In principle, equilibrium fractionation factors can be predicted from the partition function ratios of the two substances in equilibrium (Bigeleisen and Mayer, 1947; Urey, 1947). Furthermore, the comparison of different H positions has significantly improved our understanding of the relationship between chemical environments and the value of αeq.
INTRODUCTION
This is typically achieved experimentally by forcing isotopic exchange with water vapors of different 2H/1H values (Schimmelmann et al., 1999; Sauer et al., 2009). Calculation of the reduced partition function ratio (β factor) requires known molecular vibrational frequencies—generally based on spectroscopic measurements—which are not available for most 2H-substituted organic molecules relevant to natural systems (Richet et al., 1977; O' Neil, 1986).
NOTATION AND NOMENCLATURE
We discuss potential sources of error in the experimental measurements and theoretical calculations, and possible applications of the experimental-theoretical calibration. 1-2 show that the β factor of the entire molecule takes the form of the arithmetic mean of the individual beta factors weighted by the number of equivalent H atoms in each group.
METHODS
E XPERIMENTAL METHODS
- Materials
- Isotope exchange experiments
- Isotopic analysis of ketones
- Isotopic analysis of water
- Calculation of equilibrium fractionation factors
- Calculation of rate constants and activation energy for H α exchange
This range brackets the δ2H values of unchanged ketone substrates so that equilibrium can be approached from both directions. The last methane drop in each group was used to calibrate the δ2H value of the following analyte.
C OMPUTATIONAL METHODS
- Estimation of equilibrium fractionation factors
- Ab initio modeling
Vibrational frequencies for each organic molecule were calculated using the Density Functional Theory (DFT) approximation (Greeley et al., 1994) to quantum mechanics, performed using Jaguar 7.0 (Schrödinger Inc.). The solvation effect for organic molecules was simulated using the Poisson-Boltzmann continuum solvation model (Tannor et al.1994) as implemented in Jaguar.
RESULTS AND DISCUSSION
- I SOTOPE EXCHANGE EXPERIMENTS
- Exchange kinetics
- Experimental measurements of equilibrium fractionation factors
- T HEORETICAL CALCULATIONS
- Optimized molecular geometries
- Theoretical estimates of equilibrium fractionation factors
- C OMPARISON OF EXPERIMENTAL AND THEORETICAL EQUILIBRIUM FRACTIONATIONS
- S OURCES OF ERROR
- Experimental uncertainties
- Uncertainties in ab initio calculations
- Limitations in applying the calibration to other molecules
The equilibrium fractionation factors for Hα are derived from the regression slope between the δ2H values of the ketone and water in equilibrium (equation 1–3). The obtained values of δ2H of ketones did not differ from those measured for pure substances.
CONCLUSIONS
Since the effect of anharmonicity is of the same magnitude for C-bonded H in most linear compounds except for ketones, we suggest that this calibration can be used for analogous theoretical calculations for a wide variety of organic molecules with linear carbon skeletons containing five or more C -atoms, and for temperatures in the range 0-100°C. Anharmonic effects arise because the potential energy variation for molecular vibrations is not a simple quadratic function of the coordinate changes, i.e. Simple Harmonic Oscillator (SHO) approximation, widely used in ab initio methods to calculate vibrational frequencies and in the reduced partition function ratio equation (1-5) to calculate isotope fractionation factors. Note that the first term on the right-hand side (RHS) is the partition function ratio of the ZPE calculated using unscaled ab initio frequencies (βZPEunscaled).
Measured hydrogen isotopic compositions of each incubated ketone substrate (δ2HK) and the corresponding water (δ2HW) at equilibrium. a) Standard deviations from 2-3 replicates of each ketone sample and from 3-4 replicates of each water sample are shown in parentheses.
METHODS AND NOMENCLATURE
Details of this calibration are given in Chapter 1, where we show that αeq values for Hα in various linear ketone molecules can be calculated with uncertainties < 8‰ for temperatures of 0–100°C. Here, we use the same calculation methods and calibration to systematically estimate αeq values for H positions in other common organic compounds, including alkanes, alkenes, carboxylic acids, esters, alcohols and ethers. As a demonstration of this approach, we calculate αeq values for n -alkane and linear isoprenoid molecules and discuss the implications of these values for paleoenvironmental and petroleum studies.
To compactly summarize data for a large number of compounds, we average data for analogous positions (i.e. methyl, methylene group, etc.) in different molecules to report a single αeq value.
RESULTS AND DISCUSSION
C ALCULATED FRACTIONATION FACTORS
- H in alkanes
- H in alkenes
- H in ketones, carboxylic acids and esters
- H in alcohols and ethers
The secondary H atoms on the methylene group between the double bonds are each coplanar with one of the double bonds. At room temperature, the value of εeq is −160‰ for H at both ends of a conjugated double bond and −140‰ for H in between, which is comparable to H at a single double bond. 74 values ~90 ‰ higher than the value of alkane H at room temperature, which could also be attributed to the strong effect of electron donation of lone pairs of electrons on the singly bonded O atom.
