Methane clumped isotopes

The Δ notation of clumped isotopes is an analogue to δ notation of traditional isotopes (e.g. δ¹³C, δ¹⁸O, δ¹⁵N, δ³⁴S and δD).

The notation of traditional isotopes are defined as:

${\displaystyle \delta =(\left({\frac {R_{sample}}{R_{reference}}}\right)-1)\times 1000}$‰

${\displaystyle R_{sample}}$ is the ratio of the rare isotope to the abundant isotope in the sample. ${\displaystyle R_{reference}}$ is the same ratio in the reference material. Because the variation of ${\displaystyle R_{sample}}$ is rather small, in the convenience of comparison between difference samples, the notation is define as a ratio minus 1 and expressed in permil (‰).

The Δ notation is inherited from traditional δ notation. But the reference is not a physical reference material. Instead, the reference frame is defined as the stochastic distribution of isotopologues in the sample. It means the values of Δ are to denote the excess or deficit of the isotopologue relative to the amount expected if a material conforms to the stochastic distribution.[6]

The calculation of stochastic distribution of methane isotopologues:

${\displaystyle ^{^{13}CH_{3}D}R^{*}=4\times {^{2}R}\times {^{13}R}}$

${\displaystyle ^{^{12}CH_{2}D_{2}}R^{*}=6\times {^{2}R}^{2}}$

where ${\displaystyle ^{^{13}CH_{3}D}R^{*}}$ is defined as the abundance of ¹³CH₃D molecules relative to ¹²CH₄ molecules in random distribution; ${\displaystyle ^{^{12}CH_{2}D_{2}}R^{*}}$ is defined as the abundance of ¹²CH₂D₂ molecules relative to ¹²CH₄ molecules in random distribution; ${\displaystyle {^{2}R}={\frac {[D]}{[H]}}}$ calculates the abundance of deuterium relative to protium in all methane molecules; ${\displaystyle {^{13}R}={\frac {[^{13}C]}{[^{12}C]}}}$ calculates the abundance of carbon-13 relative to carbon-12 in all methane molecules.

For the random distribution (i.e. probability distribution), the probability of choosing a carbon-13 atom over a carbon-12 atom is ${\displaystyle {^{13}R}}$; the probability of choosing three protium atoms and one deuterium atom over four protium atoms is${\displaystyle {\binom {4}{1}}\times {^{2}R}}$ (see "Combination") . Therefore, the probability of the occurrence of a ¹³CH₃D molecule relative to the occurrence of a ¹²CH₄ molecule is the product of ${\displaystyle {^{13}R}}$ and ${\displaystyle 4\times {^{2}R}}$, which gets to ${\displaystyle ^{^{13}CH_{3}D}R^{*}=4\times {^{2}R}\times {^{13}R}}$. Similarly, the probability of choosing two protium atoms and two deuterium atoms over four protium atoms is ${\displaystyle {\binom {4}{2}}\times {^{2}R}^{2}}$. Therefore, the probability of the occurrence of a ¹²CH₂D₂ molecule relative to the occurrence of a ¹²CH₄ molecule is ${\displaystyle {\binom {4}{2}}\times {^{2}R}^{2}}$, which gets to ${\displaystyle ^{^{12}CH_{2}D_{2}}R^{*}=6\times {^{2}R}^{2}}$.

The calculation of deviation from the random distribution:

${\displaystyle \Delta _{^{13}CH_{3}D}=\left({\frac {^{^{13}CH_{3}D}R}{^{^{13}CH_{3}D}R^{*}}}\right)-1}$

${\displaystyle \Delta _{{}^{12}CH_{2}D_{2}}=\left({\frac {^{^{12}CH_{2}D_{2}}R}{^{^{12}CH_{2}D_{2}}R^{*}}}\right)-1}$

where the actual abundance of ¹³CH₃D molecules relative to ¹²CH₄ molecules, and the actual abundance of ¹²CH₂D₂ molecules relative to ¹²CH₄ molecules are calculated as follows:

${\displaystyle ^{^{13}CH_{3}D}R={\frac {[^{13}CH_{3}D]}{[^{12}CH_{4}]}}}$

${\displaystyle ^{^{12}CH_{2}D_{2}}R={\frac {[^{12}CH_{2}D_{2}]}{[^{12}CH_{4}]}}}$

The two Δ formulas are frequently used to report the abundance of clumped isotopologues of methane.

