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Kinetic and Equilibrium Mass-Dependent Fractionation Laws In Geochimica et Cosmochimica Acta, Vol. 66, No. 6, pp. 1095–1104, 2002 Copyright © 2002 Elsevier Science Ltd Pergamon Printed in the USA. All rights reserved 0016-7037/02 $22.00 ϩ .00 PII S0016-7037(01)00832-8 Kinetic and equilibrium mass-dependent isotope fractionation laws in nature and their geochemical and cosmochemical significance 1, ,† 2 3 EDWARD D. YOUNG, * ALBERT GALY, and HIROKO NAGAHARA 1Department of Earth Sciences, University of Oxford, Oxford OX1 3PR, UK 2Department of Earth Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EQ, UK 3Geological Institute, University of Tokyo, Hongo, Tokyo 113, Japan (Received March 30, 2001; accepted in revised form September 26, 2001) Abstract—The mass-dependent fractionation laws that describe the partitioning of isotopes are different for kinetic and equilibrium reactions. These laws are characterized by the exponent relating the fractionation ␣ ϭ ␣␤ ␤ Ϫ factors for two isotope ratios such that 2/1 3/1. The exponent for equilibrium exchange is (1/m1 Ϫ Ͻ Ͻ 1/m2)/(1/m1 1/m3), where mi are the atomic masses and m1 m2 m3. For kinetic fractionation, the masses used to evaluate ␤ depend upon the isotopic species in motion. Reduced masses apply for breaking bonds whereas molecular or atomic masses apply for transport processes. In each case the functional form of ␤ the kinetic is ln(M1/M2)/ln(M1/M3), where Mi are the reduced, molecular, or atomic masses. New high-precision Mg isotope ratio data confirm that the distinct equilibrium and kinetic fractionation laws can be resolved for changes in isotope ratios of only 3‰ per amu. The variability in mass-dependent fractionation ⌬17 ⌬17 laws is sufficient to explain the negative O of tropospheric O2 relative to rocks and differences in O between carbonate, hydroxyl, and anhydrous silicate in Martian meteorites. (For simplicity, we use integer amu values for masses when evaluating ␤ throughout this paper.) Copyright © 2002 Elsevier Science Ltd 1. INTRODUCTION yr ago. More recently, Matsuhisa et al. (1978) drew attention to the distinction between equilibrium and kinetic fractionation laws It is common practice in geochemistry and cosmochemistry with reference to the three isotopes of oxygen. Nevertheless, to describe mass-dependent isotope fractionation processes because the differences were in the noise of the measurements, the with one fractionation law represented by a single curve on a distinction has been largely ignored in subsequent literature. plot of one isotope ratio against another (the three-isotope plot With the advent of multiple collector-inductively coupled in which isotope ratios are usually expressed as fractional plasma-source mass spectrometry (MC-ICPMS) and improved differences from a standard). This practice is justified when the methods for preparing gases for introduction into gas-source analytical precision associated with light element stable isotope mass spectrometers, there is a new-found ability to measure the Ϯ ratio determinations is no better than 0.1‰ per amu. It is also relative abundances of the isotopes of O, Mg, Fe, and other common practice to approximate the non-linear fractionation elements with precision approaching 50 ppm (e.g., Miller et al., curves as straight lines in three-isotope space. When used to 1999; Galy et al., 2000; Zhu et al., 2001). This new level of characterize isotope reservoirs on Earth, these straight lines are precision is sufficient to distinguish different mass fractionation referred to as terrestrial mass fractionation lines. Deviations from laws over limited ranges in isotopic composition. such terrestrial mass fractionation lines are used as indicators of Herein we summarize the theoretical basis for mass-depen- non-mass-dependent isotope effects in geochemistry and cosmo- dent isotope fractionation during equilibrium and kinetic pro- chemistry (e.g., Matsuhisa et al., 1978; Luz et al., 1999). cesses and explore some of the implications of disparate frac- Analytical precision has been improving over the past sev- tionation laws in geochemistry and cosmochemistry. Mass eral years with the result that the assumption of a single fractionation laws for equilibrium and kinetic processes differ fractionation law is no longer valid. Fractionation laws will because kinetic fractionation results from motions that can vary in nature, and in the face of new technologies, it is worth often be described classically using effective masses whereas returning to the concepts of mass-dependent fractionation to equilibrium exchange is purely a quantum phenomenon that appreciate