12
Geol. 656 Isotope Geochemistry Lecture 26 Spring 2003 190 4/2/03 STABLE ISOTOPE THEORY: EQUILIBRIUM FRACTIONATION INTRODUCTION Stable 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 precision have 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 of their 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 ‘C 4 ’ hydrocarbon chains (that is, hydrocarbon chains 4 carbons long) as their primary photosynthetic products fractionate carbon differently than to plants that produce ‘C 3 ’ chains. This fractionation is retained up the food chain. 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 isotopes are. So, for example, we can use oxygen isotope ratios in igneous rocks to determine whether they have 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 DEFINITIONS The 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): d 18 O = ( 18 O/ 16 O) sam - ( 18 O/ 16 O) SMOW ( 18 O/ 16 O) SMOW È Î ˘ ˚ ¥ 10 3 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 in the Canyon Diablo iron meteorite. Cl isotopes are also reported relative to seawater; Li and B are reported 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

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Page 1: EQUILIBRIUM FRACTIONATION

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

È Î

˘ ˚

¥ 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

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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 ≡RA

RB26.3

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 +103)RSTD /103

(dB +103)RSTD /103 =(dA +103)(dB +103)

26.6

Subtracting 1 from each side and rearranging, we obtain:

a -1=(dA - dB )

(dB +103)@

(dA - dB )103 = D ¥10-3 26.7

since d is generally << 103. The second equation in 26.4 results from the approximation that for x ≈ 1, lnx ≈ 1 – x.As we will see, a is related to the equilibrium constant of thermodynamics by

aA-B = (K/K•)1/n 26.8where n is the number of atoms exchanged, K• is the equilibrium constant at infinite temperature, andK is the equilibrium constant is written in the usual way (except that concentrations are used ratherthan activities because the ratios of the activity coefficients are equal to 1, i.e., there are no isotopiceffects on the activity coefficient).

THEORY OF ISOTOPIC FRACTIONATIONSIsotope fractionation can originate from both kinetic effects and equilibrium effects. The former

might be intuitively expected (since for example, we can readily understand that a lighter isotopewill diffuse faster than a heavier one), but the latter may be somewhat surprising. After all, we

Table 26.1. Isotope Ratios of Stable IsotopesElement Notation Ratio Standard Absolute RatioHydrogen dD D/H (2H/1H ) SMOW 1.557 ¥ 10-4

Lithium d6Li 6li/7Li NBS L-SVEC 0.08306Boron d11B 11B/10B NBS 951 4.044Carbon d13C 13C/12C PDB 1.122 ¥ 10-2

Nitrogen d15N 15N/14N atmosphere 3.613 ¥ 10-3

Oxygen d18O 18O/16O SMOW, PDB 2.0052 ¥ 10-3

d17O 17O/16O SMOW 3.76 ¥ 10-4

Chlorine d37Cl 37Cl/35Cl seawater ~0.31978Sulfur d34S 34S/32S CDT 4.43 ¥ 10-2

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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 =

12

C18O2 + H216O

is only about 1.04 at 25°C. 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).

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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 in greater detail the basis for equilibrium fractionation, and seethat they can be predicted from statistical mechanics.

Equilibrium FractionationsUrey (1947) and Bigeleisen and Mayer (1947) pointed out the possibility of calculating the equilib-

rium constant for isotopic exchange reactions from the partition function, q, of statistical mechanics.In the following discussion, bear in mind that quantum theory states that only discrete energies areavailable to an atom or molecule.

At equilibrium, the ratio of molecules having internal energy Ei to those having the zero point en-ergy E0 is:

ni

n0

= gie-Ei / kT 26.9

where n0 is the number of molecules with ground-state or zero point energy, ni is the number of mole-cules with energy Ei and k is Boltzmann's constant, T is the thermodynamic, or absolute, temperature,and g is a statistical weight factor used to account for possible degenerate energy levels* (g is equal tothe number of states having energy Ei). The average energy (per molecule) in a system is given by theBoltzmann distribution function, which is just the sum of the energy of all possible states times thenumber of particles in that state divided by the number of particles in those states:

niEii

Â

nii

Â=

giEie-Ei / kTÂ

gie-Ei / kTÂ

26.10

The partition function, q, is the denominator of this equation:

* The energy level is said to be 'degenerate' if two or more states have the same energy level Ei.