The reason for choosing stochastic distribution as the reference frame may be historical - in the process of developing CO₂ clumped isotope measurement, the only material with known clumped isotope abundance was CO₂ heated to 1000 °C. However, this reference frame is a good choice. Because the absolute abundance of each isotopologue primarily depends on the bulk carbon and hydrogen isotope compositions (δ¹³C and δD) of the molecules, i.e. very close to stochastic distribution. Therefore, the deviation from the stochastic distribution, which is the key information embedded in the methane clumped isotopologues, is denoted by Δ values.

Under some circumstances, the abundances of ¹³CH₃D and ¹²CH₂D₂ isotopologues are only measured as a sum, which leads to the notation for isotopologues of mass-18 (i.e. ¹³CH₃D and ¹²CH₂D₂):

${\displaystyle ^{18}R=^{^{13}CH_{3}D}R+^{^{12}CH_{2}D_{2}}R={\frac {[^{13}CH_{3}D]+[^{12}CH_{2}D_{2}]}{[^{12}CH_{4}]}}}$

${\displaystyle \Delta _{18}=\left({\frac {^{18}R}{^{18}R^{*}}}\right)-1}$

Note that ${\displaystyle \Delta _{18}}$ is not just the sum of ${\displaystyle \Delta _{^{13}CH_{3}D}}$ and ${\displaystyle \Delta _{{}^{12}CH_{2}D_{2}}}$.

${\displaystyle T_{18}}$ is the inferred equilibration temperature based on ${\displaystyle \Delta _{18}}$ values; ${\displaystyle T_{^{13}CH_{3}D}}$ is the inferred equilibration temperature based on ${\displaystyle \Delta _{^{13}CH_{3}D}}$ values; and ${\displaystyle T_{^{12}CH_{2}D_{2}}}$ is the inferred equilibration temperature based on ${\displaystyle \Delta _{^{12}CH_{2}D_{2}}}$ values (see "Equilibrium thermodynamics" for details). ${\displaystyle T_{18}}$, ${\displaystyle T_{^{13}CH_{3}D}}$, and ${\displaystyle T_{^{12}CH_{2}D_{2}}}$ are also called clumped-isotope temperatures. When a Δ value is smaller than zero, there is no inferred equilibration temperature associated with it. Because at any finite temperature, the equilibrium Δ value is always positive.

When formed or re-equilibrated in reversible reactions, methane molecules can exchange isotopes with each other or with other substances present, such as H₂O, H₂ and CO₂,[4] and reach internal isotopic equilibrium. As a result, clumped isotopologues are enriched relative to the stochastic distribution. ${\displaystyle \Delta _{18}}$ and ${\displaystyle \Delta _{^{13}CH_{3}D}}$ values of methane in internal isotopic equilibrium are predicted[1][7][8][2][9] and verified[10][9] to vary as monotonic functions of temperature of equilibration as follows:

${\displaystyle \Delta _{18}=-0.0117{\left({\frac {10^{6}}{T^{2}}}\right)}^{2}+0.708\left({\frac {10^{6}}{T^{2}}}\right)-0.337}$[2]

${\displaystyle \Delta _{^{13}CH_{3}D}=-0.0141{\left({\frac {10^{6}}{T^{2}}}\right)}^{2}+0.699\left({\frac {10^{6}}{T^{2}}}\right)-0.311}$[11]

The equilibrium distribution of Δ₁₈ as a monotonic function of temperature. Redrawn from Stolper et al., 2014.

Δ values are in permil (‰).

The equilibrium distribution of Δ13CH3D as a monotonic function of temperature. Redrawn from Webb and Miller, 2014.

Similar relationship also applies to ${\displaystyle \Delta _{^{12}CH_{2}D_{2}}}$:

The equilibrium distribution of ${\displaystyle \Delta _{^{12}CH_{2}D_{2}}}$ as a monotonic function of temperature. Redrawn from Young et al., 2017.

