Chapter 8. Theory of Stable Isotope Fractionations - EAS

Isotope Geochemistry

Chapter 8: Stable Isotope Theory W. M. White

244 12/4/13

Stable Isotope Geochemistry I: Theory 8.1 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 commonly studied are H, Li, B, C, N, O, Si, S and Cl. Of these, O, H, C, and S are by far the most important. 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 fa-cilitate analysis. Elements not meeting these criteria show much smaller variations in isotopic composition. However, as new techniques offering greater sensitivity and higher precision have become available (in part through the use of the MC-ICP-MS), geochemists have begun to explore isotopic variations in 20 or so other elements, including Mg, Ca, Ti, Cr, Fe, Zn, Cu, Ge, Mo, Ti, Tl, and U. The largest isotopic varia-tions observed in these elements are produced by biologically processes, but these are still smaller than the former group of elements. Nevertheless, isotopic study of these elements has increased our under-stand of the Earth and the solar system in important ways and exploration of their isotope geochemis-try continues. We will consider some of these elements as well as Cl in Chapter 11. Stable isotopes can be applied to a variety of problems. One of the most common is geothermometry. This use derives from the extent of isotopic fractionation varying inversely with temperature: fraction-ations are large at low temperature and small at high temperature. Another application is process iden-tification. For instance, plants that produce ‘C4’ hydrocarbon chains (that is, hydrocarbon chains 4 car-bons long) as their primary photosynthetic products fractionate carbon differently than to plants that produce ‘C3’ chains. This fractionation is retained up the food chain. This allows us to draw some infer-ences about the diet of fossil mammals from the stable isotope ratios in their bones. Sometimes stable isotopes are used as 'tracers' much as radiogenic isotopes are. So, for example, we can use oxygen iso-tope ratios in igneous rocks to determine whether they have assimilated crustal material.

8.2 NOTATION AND DEFINITIONS 8.2.1 The δ Notation

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

!18O =(18O/16O)sam "(18O/16O)SMOW

(18O/16O)SMOW# $

% & ' 103 8.1

The same formula is used to report other stable isotope ratios. Hydrogen isotope ratios, δD, are re-ported relative to SMOW, carbon isotope ratios relative to Pee Dee Belemite carbonate (PDB), nitrogen isotope ratios relative to atmospheric nitrogen, and sulfur isotope ratios relative to troilite in the Can-yon Diablo iron meteorite. Cl isotopes are also reported relative to seawater; Li and B are reported rela-tive to NIST (National Institute of Standards and Technology, formerly the National Bureau of Stan-dards or NBS) standards. Unfortunately, a dual standard has developed for reporting O isotopes. Iso-



245 12/4/13

tope ratios of carbonates are reported relative to the PDB carbonate standard. This value is related to SMOW by: δ18OPDB = 1.03086δ18OSMOW + 30.86 8.2 Table 8.1 lists the values for standards used in stable isotope analysis.

8.2.2 The Fractionation Factor An important parameter in stable isotope geochemistry is the fractionation factor, α. It is defined as:

!

"A#B $RA

RB 8.3

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 = δA – δB. The rela-tionship between ∆ and α is: ∆ ≈ (α - 1)103 or ∆ ≈ 103 ln α 8.4 We derive it as follows. Rearranging equ. 8.1, we have: RA = (δA + 103)RSTD/103 8.5 where R denotes an isotope ratio. Thus α may be expressed as:

!

" =(#$ +103)RSTD /10

3

(#% +103)RSTD /103 =

(#$ +103)(#% +103)

8.6

Subtracting 1 from each side and rearranging, and since δ is generally << 103, we obtain:

!

" #1=($% # $& )($& +103)

'($% # $& )103

= ( )10#3 8.7

The second equation in 8.4 results from the approximation that for x ≈ 1, ln x ≈ x - 1. As we will see, α is related to the equilibrium constant of thermodynamics by αA-B = K1/n 8.8 where n is the number of atoms exchanged.

