circumstance of δD(1)= δD(2)= −850‰ and δD(3)= δD(4)= −350‰ can explain the bulk methane δD (~ −600‰), and the methane clumped isotope values. According to the theoretical treatments in (Cao et al., 2019) and (Young, 2019), D/H fractionations associated with the assembly of methane would plausibly cause such variations in D/H from step to step. For example, a mixture of early steps involving D/H equilibrium followed by later H addition steps with purely kinetic isotope effects could produce the requisite large differences in D/H of > 400‰ among hydrogens (Young, 2019) that can reproduce the observed low Δ12CH2D2 values.
The question arises as to whether our interpretation of the source of low Δ12CH2D2 values is affected by the of the source of carbon, i.e. CO versus CO2. CO2 methanation begins by reduction to CO (i.e., the reverse of reaction 4). That additional step occurs prior to the onset of CO reduction and does not involve hydrogen additions to carbon. There is no clear physical reason to anticipate variable Δ12CH2D2signatures as the result of the additional step of CO2reduction to CO prior to further reduction by hydrogen. This likely explains why the Sabatier experiments (run with CO2) versus the FTT experiments here (with CO) can be interpreted together (Fig. 3b). Indeed, our data suggests the two pathways, from CO to methane and from CO2 to methane, may not be very different at all. Our carbon isotope data suggesting that CO2 is the precursor for the13C/12C ratios for product CH4 in our experiments in which CO is the initial carrier of carbon attest to this. Together, the data indicate that it may be valid to extrapolate our experiments to natural systems, where CO2 is likely the primary carbon source for abiotic methane synthesis.