Relationship between Δ and Δ 𝑒 𝑞

Chapter VI: 𝐷 and 13 𝐶 isotopic equilibria in alkanes

A.5 Relationship between Δ and Δ 𝑒 𝑞

Throughout the study we report the Δ^{𝑒 𝑞} values defined in Eq. (3.20). These are independent of heavy isotope abundances and express only the thermodynamic preference for two heavy isotopes to occupy the same site of the molecule at a given temperature. In contrast, experimentally measured Δ values (defined in Eq. 3.5) depend on isotopic abundance and are relatable to the corresponding Δ^{𝑒 𝑞} values only in the limit of infinite dilution:

lim

[𝐷]→0,[¹³𝐶]→0

1000×ln Δ

1000 +1 . (A.26)

Figure A.1: Effect of the abundance of heavy isotopes on (a)D+Dand (b) ¹³C+D clumped isotope effects in methane.Dotted horizontal lines are concentration-independent Δ^𝑒𝑞 values (i.e., infinite dilution limit, see Eq. A.26). Vertical lines label the VPDB and VSMOW standards for carbon-13 and deuterium, respectively. When abundance of deuterium is varied (dots), carbon-13 is set to VPDB; when abundance of carbon-13 is varies (open circles), deuterium is set to VSMOW.

[D] 1000×ln Δ_CH2D2/1000+1

1000×ln

Δ¹³_CH3D/1000+1

0°C 50°C 100°C 200°C 400°C 0°C 50°C 100°C 200°C 400°C Δ^{𝑒 𝑞} 23.7 16.177 11.284 5.829 1.909 6.637 5.048 3.931 2.503 1.142 10⁻⁶ 23.552 16.064 11.197 5.773 1.884 6.563 4.992 3.888 2.476 1.129 10⁻⁵ 23.551 16.064 11.197 5.773 1.883 6.562 4.992 3.887 2.475 1.129 10⁻⁴ 23.538 16.055 11.191 5.77 1.882 6.553 4.985 3.883 2.473 1.128 10⁻³ 23.409 15.967 11.13 5.738 1.872 6.465 4.923 3.838 2.448 1.119 10⁻² 22.134 15.1 10.526 5.428 1.771 5.592 4.311 3.397 2.204 1.028 0.05 16.758 11.441 7.979 4.115 1.342 1.934 1.743 1.548 1.182 0.647 0.1 10.68 7.296 5.09 2.625 0.855 -2.155 -1.128 -0.518 0.04 0.222

Table A.2: Dependence of clumped isotope in methane on abundance of D, while keeping

13C at VPDB (F12/ATZ potential used).

[¹³C] 1000×ln Δ_CH2D2/1000+1

1000×ln

Δ13CH3D/1000+1 0°C 50°C 100°C 200°C 400°C 0°C 50°C 100°C 200°C 400°C Δ^{𝑒 𝑞} 23.7 16.177 11.284 5.829 1.909 6.637 5.048 3.931 2.503 1.142 10⁻⁴ 23.676 16.16 11.273 5.823 1.907 6.621 5.037 3.923 2.499 1.14 10⁻³ 23.664 16.151 11.266 5.818 1.905 6.615 5.032 3.919 2.497 1.139 10⁻² 23.544 16.06 11.195 5.773 1.884 6.555 4.987 3.884 2.474 1.129 0.05 23.012 15.655 10.88 5.573 1.793 6.289 4.785 3.727 2.374 1.083 0.1 22.347 15.15 10.487 5.322 1.679 5.956 4.532 3.53 2.249 1.026 0.2 21.017 14.139 9.7 4.821 1.45 5.291 4.026 3.136 1.998 0.912 Table A.3: Dependence of clumped isotope in methane on abundance of¹³C, while keeping D at VSMOW (F12/ATZ potential used).

Fig. A.1 addresses the effect of heavy isotope abundance on the strength of experi- mentally measured excess of doubly substituted species (i.e.,Δvalues) in methane.

The values plotted are mirrored in the Tables A.2 and A.3. At low (near-natural) abundances the two are close to each other, but increasing the abundance of heavy isotopes decreases the strength of measured clumping. Notably, at surprisingly low abundance of deuterium of 10% the apparent ¹³C+D clumped isotope effect re- verses direction, i.e., the doubly substituted ¹³CH₃D species is less abandoned than at random distribution of isotopes as evidenced by the negative values in the last rows of Table A.2. This is because the D+D clumped isotope effect in methane is significantly stronger than the¹³C+D clumping, so it leads to the depletion of singly deuterated methane that is large enough to force “reverse clumping” of ¹³C+D.