Consequently, the effect on neighboring C−H bonds seems to depend strongly on the spatial orientation of the C−H bond relative to the hydroxyl group.
T HE EFFECTS OF FUNCTIONAL GROUP ON EQUILIBRIUM FRACTIONATIONS OF C- BOUND H
For example, in secondary alcohols where the hydroxyl group is on one side of the carbon chain, Hγ is more affected than Hβ. In contrast, the Cβ-Hβ bonds point in the opposite direction due to the zigzag structure of the aliphatic chain. This pattern is analogous to the well-known effects of substituents on the aromatic ring, which arise from a combination of resonance and inductive effects of the functional group.
In contrast, the π-π conjugation between the C=O double bond and the benzene ring results in electron withdrawal from the arene system due to the high electronegativity of oxygen.
T EMPERATURE DEPENDENCE AND THE EQUILIBRIUM VAPOR - AQUEOUS FRACTIONATIONS
In other words, there is no simple relationship between the electronic properties of a functional group and the shifts in εeq values they induce for Hβ or Hγ. As a result, the effects of functional groups on fractionation at Hβ are generally observed to be uncorrelated with those at Hα (Fig. 2-6).
A PPLICATION TO ORGANIC GEOCHEMICAL STUDIES
- δ 2 H changes during maturation of organic matter in natural systems
- Constraints on the use of organic δ 2 H as a paleoenvironmental proxy
Many recent studies have attempted to use the δ2H values of sedimentary lipids as a proxy for paleoenvironmental conditions (Huang et al., 2002; Dawson, 2004; Sachse et al., 2006). Implicit in this approach are the assumptions that (i) biosynthetic fractionations between water and lipids are known, and (ii) the δ2H values of lipids have not been significantly altered by isotopic exchange. A second approach to evaluate isotopic exchange in ancient lipids is to compare δ2H values between hydrocarbons with n-alkyl and linear isoprenoid carbon skeletons (Andersen et al., 2001; Sessions et al., 2004).
Isotope exchange thus leads to a convergence of δ2H values over time in linear isoprenoids and n-alkanes.
CYCLIC MOLECULES
METHODS
- E XPERIMENTAL METHODS
- Materials
- Isotope exchange experiments
- C OMPUTATIONAL METHODS
- I SOTOPE EXCHANGE EXPERIMENTS
- Exchange kinetics
- Experimental measurements of equilibrium fractionation for H α
- T HEORETICAL CALCULATIONS OF EQUILIBRIUM FRACTIONATION FACTORS
- Conformational change and ring flip
- Theoretical estimates of equilibrium fractionation for H α
- Theoretical estimates of equilibrium fractionation for non-alpha H
- C OMPARISON OF E XPERIMENTAL AND T HEORETICAL E QUILIBRIUM F RACTIONATIONS
- A PPLICATION TO CYCLIC BIOMARKER MOLECULES
- Estimate of equilibrium fractionation for cyclic biomarker molecules
- Application to organic geochemical studies
Similar to acyclic ketones, the δ2H values of cyclic ketones varied systematically with time, clearly recording the progress of 2H/1H exchange with water. For the same ketone samples, using the second normalization gives equilibrium fractionation (εeq values) that are consistently 40–60‰ higher than the εeq values when using the first. Derived equilibrium fractionation values (εeq values) for Hα positions in cyclic ketone substrates are summarized in Table 3-2 and the figure.
ΔG (kJ/mol) is reported as compensation relative to the highest energy conformation.
CONCLUSIONS
For two consecutive GC peaks, the measured isotopic memory is typically 2–4% of the difference in δ2H values between the two. A detailed description of the model algorithm and selection of appropriate parameters is in the appendix. Fg is calculated by mass balance from the prescribed δ2H values of the analyte (assuming no chromatographic separation of isotopes) and background hydrogen.
Calculation of the fractional abundance of 2H in the adsorbed and gas phase is based on isotopic mass balance.
M ATERIALS
Each 2H-labeled ester was then mixed quantitatively with a stock solution of the same unlabeled ester (Sigma-Aldrich) in different ratios to give 6 standards each for EP and PP with δ2H values ranging between -230 and +800‰. Values of δ2H for CH4 and H2 reference gases were determined using the GC/P/IRMS system by comparison with a series of n-alkanes with known δ2H values (Sessions et al., 2001a). It is important that the δ2H values for the n-alkanes - determined independently of offline combustion/reduction - and CH4 and H2 gases are similar, so that memory effects will have little influence on the measurements.
The approximate δ2H values of synthetic esters were then measured against the CH4 reference gas.