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q = gie-Ei / kTÂ 26.11

Substituting 26.11 into 26.10, we can rewrite 26.10 in terms of the partial derivatives of q:

E = kT 2 ∂ lnq∂T

26.12

We will return to these equations shortly, but first let’s see how all this relates to some parametersthat are more familiar from thermodynamics and physical chemistry. It can also be shown (but wewon't) from statistical mechanics that entropy† is related to energy and q by

S =UT

R lnq 26.13

Where R is the ideal gas constant and U is the internal energy of a system. We can rearrange this as:

U - TS = -R lnq 26.14And for the entropy and energy changes of a reaction, we have:

DU - TDS = -R ln qNx’ 26.15

where x in this case is the stoichiometric coefficient. In this notation, the stoichiometric coefficientis taken to have a negative sign for reactants (left side of reaction) and a positive sign for products(right side of reaction). The left hand side of this equation is simply the Gibbs Free Energy change ofreaction under conditions of constant volume (as would be the case for an isotopic exchange reaction),so that

DG = -R ln qNx’ 26.16

The Gibbs Free Energy change is related to the equilibrium constant, K, by:

DG = -RT lnK 26.17so the equilibrium constant for an isotope exchange reaction is related to the partition function as:

K = qNx’ 26.18

For example, in the reaction involving exchange of 18O between H2O and CO2, the equilibrium constantis simply:

K =qC 16Oq

H218O

qC 18OqH216O

26.19

The point of all this is simply that: the usefulness of the partition function is that it can be calcu-lated from quantum mechanics, and from it we can calculate equilibrium fractionations of isotopes.

The partition function can be written as:

qtotal = qtrqvibqrot 26.20

† Entropy is defined in the second law of thermodynamics, which states:

dS =dQrev

Twhere Qrev is heat gained by a system in a reversible process. Entropy can be thought of as a measure of the randomnessof a system.

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i.e., the product of the translational, rotational and vibrational partition functions. It is convenientto treat these three modes of motion separately. Let's now do so.

Translational Partition FunctionWriting a version of equation 26.11 for translational energy, qtrans is expressed as:

qtrans = gtr,ie-Etr ,i / kT

i 26.21

Now all that remains is to find and expression for translational energy and a way to do the summa-tion.At temperatures above about 2 K, translational energy levels are so closely spaced that they essen-tially form a continuum, so we can use a classical mechanical approach to calculating the energy. Thequantum translational energy of a particle in a cubical box is given by:

Etrans =n2h2

8md 2 26.22

where n is the quantum energy level, h is Planck’s constant, d is the length of the side of the cube, andm is mass of the particle. Substituting 26.22 into 26.21 and integrating:

qtrans = e-n 2h 2 8md 2kT dn0

Ú =2pmkT( )1/ 2

hd 26.23

gives an expression for qtrans for each dimension. The total three-dimensional translational partitionfunction is then:

qtrans =2pmkT( )3/2

hV 26.24

where V is volume and is equal to d3. (It may seem odd that the volume should enter into the calcula-tion, but since it is the ratio of partition functions that are important in equations such as 26.19, a l lterms in 26.24 except mass will eventually cancel.) If translation motion were the only component ofenergy, the equilibrium constant for exchange of isotopes would be simply the ratio of the molecularweights raised to the 3/2 power. If we define the translational contribution to the equilibrium con-stant as Ktr as:

Ktr = qtrx’ 26.25

Ktr reduces to the product of the molecular masses raised to the stoichiometric coefficient times three-halves:

Ktr = Mixi 3/2

i’ 26.26

where we have replace m with M, the molecular mass. Thus the translational contribution to thepartition function and fractionation factor is independent of temperature.

Rotational Partition FunctionThe allowed quantum rotational energy states are:

Etrans =j( j +1)h2

8p 2I26.27

where j is the rotational quantum number and I is the moment of inertia. For a diatomic molecule, I=µd2, where d is the bond length, mi is the atomic mass of atom i, and µ is reduced mass:

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m =m1m2

m1 + m2

26.28

A diatomic molecule will have two rotational axes, one along the bond axis, the other perpendicularto it. Hence in a diatomic molecule, j quanta of energy may be distributed 2j+1 ways because there aretwo possibilities for every value of j except j = 0, for which there is only one possible way. The statis-tical weight factor is therefore 2j + 1. Hence:

qrot = (2 j +1)e j( j+1)h2 /8p 2 Ii kT

i 26.29

Again the spacing between energy levels is relatively small (except for hydrogen) and 26.29 may beevaluated as an integral. For a diatomic molecule, the partition function for rotation is given by:

qrot =8p 2IkT

sh2 26.30

where s is a symmetry factor. It is 1 for a heteronuclear diatomic molecule (such as CO or 18O16O), and2 for a homonuclear diatomic molecule such as 16O2. Equ. 26.30 also holds for linear polyatomic mole-cules with the symmetry factor equal to 2 if the molecule has a plane of symmetry (e.g., CO2) and 1 i fit does not. For non-linear polyatomic molecules, the partition function is given by:

qrot =8p 2(8p 2ABC)1/ 2(kT)3 / 2

sh3 26.31

where A, B, and C are the principal moments of inertia of the molecule and s is equal to the number ofequivalent ways of orienting the molecule in space (e.g., 2 for H2O, 12 for CH4). In calculating the ro-tational contribution to the partition function and equilibrium constant, all terms cancel except formoment of inertia and the symmetry factor, and the contribution of rotational motion to isotope frac-tionation is also independent of temperature. For diatomic molecules we may write:

Ktr =Ii

s i

Ê

Ë Á

ˆ

¯ ˜

x

i’ 26.32

In general, bond lengths are also independent of the isotope involved, so the moment of inertia termmay be replaced by the reduced masses.

Vibrational Partition FunctionWe will simplify the calculation of the vibrational partition function by treating the diatomic

molecule as a harmonic oscillator (as Fig. 26.1 suggests, this is a good approximation in most cases). Inthis case the quantum energy levels are given by:

Evib = n +12

Ê

Ë Á

ˆ

¯ ˜ hn 26.33

where n is the vibrational quantum number and n is vibrational frequency. Unlike rotational and vi-brational energies, the spacing between vibrational energy levels is large at geologic temperatures, sothe partition function cannot be integrated. Instead, it must be summed over all available energy lev-els. For diatomic molecules the summation is simply equal to:

qvib =e-hn /2kT

1- e-hn /2kT 26.34

For a non-linear polyatomic molecule consisting of i atoms and the product is performed over all vi -brational modes, l (there are only 3i-5 modes of motion for a linear polyatomic molecule, hence theproduct is carried out only to 3i-5):

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qvib =e-hn l /2kT

1- e-hn l /2kTl

3i-6

’ 26.35

At room temperature, the exponential term in the denominator approximates to 0, and the denomina-tor therefore approximates to 1, so the relation simplifies to:

qvib @ e-hn /2kT 26.36Thus at low temperature, the vibrational contribution to the equilibrium constant approximates to:

Kvib = e-xlhn l /2kT

l

’ 26.37

which has an exponential temperature dependence.The full expression for the equilibrium constant calculated from partition functions for diatomic

molecules is then:

K = qitrqi

rotqivib( )xi

i’ = Mi

3/2 Ii

s i

È

Î Í

˘

˚ ˙

e-xhn i /2kT

1- e-xhn i /2kT

Ê

Ë Á

ˆ

¯ ˜

xi

i’ 26.38

By use of the Teller-Redlich spectroscopic theorem*, this equation simplifies to:

K =1

s i

mi3/2 e-U /2

1- e-U

Ê

Ë Á

ˆ

¯ ˜

x

i’ 26.39

where m is the mass of the isotope exchanged and U is defined as:

U =hnkT

=hcwkT

26.40

and w is the vibrational wave number.Example of fractionation factor calculated from partition functions

To illustrate the use of partition functions in calculating theoretical fractionation factors, we willdo the calculation for a very simple reaction: the exchange of 18O and 16O between O2 and CO:

C1 6O + 1 8O1 6O = C1 8O + 1 6O2 26.41The choice of diatomic molecules greatly simplifies the equations. Choosing even a slightly morecomplex model, such as CO2 would complicate the calculation because there are more vibrationalmodes possible. Valley (2001) provides an example of the calculation for more complex molecules.The equilibrium constant is:

K =[16O2 ][C18O]

[18O16O][C16O]26.42

where we are using the brackets in the unusual chemical sense to denote concentration. We can use con-centrations rather than activities or fugacities because the activity coefficient of a phase is inde-pendent of its isotopic compositions. The fractionation factor, a, is defined as:

* The Teller-Redlich Theorem relates the products of the frequencies for each symmetry type of the two isotopes to theratios of their masses and moments of inertia:

m2

m1

Ê

Ë Á

ˆ

¯ ˜

3/2I1

I2

M1

M 2

Ê

Ë Á

ˆ

¯ ˜

3/2

=U1

U2

where m is the isotope mass and M is the molecular mass. We need not concern ourselves with its details.