8.3 THEORY OF MASS DEPENDENT ISOTOPIC FRACTIONATIONS Isotope fractionation can originate from either kinetic or equilibrium effects or both. The former might be intuitively expected (since for example, we can readily understand that a lighter isotope will diffuse faster than a heavier one), but the latter may be somewhat surprising. After all, we were taught in in-troductory chemistry that oxygen is oxygen, and its properties are dictated by its electronic structure. In the following sections, we will see that quantum mechanics predicts that mass affects the strength of chemical bonds and the vibrational, rotational, and translational motions of atoms. These quantum me-chanical effects predict the small differences in the chemical properties of isotopes quite accurately. We shall now consider the manner in which isotopic fractionations arise.

Table 8.1. Values of Commonly Analyzed Stable Isotope Ratios Element Notation Ratio Standard Absolute Ratio Hydrogen δD D/H (2H/1H) SMOW 1.557 × 10-4 Carbon δ13C 13C/12C PDB 1.122 × 10-2 Nitrogen δ15N 15N/14N atmosphere 3.613 × 10-3 Oxygen δ18O 18O/16O SMOW, PDB 2.004 × 10-3 δ17O 17O/16O SMOW 3.71 × 10-4 Sulfur δ34S 34S/32S CDT 4.43 × 10-2



246 12/4/13

The electronic structures of all isotopes of an element are iden-tical and since the electronic structure governs chemical properties, these properties are generally identical as well. Nevertheless, small differences in chemical behavior arise when this behavior depends on the frequencies of atomic and mo-lecular vibrations. The energy of a molecule can be described in terms of several components: electronic, nuclear spin, transla-tional, rotational and vibra-tional. The first two terms are negligible and play no role in isotopic fractionations. The last three terms are the modes of motion available to a molecule 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 me-chanics, but an adequate de-scription of vibrational motions of atoms in a lattice or molecule requires the application of quantum theory. As we shall see, temperature-dependent equilibrium isotope fractionations arise from quantum mechanical effects on vibrational motions. These effects are, as one might expect, generally small. For example, the equilibrium constant for the reaction: ½ C16O2 + H2

18O ⇋ ½C18O16O + H21O

is only about 1.04 at 25°C. Figure 8.1 is a plot of the potential energy of a diatomic molecule as a function of distance between the two atoms. This plot looks broadly similar to one we might construct for two masses connected by a spring. When the distance between masses is small, the spring is compressed, and the potential energy of the system correspondingly high. At great distances between the masses, the spring is stretched and the energy of the system also high. At some intermediate distance, there is no stress on the spring, and the potential energy of the system is at a minimum (energy would be nevertheless be conserved be-cause kinetic energy is at a maximum when potential energy is at a minimum). The diatomic oscillator, for example consisting of a Na and a Cl ion, works in an analogous way. At small interatomic distances, the electron clouds repel each other (the atoms are compressed); at large distances, the atoms are at-tracted 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 are related to temperature. In quantum theory, a diatomic oscillator cannot assume just any energy: only discrete energy levels may be occupied. The permissible energy levels, as we shall see, depend on mass. Quantum theory also

Figure 8.1. Energy-level diagram for the hydrogen atom. Fundamental vibration frequencies are 4405 cm-1 for H2, 3817 cm-1 for HD, and 3119 cm-1 for D2. The zero-point energy of H2 is greater than that for HD which is greater than that for D2. After O'Neil (1986).



247 12/4/13

tells us that even at absolute 0 the atoms will vibrate at a ground frequency ν0. The system will have en-ergy of 1/2hν0, where h is Planck's constant. This energy level is called the Zero Point Energy (ZPE). Its value depends the electronic arrangements, the nuclear charges, and the positions of the 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 for different for isotopes. The vibrational frequency will be lower for a bond involving a heavier isotope of an element, which in turn lowers the vibrational energy of the molecule or crystal, as suggested in Figure 8.1. Thus bonds involving heavier iso-topes will be stronger. If a system consists of two possible atomic sites with different bond energies and two isotopes of an element available to fill those sites, the energy of the system is minimized when the heavy isotope occupies the site with the stronger bond. Thus at equilibrium, the heavy isotope will tend to occupy the site with the stronger bond. This, in brief, is why equilibrium fractionations arise. Because bonds involving lighter isotopes are weaker and more readily broken, the lighter isotopes of an element par-ticipate more readily in a given chemical reaction. If the reaction fails to go to completion, which is of-ten the case, this tendency gives rise to kinetic fractionations of isotopes. There are other causes of ki-netic fractionations as well, and will consider them in due course. We will now consider in greater de-tail the basis for equilibrium fractionation, and see that they can be predicted from statistical mechanics.

