. (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.