This is made more concrete by Table A.4. At low abundance of deuterium,Δ_CH3D is tiny, so the two clumped isotope effects are essentially independent with each of them close to the infinite dilution limit. However, at high deuterium abundance the 𝐷+𝐷 clumped isotope effect leads to significant depletion of CH3D with the Δ_CH3D comparable in size toΔ^{𝑒 𝑞}₁₃

CH3D. The isotopologues with carbon-13 and more than one deuterium substitution also deplete the equilibrium amount of ¹³CH₄. As a result, at equilibrium the denominator of the right-hand side of the Eq. A.27 is sufficiently small that the numerator needs to be smaller than unity to obtain the

[D] CH3D CH2D2 CHD3 CD4 13CH4 13CH3D ¹³CH2D2 13CHD3 13CD4

10⁻⁵ -0.074 23.83 48.31 73.37 -0.00027 6.58 30.65 55.29 80.52 0.1 -6.49 10.74 28.26 46.09 -2.28 -2.15 15.15 32.75 50.66 Table A.4: Δvalues of all isotopologues of methane at low vs high abundance of deuterium, both at 0°C.

left-hand side of the equation, which is only very slightly larger than one.

1≲ 𝐾₁₃

CH3D

𝐾^random

13CH3D

= Δ13CH3D+1 Δ_CH3D

× Δ13CH₄

(A.27)

Propane is distinct compared to other molecules here for clumped isotopes because it has two non-equivalent sites for placing carbon-13 or deuterium. The presence of these non-equivalent sites leads to significant deviations from a random distribution for the equilibrium concentrations of singly-substituted isotopologues of propane at each site. This is distinct from other molecules where, at natural abundances, the concentration of the single substituted isotopologues is effectively the same as that for the random distribution. As a result, theΔvalues for clumping on the individual sites of propane represent a combination of the clumped and position-specific heavy isotope effects. To be more specific, the Δ value for clumping in the methylene group of propane is significantly larger than expected based on the clumped heavy isotope effect considerations alone, while the same value for the methyl group is negative. Both of these are due to the site-specific heavy isotope effect in propane, which makes the methylene group significantly heavier than the methyl group. In contrast, Δ^{𝑒 𝑞} describe the heavy isotope clumped effect alone. These values are always positive and have similar magnitudes to analogous reactions for methane and ethane – provided one takes care to consider the equilibrium reactions that do not involve shifts in heavy isotope positions.

Table A.5 relates the Δ^{𝑒 𝑞} values for up to doubly substituted isotopologues of propane to experimentally measurableΔvalues (calculated with carnon-13 at VPDB and deuterium at VSMOW abundances).^82,211 For propane, the Δ^{𝑒 𝑞} values do not approximate 1000×ln(Δ/1000+1) values even in the limit of infinite dilution, other than for singly substituted species. This is because the Δ^{𝑒 𝑞} value of each clumped isotopologue decouples clumping from the site preference and reports the thermodynamic preference of clumping. In contrast, the experimentally measuredΔ values combine the (strong) site preference and the (much weaker) clumped effect.

Thus, theΔvalues and corresponding 1000×ln(Δ/1000+1)values are negative for all doubly substituted D+D and¹³C+D isotopologues where no deuterium atom is in the center position. As suggested by [91], addition of the corresponding pair ofΔ^{𝑒 𝑞} of singly substituted isotopologues recovers the 1000×ln(Δ/1000+1)as a crude approximation (see second to last column of the table).

Species Heavy atoms [𝐴] [𝐴]_random 1000×ln Δ

1000+1

Δ^{𝑒 𝑞}₂ +2Δ^{𝑒 𝑞}₁ Δ^{𝑒 𝑞}

CH3−CH2−CH3 None 0.965992 0.965992 0 0

CH2D−CH2−CH3 D 0.000879 0.000903 -26.22 -28.37

CH3−CHD−CH3 D 0.000324 0.000301 74.48 81.69

CHD2−CH2−CH3 D&D 1.36E-07 1.41E-07 -31.64 -35.94 20.8 CH3−CD2−CH3 D&D 2.77E-08 2.34E-08 167.7 182.1 18.7 CH2D−CHD−CH3 D&D 2.95E-07 2.81E-07 48.72 53.77 0.4594 CH2D−CH2−CH2D D&D 2.00E-07 2.11E-07 -52.41 -56.72 0.0265

13CH₃−CH₂−CH₃ ¹³C 0.021477 0.0216 -5.672 N/A -5.516 CH3−¹³CH2−CH3 13C 0.010922 0.0108 11.23 N/A 10.97

13CH₃−¹³CH₂−CH₃ ¹³C &¹³C 0.000243 0.000241 5.821 5.719 0.263

13CH3−CH2−¹³CH3 13C &¹³C 0.000119 0.000121 -11.33 -11.02 0.0166

13CH₂D−CH₂−CH₃ ¹³C & D 9.84E-06 1.01E-05 -25.78 -27.95 6.106 CH3−¹³CHD−CH3 13C & D 3.69E-06 3.36E-06 91.33 81.79 5.621

13CH3−CHD−CH3 13C & D 7.21E-06 6.73E-06 69.27 76.63 0.4595 CH₂D−¹³CH₂−CH₃ ¹³C & D 9.95E-06 1.01E-05 -14.58 -16.99 0.4072

13CH3−CH2−CH2D ¹³C & D 9.78E-06 1.01E-05 -31.87 -33.87 0.0179

Table A.5: Equilibrium concentrations of up to doubly substituted propane at 0°C and at random distribution of isotopes (i.e., infinite temperature). The last three columns compare different measures of isotopic enrichment.

A.6 Averaging of clumped heavy isotope effect due to different isotopologues