EQUILIBRIUM FRACTIONATION

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EQUILIBRIUM FRACTIONATION

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  • Geol. 656 Isotope Geochemistry

    Lecture 26 Spring 2003

    190 4/2/03

    STABLE ISOTOPE THEORY: EQUILIBRIUMFRACTIONATION

    INTRODUCTIONStable isotope geochemistry is concerned with variations of the isotopic compositions of light ele-

    ments arising from chemical fractionations rather than nuclear processes. The elements most com-monly studied are H, Li, B, C, N, O, Si, S and Cl. Of these, O, H, C and S are by far the most impor-tant. Most of these elements have several common characteristics:

    They have low atomic mass. The relative mass difference between the isotopes is large. They form bonds with a high degree of covalent character. The elements exist in more than one oxidation state (C, N, and S), form a wide variety of com-

    pounds (O), or are important constituents of naturally-occurring solids and fluids. The abundance of the rare isotope is sufficiently high (generally at least tenths of a percent) to

    facilitate analysis.It was once thought that elements not meeting these criteria would not show measurable variation

    in isotopic composition. However, as new techniques offering greater sensitivity and higher precisionhave become available (particularly use of the MC-ICP-MS), geochemists have begun to explore iso-topic variations of metals such as Mg, Ca, Ti, Cr, Fe, Zn, Cu, Ge, Mo, Ti, and Tl. The isotopic varia-tions observed in these metals have generally been quite small, except in materials affected or pro-duced by biologically processes, where fractionations are generally larger. Nevertheless, some geo-logically useful information has been obtained from isotopic study of these metals and exploration oftheir isotope geochemistry continues.

    Stable isotopes can be applied to a variety of problems. One of the most common is geothermome-try. Another is process identification. For instance, plants that produce C4 hydrocarbon chains(that is, hydrocarbon chains 4 carbons long) as their primary photosynthetic products fractionatecarbon differently than to plants that produce C3 chains. This fractionation is retained up the foodchain. This allows us to draw some inferences about the diet of fossil mammals from the stable iso-tope ratios in their bones. Sometimes stable isotopes are used as 'tracers' much as radiogenic isotopesare. So, for example, we can use oxygen isotope ratios in igneous rocks to determine whether theyhave assimilated crustal material. The extent of isotopic fractionation varies inversely with tem-perature: fractionations are large at low temperature and small at high temperature.

    NOTATION AND DEFINITIONSThe d Notation

    Variations in stable isotope ratios are typically in the parts per thousand range and hence are gen-erally reported as permil variations, d, from some standard. Oxygen isotope fractionations are gener-ally reported in permil deviations from SMOW (standard mean ocean water):

    d18O = (18O/16O)sam -(18O/16O)SMOW

    (18O/16O)SMOW

    103 26.1

    The same formula is used to report other stable isotope ratios. Hydrogen isotope ratios, dD, are re-ported relative to SMOW, carbon isotope ratios relative to Pee Dee Belemite carbonate (PDB), ni-trogen isotope ratios relative to atmospheric nitrogen, and sulfur isotope ratios relative to troilite inthe Canyon Diablo iron meteorite. Cl isotopes are also reported relative to seawater; Li and B arereported relative to NBS (which has now become NIST: National Institute of Standards and Tech-nology) standards. Unfortunately, a dual standard has developed for reporting O isotopes. Isotope

  • Geol. 656 Isotope Geochemistry

    Lecture 26 Spring 2003

    191 4/2/03

    ratios of carbonates are reported relative to the PDB carbonate standard. This value is related toSMOW by:

    d1 8OPDB = 1.03086d1 8OSMOW + 30.86 26.2Table 26.1 lists the values for standards used in stable isotope analysis.

    The Fractionation FactorAn important parameter in stable isotope geochemistry is the fractionation factor, a. It is defined

    as:

    aA- B RARB

    26.3

    where RA and RB are the isotope ratios of two phases, A and B.The fractionation of isotopes between two phases is often also reported as A-B = dA dB. The rela-

    tionship between and a is: (a - 1)103 or 103 ln a 26.4

    We derive it as follows. Rearranging equ. 26.1, we have:RA = (dA + 103)RSTD/103 26.5

    where R denotes an isotope ratio. Thus a may be expressed as:

    a =(dA +10

    3)RSTD /103

    (dB +103)RSTD /10

    3 =(dA +10

    3)(dB +10

    3)26.6

    Subtracting 1 from each side and rearranging, we obtain:

    a -1= (dA - dB )(dB +10

    3)@

    (dA - dB )103

    = D 10-3 26.7

    since d is generally

  • Geol. 656 Isotope Geochemistry

    Lecture 26 Spring 2003

    192 4/2/03

    have been taught that oxygen is oxygen, and its properties are dictated by its electronic structure. Inthe following sections, we will see that quantum mechanics predicts that mass affects the strength ofchemical bonds and the vibrational, rotational, and translational motions of atoms. These quantummechanical effects predict the small differences in the chemical properties of isotopes quite accu-rately. We shall now consider the manner in which isotopic fractionations arise.