${\displaystyle \Delta _{^{12}CH_{2}D_{2}}={\left({\frac {0.183798}{T}}\right)}-{\left({\frac {785.483}{T^{2}}}\right)}+\left({\frac {1056280.0}{T^{3}}}\right)+\left({\frac {9.37307\times 10^{7}}{T^{4}}}\right)-\left({\frac {8.919480\times 10^{10}}{T^{5}}}\right)+\left({\frac {9.901730\times 10^{12}}{T^{6}}}\right)}$[12]

The equilibrium distribution of ${\displaystyle \Delta _{^{12}CH_{2}D_{2}}}$ and ${\displaystyle \Delta _{^{13}CH_{3}D}}$. Redrawn from Young et al., 2017.

Based on these correlations, ${\displaystyle \Delta _{18}}$, ${\displaystyle \Delta _{^{13}CH_{3}D}}$ and ${\displaystyle \Delta _{^{12}CH_{2}D_{2}}}$ can be used as a geothermometer to indicate the formation temperature of methane (${\displaystyle T_{18}}$, ${\displaystyle T_{^{13}CH_{3}D}}$ and ${\displaystyle T_{^{12}CH_{2}D_{2}}}$). And the correlation of ${\displaystyle \Delta _{^{13}CH_{3}D}}$ and ${\displaystyle \Delta _{^{12}CH_{2}D_{2}}}$can help to determine whether methane is formed in internal isotopic equilibrium.[12]

Kinetic isotope effect (KIE) occurs in irreversible reactions, such as methanogenesis, and can deviate methane clumped isotopologue composition from its thermodynamic equilibrium. Normally, KIE significantly drives ${\displaystyle \Delta _{18}}$ and ${\displaystyle \Delta _{^{13}CH_{3}D}}$ lower than their equilibrium states and even to negative values (i.e. more depleted of clumped isotopologues than stochastic distribution.[9][13][14][12][5] Such lower ${\displaystyle \Delta _{18}}$ and ${\displaystyle \Delta _{^{13}CH_{3}D}}$ values correspond to apparent formation temperatures that are significantly higher than actual formation temperature, or to no possible temperatures (when a Δ value is smaller than zero, there is no inferred equilibration temperature associated with it).

Mixing between end-members with different conventional carbon and hydrogen isotope compositions (i.e. δ¹³C, δD) results in non-linear variations in ${\displaystyle \Delta _{18}}$ or ${\displaystyle \Delta _{^{13}CH_{3}D}}$. This non-linearity results from the non-linear definition of ${\displaystyle \Delta _{18}}$ and ${\displaystyle \Delta _{^{13}CH_{3}D}}$ values in reference to the random distributions of methane isotopologues (${\displaystyle ^{^{13}CH_{3}D}R^{*}=4\times {^{2}R}\times {^{13}R}}$ and ${\displaystyle ^{18}R^{*}=^{^{13}CH_{3}D}R^{*}+^{^{12}CH_{2}D_{2}}R^{*}=4\times {^{2}R}\times {^{13}R}+6\times {^{2}R}^{2}}$, as in "Notation"), which are non-linear polynomial functions of δD and δ¹³C values. Such non-linearity can be a diagnostic signature for mixing if multiple samples of various mixing ratios can be measured. When end-members have similar δ¹³C or δD compositions, the non-linearity is negligible.[4]

Examples of mixing effects for ${\displaystyle \Delta _{18}}$ values. Mixing relationships in δ¹³C-${\displaystyle \Delta _{18}}$ space and δD-${\displaystyle \Delta _{18}}$ space for mixtures of methane with varying end-member compositions. The end-member ${\displaystyle \Delta _{18}}$ values remain fixed, but the end-member δ¹³C and δD values vary. Redrawn from Douglas et al., 2017.