the magnitudes and sources of these variations. Most depends on the atomic masses alone. The differences between importantly, the existence of different fractionation laws inval- these mass-dependent fractionation laws may explain some idates the concept of a single terrestrial fractionation line for a effects previously attributed to non-mass-dependent fraction- given isotope system in three-isotope space. ation, and in any case, must be considered when evaluating the The theoretical basis for mass-dependent isotope fractionation significance of apparent departures from a reference mass frac- resulting from equilibrium and kinetic processes was established tionation line at the sub-per mil level. in a series of papers by Bigeleisen, Urey, and others more than 50 2. EQUILIBRIUM MASS-DEPENDENT ISOTOPE FRACTIONATION * Author to whom correspondence should be addressed (eyoung@ ess.ucla.edu). † Present address: Institute of Geophysics and Planetary Physics and Isotope exchange between two compounds can be written as Department of Earth & Space Sciences, 595 Charles Young Dr. E., ϩ ϭ ϩ UCLA, Los Angeles, CA 90095, USA. aX1 bX2 aX2 bX1 (1) 1095 1096 E. D. Young, A. Galy, and H. Nagahara in which the subscript 2 refers to a heavy isotope and the 1 1 K ͸͑␷2 Ϫ ␷2 ͒ ϭ ͸ͩ Ϫ ͪ f, j subscript 1 refers to the light isotope of element X. The equi- 1, j 2, j ␲2. (7) m1, j m2, j 4 librium constant for Eqn. 1 is j j Q(aX )Q(bX ) Since the masses refer to the atoms in the sum over j, masses ϭ 2 1 Keq (2) can be factored to give for f: Q(aX1)Q(bX2) 1 h 2 1 1 K where Q is the partition function for the isotopic species (iso- ϭ ϩ ͩ ͪ ͩ Ϫ ͪ͸ f, j f 1 ␲2. (8) 24 kb T m1 m2 4 topologue) indicated. j Bigeleisen and Mayer (1947) showed that the partition func- tion ratio for two isotopologues can be written in terms of a The equilibrium isotope fractionation factor ␣a-b for the two classical part and a quantum mechanical part: substances aX and bX is defined as the ratio of the isotope ratios for these two substances at equilibrium and is equivalent 3/ 2 Q2 m2,l to the equilibrium constant for the exchange reaction: ϭ f͹ͩ ͪ (3) Q1 m1,l l ͩX2ͪ where Q refers to the partition function of the particular X1 i ϭ ␣aϪb ϭ a Keq (9) isotopologue i, mi is the mass of the isotope i, and the product X2 is taken for all isotope atoms l. (In this treatment, symmetry ͩ ͪ X1 numbers that would otherwise clutter the presentation, but are b required when evaluating Eqn. 3 for individual molecules, can where each X represents the numbers of isotopes indicated. be ignored without loss of generality because they cancel in the Eqn. 9, 8, and 4 can be combined to yield expressions for the end results where ratios of partition function ratios are taken. equilibrium isotope fractionation factor in terms of mass: Moments of inertia are also ignored for similar reasons.) The ␣aϪb ϭ Ϫ parameter f comprises the quantum part of the partition func- ln ln fa ln fb. (10) tion ratio for two compounds that differ only in their isotopic ϩ Х composition. The classical term (m /m )3/2 represents the The approximation ln(1 x) x can be used to evaluate Eqn. 2,l 1,l ϳ differences in momentum of the isotopes factored out of the 10 (f 1.0x at 298K), leading to partition function ratio. Although it is central to kinetic frac- 1 h 2 1 1 K K tionation effects, the classical term cancels in balanced reac- ␣aϪb ϭ ͩ ͪ ͩ Ϫ ͪ͸ͩ f, j,a Ϫ f, j,bͪ ln ␲2 ␲2 (11) 24 kb T m1 m2 4 4 tions where the numbers of each isotope are conserved, and it j can be omitted from Q2 /Q1 when constructing equilibrium constants. For Eqn. 1 the equilibrium constant is then where Kf,j,a and Kf,j,b are the force constants appropriate for atomic species X in compounds a and b, respectively (Weston, f (aX) 1999). K ϭ (4) eq f (bX) Eqn. 11 shows that for a given pair of isotopes, the fraction- ation factor varies from substance to substance according to where f (aX) and f (bX) are the quantum parts of the isotopo- bond strength. A similar derivation for isotopes 3 and 1 leads to logue partition function ratios for compounds aX and bX, an equation analogous to Eqn. 11. The ratio of the fractionation respectively. factors for the two isotope ratios 2/1 and 3/1 is Urey (1947) and Bigeleisen (1955) made use of the fact that f approaches 1 h 2 1 1 K K ͩ ͪ ͩ Ϫ ͪ͸ͩ f, j,a Ϫ f, j,bͪ 2 2 Ϫ ␲ ␲ 2 ␣a b 24 kb T m1 m2 4 4 1 h ln 2/1 j 2 2 ϭ ϩ ͩ ͪ ͸͑␷ Ϫ ␷ ͒ Ϫ ϭ . (12) f 1 1, j 2, j (5) ␣a b 2 24 kb T ln 3/1 1 h 1 1 Kf, j,a Kf, j,b j ͩ ͪ ͩ Ϫ ͪ͸ͩ Ϫ ͪ ␲2 ␲2 24 kb T m1 m3 4 4 j to evaluate equilibrium constants at temperatures of several hundred degrees K and higher where quantum effects are small.
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