I SOTOPIC A NALYSES
Various combinations and amounts of the two esters were then further mixed and diluted to produce sample solutions analyzed by GC/P/IRMS. This valve was used to feed discrete spikes of reference gas into the GC system at a point directly downstream of the analytical column (Figure 4-1). These peaks, which are also variable in magnitude and time, do not experience any potential memory effects associated with GC and pyrolysis systems.
To minimize variations in δ2H values due to changing background conditions during GC procedures, the following protocol was used for all sample analyses.
R AMAN SPECTROSCOPY
The value of the H3+ factor was very stable at 2.9–3.2 ppm/mV, and all H3+ corrections were performed via ISODAT software. Each chromatogram consisted of four peaks of the reference gas CH4, followed at an interval of 100 s by the analytes to be measured. All peak sizes were held constant at 16 ± 2 Vs for CH4 and 20 ± 2 Vs for esters (expressed as integrated mass-2 peak area), except where noted below.
The final CH4 peak from the first set — before the esters — was used as a calibration peak for all experiments with an assigned value of -148‰.
N UMERICAL MODEL
The concentrations and 2H/1H ratios of gas packets in the input stream are varied sequentially to simulate chromatographic peaks. H2 in the input stream is allowed to interact with adsorption sites uniformly coated on the inner wall of the pyrolysis reactor during its transit. The concentration and 2H/1H ratio of H2 gas leaving the reactor is calculated and recorded as the output signal (see Appendix for details on the model and algorithm).
Parameters required by the model include: (i) the number of adsorption sites surrounding the reactor tube, which is estimated as a product of the available carbonaceous material (typically 0.8–1.0 mg) in the pyrolysis tubes used and the concentration of strong (~20 ppm) ) or weak (~200 appm) adsorption sites in pyrolytic carbon (Kanashenko et al., 1996); (ii) the desorption rate constant for hydrogen adsorbed on these sites, estimated to be ~1.1 s-1 based on the corresponding C-H bond dissociation energy (see Theory and Model section); (iii) average H2 concentration in a typical chromatographic peak, estimated to be ~890 Pa using typical carrier gas flow rate, reactor tube dimensions and temperature, peak shape (assumed to be Gaussian with σ = 5 s), and the amount of analyte injected (50 nmol H2); and (iv) background concentration and ratio of 2H/1H hydrogen in the carrier gas.
EXPERIMENTAL SECTION
- V ARIATIONS IN 2 H/ 1 H RATIOS
- V ARIATION IN TIME SEPARATION
- V ARIATION IN RELATIVE ABUNDANCE
In the first experiment (A in Table 4-1), each chromatogram contained ethyl palmitate (EP) and propyl palmitate (PP) at equal concentrations and constant time separation. B (shifted up from exp. B data for clarity), as discussed in the Theory and Model section. In this experiment, the value of M was measured to be approximately two times greater than that measured in the EP + CH4 test during the same period of the reactor (Table 4-2).
Clearly this is not the case, and the participation of background hydrogen in the memory phenomenon offers a possible explanation.
THEORY AND MODEL SECTION
- P HYSICAL BASIS FOR MEMORY EFFECTS
- H YDROGEN - GRAPHITE INTERACTIONS
- M ODEL OF MEMORY EFFECTS
On the surface of the aromatic plane, adsorption occurs both through molecular or dissociative adsorption. It continuously equilibrates with H2 in the gas phase, but on a time scale similar to that of the chromatographic separation. The size of the adsorbed pool and thus the resulting memory effect is controlled by the amount and structure of carbonaceous material in the pyrolysis reactor.
4-2 is the result of model tuning, the prediction of a linear dependence of the δ2H value for peak 2 on the δ2H value of peak 1 is robust, regardless of the parameters chosen.
DISCUSSION
- M ODEL SIMULATIONS OF EXPERIMENTAL DATA
- I MPACT AND MITIGATION OF MEMORY EFFECTS IN COMPLEX SAMPLES
Given a typical analytical precision for δ2H values of ~4‰, we suggest that—as a rule of thumb—memory effects will become significant when peaks differ in δ2H values by more than 100‰. Memory effects will tend to reduce the measured differences in δ2H values between these peaks, leading to the appearance of scale compression. A simple calculation assessing the impact of memory effects on these analyzes shows that the trend may derive almost entirely from isotopic memory, rather than true scale compression.
A simple and robust strategy to separately assess memory effects and scale compression is currently lacking.
CONCLUSIONS
The molar amount and isotopic composition of hydrogen in the graphite phase are maintained from one gas parcel to another. For similar reasons, diffusion within and between gas parcels and the graphite surface is ignored in the model. In the model, the molar amount of hydrogen (n) and the fractional deuterium abundance (F . = [2H]/[2H+1H]) are tracked for each gas parcel (ng, Fg) and for the graphite surface (na, Fa ).
There is no change in the abundance of hydrogen in any phase because - for strong adsorption sites - surface coverage remains complete.