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Geol. 656 Isotope Geochemistry

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a =(18O /16 O)CO

(18O /16 O)O2

26.43

We must also consider the exchange reaction:1 8O1 8O + 1 6O1 6O = 21 6O1 8O

for which we can write a second equilibrium constant, K2. It turns out that when both reactions areconsidered, a ≈ 2K. The reason for this is as follows. The isotope ratio in molecular oxygen is relatedto the concentration of the 2 molecular species as:

18O16O

Ê

Ë Á

ˆ

¯ ˜

O2

=[18O16O]

[18O16O]+ 2[16O2 ]26.44

(16O2 has 2 16O atoms, so it must be counted twice) whereas the ratio in CO is simply:

18O16O

Ê

Ë Á

ˆ

¯ ˜

CO

=[C18O][C16O]

26.45

Letting the isotope ratio equal R, we can solve 26.44 for [18O16O]:

[18O /16 O] = 2[16O2 ]RO2

1- RO2

26.46

and substitute it into 26.42:

K =(1- RO2

)[C18O]2RO2

[C16O]= 2

(1- RO2)RCO

2RO2

26.47

Since the isotope ratio is a small number, the term (1 –#R) ≈ 1, so that:

K @ 2 RCO

2RO2

=a2

26.48

We can calculate K from the partition functions as:

K =q16 O2

qC18O

q18 O16OqC16O

26.49

where each partition function is the product of the translational, rotational, and vibrational parti-tion functions. However, we will proceed by calculating an equilibrium constant for each mode of mo-tion. The total equilibrium constant will then be the product of all three partial equilibrium con-stants. For translational motion, we noted the ratio of partition functions reduces to the ratio of mo-lecular masses raised to the 3/2 power. Hence:

Ktr =q16 O2

qC18O

q18 O16OqC16O

=M 16 O2

MC18O

M 18 O16OMC16O

Ê

Ë Á Á

ˆ

¯ ˜ ˜

3/2

=32 ¥ 3034 ¥ 28

Ê

Ë Á

ˆ

¯ ˜

3/2

=1.0126 26.50

We find that CO would be 12.6‰ richer in 18O if translational motions were the only modes of energyavailable.

In the expression for the ratio of rotational partition functions, all terms cancel except the momentof inertia and the symmetry factors. The symmetry factor is 1 for all the molecules involved except16O2. In this case, the terms for bond length also cancel, so the expression involves only the reducedmasses. So the expression for the rotational equilibrium constant becomes:

Ktr =q16 O2

qC18O

q18 O16OqC16O

=I 16 O2

IC18O

2I 18 O16OIC16O

Ê

Ë Á Á

ˆ

¯ ˜ ˜

3/2

=12

16 ¥1616 +16

¥12 ¥1812 +18

18 ¥1618 +16

¥12 ¥1612 +16

Ê

Ë

Á Á Á

ˆ

¯

˜ ˜ ˜

3/2

=0.9916

226.51

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(ignore the 1/2, it will cancel out later). If rotational were the only mode of motion, 18O would be 8per mil more abundant in O2.

The vibrational equilibrium constant may be expressed as:

Kvib =q16 O2

qC18O

q18 O16OqC16O

= e-h(n16 O2

+nC18O

-nC16O

-n18 O16O)

2KT

Kvib = e-xlhn l /2kT

l

’ 26.52

Since we expect the difference in vibrational frequencies to be quite small, we may make the ap-proximation ex = x + 1. Hence:

Kvib @1+h

2KTnC16O -nC18O{ }- n 16 O2

-n 18 O16O{ }[ ] 26.53

Let's make the simplification that the vibration frequencies are related to reduced mass as in a sim-ple Hooke's Law harmonic oscillator:

n =1

2pkm

26.54

where k is the forcing constant, and depends on bond strength, and will be identical for all isotopes ofan element. In this case, we may write:

nC18O = nC16O

mC18O

mC16O

= nC16O

2021

= 0.976nC16O 26.55

A similar expression may be written relating the vibrational frequencies of the oxygen mole-cule: n O1816 O = 0.9718n O2

16 Substituting these expressions in the equilibrium constant expression, we have:

Kvib =1+h

2kTnC16O[1- 0.976]-n 16 O2

[1- 0.9718]( )The measured vibrational frequencies of CO and O2 are 6.50 ¥ 1013 sec-1 and 4.74 ¥ 1013 sec-1. Substi-tuting these values and values for the Planck and Boltzmann constants, we obtain:

K =1+5.976

TAt 300 K (room temperature), this evaluates to 1.0199.