On an isotope-ratio mass spectrometer, the measurement of clumped isotopologues has to be conducted on intact methane molecules, instead of converting methane to CO_2, H₂ or H₂O. High mass resolution is required to distinguish different isotopologues of very close relative molecular mass (same "cardinal mass", e.g. ¹³CH₄ and ¹²CH₃D (17.03465 Da (Daltons) versus 17.03758 Da), ¹³CH₃D and ¹²CH₂D₂ (18.04093 Da versus 18.04385 Da). Currently, two commercial models capable of such measurement are Thermo Scientific 253 Ultra[15] and the Panorama by Nu Instruments.[16]

Tunable infrared laser direct absorption spectroscopy (TILDAS) has been developed to measure the abundance of ¹³CH₃D with two continuous wave quantum cascade lasers.[17]

Many studies have observed composition of thermogenic methane in equilibria.[10][13][12] The reported ${\displaystyle T_{18}}$ and ${\displaystyle T_{^{13}CH_{3}D}}$ are normally distributed within the range of 72 to 298 °C (peak value: ${\displaystyle 175\pm 47}$°C), which aligns well with modeled results of methane formation temperature and yield.[3] However, some thermogenic methane samples have clumped-isotope temperatures that are unrealistically high.[10][3] Possible explanations for exceedingly high clumped isotope temperatures include natural gas migration after formation, mixing effect, and kinetic isotope effect of secondary cracking.

Methanogenesis is a form of anaerobic respiration used by microbes, and microbial methanogenesis can occur in deep subsurface, marine sediments, freshwater bodies, etc. It appears that methane from deep subsurface and marine sediment is generally in internal isotopic equilibrium.,[10][18][13][14] while freshwater microbial methanogenesis expresses large kinetic isotope effect on methane clumped isotope composition.[13][9][14][12][5]

There are two possible explanations for this variance: firstly, substrate limitation may enhance the reversibility of methanogenesis, thus allowing methane to achieve internal isotopic equilibrium via rapid hydrogen exchange with water;[13][9] secondly, activation of C-H bonds during anaerobic oxidation precedes reversibly such that C-H bonds are broken and reformed faster than the net rate of methane consumption and methane can be reequilibrated.[13]

Theoretical calculations have predicted ${\displaystyle \Delta _{18}}$ and ${\displaystyle \Delta _{^{13}CH_{3}D}}$ values of methane in internal isotopic equilibrium.[1][7][8][2][9] As there are assumptions and approximations in calculations, the equilibrium distribution is only experimentally validated after the analysis of samples brought to thermodynamic equilibrium.[10][9] Nickel and platinum catalysts have been used to equilibrate methane C-H bonds at various temperatures from 150 to 500 °C in laboratory.[17][2][9][14] Currently, catalytic equilibration is also the practice to develop the reference material for clumped isotope analysis .

Hydrogenotrophic methanogens utilize CO₂ and H₂ to produce methane by the following reaction:

CO₂ + 4H₂ → CH₄ + 2H₂O

Acetoclastic methanogens metabolize acetate acid and produce methane:

CH₃COOH → CH₄ + CO₂

In laboratories, clumped isotope compositions of methane generated by hydrogenotrophic methanogens,[10][9][12][5] acetoclastic methanogens (biodegradation of acetate),[14][12][5] and methylotrophic methanogens[5] are universally out of equilibria. It has been proposed that the reversibility of methanogenic enzyme is key to the kinetic isotope effect expressed in biogenic methane.[13][9]

Both pyrolysis of propane and closed-system hydrous pyrolysis of organic matter generate methane of ${\displaystyle T_{18}}$ consistent with experimental temperatures.[10] Closed-system nonhydrous pyrolysis of coal yields non-equilibrium distribution of methane isotopologues.[19]

Methane synthesized by Sabatier reaction is largely depleted in CH₂D₂ and slightly depleted in ¹³CH₃D relative to the equilibrium state. It has been proposed that quantum tunneling effects result in the low ${\displaystyle \Delta _{^{12}CH_{2}D_{2}}}$ observed in the experiment.[12]

Biogenic, thermogenic and abiotic methane is formed at different temperatures, which can be recorded in clumped isotope compositions of methane.[10][13][14][20][21] Combined with conventional carbon and hydrogen isotope fingerprints and gas wetness (the abundance of low molecular weight hydrocarbon),[22] methane clumped isotope can be used to identify the origins of methane in different types of natural gas accumulations.[3]

In freshwater environments, significant kinetic isotope effect leads to a wide range of observed ${\displaystyle \Delta _{18}}$ and ${\displaystyle \Delta _{^{13}CH_{3}D}}$ values, which has the potential to provide insights into methanogenesis rate and chemical condition in the corresponding environments.[4][5]