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Hydrogen Isotope Fractionation in the Epidote–Hydrogen and Epidote–Water Systems: Theoretical Study and Implications Abu Md. Asaduzzaman, and Jibamitra GANGULY ACS Earth Space Chem., Just Accepted Manuscript • DOI: 10.1021/ acsearthspacechem.8b00076 • Publication Date (Web): 06 Sep 2018 Downloaded from http://pubs.acs.org on September 12, 2018
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ACS Earth and Space Chemistry
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Hydrogen Isotope Fractionation in the Epidote–Hydrogen and Epidote–Water Systems:
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Theoretical Study and Implications
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Abu Asaduzzaman1,2 and Jibamitra Ganguly*,3
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Middletown, PA 17057, USA
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Materials Science and Engineering, University of Arizona, Tucson, AZ 85721, USA
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Department of Geosciences, University of Arizona, Tucson, AZ 85721, USA
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*Corresponding author
School of Science, Engineering and Technology, Pennsylvania State University - Harrisburg,
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Key words: Hydrogen Isotope, Density Functional Theory, Partition Function, Epidote, Equilibrium Constant
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ABSTRACT Experimental data for hydrogen isotope and in general for stable isotope
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fractionation between mineral and water often tend to be widely divergent primarily because of
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solution and re-precipitation of finely ground mineral grains in the experimental charges. With
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the objective of development of a theoretical methodology for calculation of stable isotope
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fractionation that may be suitably integrated with experimental data to produce better constraints
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on the fractionation behavior, we have calculated hydrogen isotope fractionation in the epidote-
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water and epidote-hydrogen systems for which reliable experimental data are available. The
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calculations were carried out using a combination of classical and statistical thermodynamics and
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quantum chemical methods. The calculated fractionation factors vs. temperature in both systems
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have been found to be in very good agreement with the experimental data and provide a
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theoretical framework for extrapolation of experimental data as a function of temperature.
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1. INTRODUCTION
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Quantitative data about deuterium and hydrogen isotope fractionation between hydrous minerals
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and their fluid environments as function of temperature and pressure permit indirect
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determination of the D/H ratio of the fluid phase from the measured isotopic ratios of the
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hydrous minerals that have important implications for a variety of geological processes. Thus,
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there has been a sustained effort in the geological community to experimentally determine the
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hydrogen isotopic composition of hydrous minerals in equilibrium with water. Additionally,
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knowledge of the hydrogen isotopic fractionation between the solar nebula and nominally
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anhydrous minerals, such as olivine, is of critical importance to understand if the nebular water
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was responsible for a significant component of the Earth’s water via adsorption mediated
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accretion on mineral surfaces during nebular condensation process1,2. The experimental studies
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in the mineral-water systems have, however, suffered from several problems such as failure to
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attain equilibrium fractionation, especially at relatively low temperatures, and solution-
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precipitation that often resulted in wide divergence of the published data. This has led to revival
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of efforts for theoretical calculations using statistical thermodynamics, following the groundwork
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laid by Urey3 and subsequent developments4,5,6,7,8,9.
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The primary purpose of this paper is to calculate hydrogen isotope fractionation between
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epidote and water/molecular hydrogen using a combination of thermodynamics and quantum
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chemical methodologies, and to compare the results with reliable experimental data that are
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available for these systems in order to assess the validity of the theoretical method. Additionally,
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we suggest a method of fitting of the experimental lnα vs. 1/T2 relation for mineral-water
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systems (where α is the bulk isotopic fractionation factor) that is better suited to extrapolation
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than the conventional practice of linear regression and extrapolation. The demonstrated
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agreement between the calculated and experimental data would also lay the groundwork for a
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follow-up study on the hydrogen isotope fractionation between solar nebula and chemisorbed
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water on olivine in the nebular environment.
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2. THERMODYNAMICS OF HYDROGEN ISOTOPE FRACTIONATION IN
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MINERAL-WATER/H2 SYSTEMS
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Let us first consider the general case of hydrogen isotope fractionation between a crystal (x’l)
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and water. The problem may be treated in terms of an exchange reaction as follows.
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x’l(OH) + ½ D2O = x’l(OD) + ½ H2O
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for which the equilibrium constant is
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K =
(a)
(1)
/
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where the parenthetical terms indicate activities of the specified components. From the available
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thermodynamic data on the excess Gibbs free energy of mixing in the CH4-CD4 system10, which
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is found to be symmetrical to composition, the activity coefficient of either species at 50%
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composition is found to be 1.0004 at 120 K. Thus, the deviation from ideality of mixing of the
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isotopic species in a given phase should be expected to be negligible at temperatures of
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geological and planetary interests, and hence the activity of a species may be equated to its mole
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fraction (using a standard state of pure species at P-T condition of interest).
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The experimentally measured mineral/water bulk fractionation factor is given by α=
(2)
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This is equivalent to Ka if D2O and H2O are the only species in water since in that case, a(D2O) =
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(XD)2 and a(H2O)= (XH)2 for the case of ideal mixing, where a and X stand, respectively, for the
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activity and mole fraction of the specified species. However, in general, this is not the case
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because of the presence of other H-D-O species in the system.
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In this work, we are interested in comparing the results of theoretical calculation with the
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experimentally determined bulk isotopic fractionations. The experiments were carried out at ~2-
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4 kb pressure in sealed gold capsules that are effectively impervious to hydrogen diffusion at
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such pressures. If the molar H2 to O2 ratio within the gold capsules remains close to 2 (as would
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be the case for a closed system in which H2 and O2 are derived only from the stoichiometric
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dissociation of water), then H2 content of the vapor phase becomes vanishingly small11 and
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hence may be easily neglected in the mass balance calculations. Thus, we need to consider only
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three species in the H-D-O system, viz. H2O, D2O and HDO. These species are related by the
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equilibrium ½ H2O + ½ D2O = HDO
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for which the equilibrium constant is
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K =
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⁄
(b)
(3)
For the molar abundance of D and H in water we can write
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D ≈ 2D O + HDO = 2D O + K $H OD O%&/
(4a)
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H ≈ 2H O + HDO = 2H O + K $H OD O%&/
(4b)
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(D, H, D2O and HDO in these equations represent the molar abundances of the respective
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species) so that
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'()*
&⁄
=
φ
(5)
where φ
/ , -. / = / , -. /
Thus, α, and Ka are related according to
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αx 0 l − water ≡
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=
-9 :
(7)
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It is easy to see from the last two equations that α equals Ka when Kb = 2. From available
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thermodynamic data in the JANAF Tables12, Kb is found to increase from 1.9239 at 300 K to
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1.9892 at 1200 K. Thus, since Kb/2 ~1 and (D2O) 0 for product,