We may now evaluate the total exchange equilibrium constant at 300 K as:

K = KtrKrotKvib =1.0126 ¥12

0.9916 ¥1.0199 =1.024

2Since a = 2K, the fractionation factor is 1.024 at 300 K and would decrease by about 6 per mil per 100°temperature increase (however, we must bear in mind that our approximations hold only at low tem-perature). This temperature dependence is illustrated in Figure 26.2. Thus CO would be 24 permilricher in the heavy isotope, 18O, than O2. This illustrates an important rule of stable isotope frac-tionations:The heavy isotope goes preferentially in the chemical compound in which the element is moststrongly bound.

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Geol. 656 Isotope Geochemistry

Lecture 26 Spring 2003

200 4/2/03

Translational and rotational energy modesare, of course, not available to solids. Thus iso-topic fractionations between solids are entirelycontrolled by the vibrational partition function.In principle, fractionations between coexistingsolids could be calculated as we have doneabove. The task is considerably complicated bythe variety of vibrational modes available to alattice. The lattice may be treated as a largepolyatomic molecule having 3N-6 vibrationalmodes, which for large N approximates to 3N.Vibrational frequency and heat capacity areclosely related because thermal energy in acrystal is stored as vibrational energy of the a t -oms in the lattice. Einstein and Debye inde-pendently treated the problem by assuming thevibrations arise from independent harmonic os-cillations. Their models can be used to predictheat capacities in solids.

The vibrational motions available to a lat-tice may be divided into 'internal' or 'optical'vibrations between individual radicals oratomic groupings within the lattice such asCO3, and Si–O. The vibrational frequencies ofthese groups can be calculated from the Einsteinfunction and can be measured by optical spec-troscopy. In addition, there are vibrations ofthe lattice as a whole, called 'acoustical' vi-

brations, which can also be measured, but may be calculated from the Debye function. From eithercalculated or observed vibrational frequencies, partition function ratios may be calculated, which inturn are directly related to the fractionation factor. Generally, the optical modes are the primarycontribution to the partition function ratios. For example, for partitioning of 18O between water andquartz, the contribution of the acoustical modesis less than 10%. The ability to calculate frac-tionation factors is particularly important a tlow temperatures where reaction rates are quiteslow, and experimental determination of frac-tionation therefore difficult. Figure 26.3 showsthe calculated fractionation factor betweenquartz and water as a function of temperature.

REFERENCES AND SUGGESTIONS FORFURTHER READING

Chacko, T., D. Cole and J. Horita, Equilibriumoxygen, hydrogen and carbon isotope frac-tionation factors applicable to geologic sys-tems, in J. W. Valley (ed.), Stable Isotope Geo-chemistry, pp. 1-61, 2001.

Bigeleisen, J. and M. G. Mayer, Calculation ofequilibrium constants for isotopic exchange re-actions, J. Chem. Phys., 15: 261-267, 1947.

aaa

T , K

a

1.00

1.02

1.041.06

1.081.101.121.14

0 200 400 600 800

Figure 26.2. Fractionation factor, a= (18O/16O)CO/(18O/16O)O2, calculated from partition functions as afunction of temperature.

..

1000 400 200 100 50 30 10 0

-10

0

10

20

30

40

50

60

106/T2 K-20 2 4 6 8 10 12 14

1000

lna

Qtz–

H2O

T (°C)

b–quartza–quartz

Figure 26.3. Calculated temperature depend-encies of the fractionation of oxygen betweenwater and quartz. From Kawabe (1978).

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Geol. 656 Isotope Geochemistry

Lecture 26 Spring 2003

201 4/2/03

Broecker, W. and V. Oversby, Chemical Equilibria in the Earth, McGraw-Hill, New York, 318 pp.,1971.

Kawabe, I., Calculation of oxygen isotopic fractionation in quartz-water system with special refer-ence to the low temperature fractionation, Geochim. Cosmochim. Acta, 42: 613-621, 1979.

O'Neil, J. R., Theoretical and experimental aspects of isotopic fractionation, in Stable Isotopes inHigh Temperature Geologic Processes, Reviews in Mineralogy Volume 16, J. W. Valley, et al., eds.,Mineralogical Society of America, Washington, pp. 1-40, 1986.

Urey, H., The thermodynamic properties of isotopic substances, J. Chem. Soc. (London), 1947: 562-581,1947.