    The electronic structures of all isotopes of an element are identical and since the electronic structuregoverns chemical properties, these properties are generally identical as well. Nevertheless, smalldifferences in chemical behavior arise when this behavior depends on the frequencies of atomic andmolecular vibrations. The energy of a molecule can be described in terms of several components: elec-tronic, nuclear spin, translational, rotational and vibrational. The first two terms are negligible andplay no role in isotopic fractionations. The last three terms are the modes of motion available to amolecule and are the cause of differences in chemical behavior among isotopes of the same element.Of the three, vibration motion plays the most important role in isotopic fractionations. Transla-tional and rotational motion can be described by classical mechanics, but an adequate description ofvibrational motions of atoms in a lattice or molecule requires the application of quantum theory. Aswe shall see, temperature-dependent equilibrium isotope fractionations arise from quantum mechani-cal effects in vibrational motions. These effects are, as one might expect, generally small. For exam-ple, the equilibrium constant for the reaction

    12

    C16O2 + H218O = 1

    2C18O2 + H2

    16O

    is only about 1.04 at 25C. Figure 26.1 is a plot of the

    potential energy of a dia-tomic molecule as a functionof distance between the twoatoms. This plot looksbroadly similar to one wemight construct for twomasses connected by a spring.When the distance betweenmasses is small, the spring iscompressed, and the po-tential energy of the systemcorrespondingly high. Atgreat distances between themasses, the spring isstretched and the energy ofthe system also high. Atsome intermediate distance,there is no stress on thespring, and the potential en-ergy of the system is at aminimum (energy would benevertheless be conservedbecause kinetic energy is at amaximum when potentialenergy is at a minimum).The diatomic oscillator, forexample consisting of an N aand a Cl ion, works in ananalogous way. At small in-

    Figure 26.1. Energy-level diagram for the hydrogen atom. Fundamen-tal vibration frequencies are 4405 cm-1 for H2, 3817 cm-1 for HD, and 3119cm-1 for D2. The zero-point energy of H2 is greater than that for HDwhich is greater than that for D2. From O'Neil (1986).

  • Geol. 656 Isotope Geochemistry

    Lecture 26 Spring 2003

    193 4/2/03

    teratomic distances, the electron clouds repel each other (the atoms are compressed); at large dis-tances, the atoms are attracted to each other by the net charge on atoms. At intermediate distances,the potential energy is at a minimum. The energy and the distance over which the atoms vibrate arerelated to temperature.

    In quantum theory, a diatomic oscillator cannot assume just any energy: only discrete energy levelsmay be occupied. The permissible energy levels, as we shall see, depend on mass. Quantum theoryalso tells us that even at absolute 0 the atoms will vibrate at a ground frequency n0. The system willhave an energy of 1/2h n 0, where h is Planck's constant. This energy level is called the Zero Point En-ergy (ZPE). Its energy depends the electronic arrangements, the nuclear charges, and the positions ofthe atoms in the molecule or lattice, all of which will be identical for isotopes of the same element.However, the energy also depends on the masses of the atoms involved, and thus will be different fordifferent for isotopes. The vibrational energy level for a given quantum number will be lower for abond involving a heavier isotope of an element, as suggested in Figure 26.1. Thus bonds involvingheavier isotopes will be stronger. If a system consists of two possible atomic sites with different bondenergies and two isotopes of an element available to fill those sites, the energy of the system is mini-mized when the heavy isotope occupies the site with the stronger bond. Thus at equilibrium, theheavy isotope will tend to occupy the site with the stronger bond. This, in brief, is why equilibriumfractionations arise. Because bonds involving lighter isotopes are weaker and more readily broken,the lighter isotopes of an element participate more readily in a given chemical reaction. If the reac-tion fails to go to completion, which is often the case, this tendency gives rise to kinetic fractionationsof isotopes. There are other causes of kinetic fractionations as well, and will consider them in thenext lecture. We will no consider i