8.3.1 Equilibrium Fractionations Urey (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 are avail-able to an atom or molecule. At equilibrium, the ratio of the number of molecules having internal energy Ei to the number having the zero point energy E0 is:

!

nin0

= gie"Ei / kT 8.9

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 to the number of states having energy Ei). The average energy (per molecule) in a system is given by the Boltzmann distribution function, which is just the sum of the energy of all possible states times the num-ber of particles in that state divided by the number of particles in those states:

E =niEi

i!

nii!

=giEie

"Ei /kT!

gie"Ei /kT!

8.10

The denominator of this equation, which is the sum of all energy states accessible to the system is called the partition function, q:

!

q = gie"Ei / kT# 8.11

Substituting 8.11 into 8.10, we can rewrite 8.10 in terms of the partial derivatives of q:

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



248 12/4/13

!

E = kT 2 " lnq"T

8.12

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

!

S =UTR lnq 8.13

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 8.14 And for the entropy and energy changes of a reaction, we have:

!U " T!S = "R ln qn#$ 8.15

where ξ in this case is the stoichiometric coefficient. In this notation, the stoichiometric coefficient is 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 of reaction under conditions of constant volume (as would be the case for an isotopic exchange reaction), so that

!G = "R ln qn#$ 8.16

The Gibbs Free Energy change is related to the equilibrium constant, K, by: ∆G = −RT lnK 8.17 so the equilibrium constant for an isotope exchange reaction is related to the partition function as:

K = qnξ

n∏ 8.18

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

K =q

C18O16O

1/2 qH2

16O

qC16O2

1/2 qH2

18O

8.19

The point of all this is simply that the usefulness of the partition function is that it can be calculated from quantum mechanics, and from it we can calculate equilibrium fractionations of isotopes. The partition function involving energies of interest can be written as approximately: qtotal = qtransqrotqvib 8.20 i.e., the product of the translational, rotational and vibrational partition functions (we are ignoring con-tributions from anharmonic vibrations, rotational-vibrational interactions, and electronic energies). We should note here that since rotational and translations motions are not available to atoms in a solid, the partition function reduces to the vibrational partition function only. It is convenient to treat these three modes of motion separately. Let's now do so.

8.3.1.1 Translational Partition Function † Entropy is defined in the second law of thermodynamics, which states:

dS = dQrev

T

where Qrev is heat gained by a system in a reversible process. Entropy can be thought of as a measure of the random-ness of a system.



249 12/4/13

Writing a version of equation 8.11 for translational energy, qtrans is expressed as:

!

qtrans = gtr,ie"Etr ,i / kT

i# 8.21

Now all that remains is to find an expression for translational energy and a way to do the summation. At temperatures above about 2 K, translational energy levels are so closely spaced that they essentially form a continuum. The quantum translational energy of a particle in a cubical box is given by:

Etrans =n2h2

8Md 2 8.22

where n is the quantum energy level, h is Planck’s constant, d is the length of the side of the cube, and M is mass of the particle. Substituting 8.22 into 8.21 and integrating:

qtrans = e−n2h2 /8Md2kT

0

∞

∫ = (2πMkT )1/2

hd 8.23

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

qtrans =(2πMkT )3/2

h3 V 8.24

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 8.19, all terms in 8.24 except mass will eventually cancel.) If translation motion were the only component of energy, the equilibrium constant for exchange of isotopes would be simply the ratio of the molecular weights raised to the 3/2 power. If we define the translational contribution to the equilibrium constant as Ktr as:

K tr = qtrξ∏ 8.25

Ktr reduces to the product of the molecular masses raised to three-halves of the stoichiometric coeffi-cient:

K tr = Mtrξ( )3/2∏ 8.26

(note that M is the mass of the molecule, not the isotope). Thus the translational contribution to the par-tition function and fractionation factor is independent of temperature.

8.3.1.2 Rotational Partition Function The allowed quantum rotational energy states are:

Erot =j( j +1)h2

8! 2I 8.27

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 and µ is reduced mass.

!

µ =m1m2

m1 + m2

8.28

where mi is the atomic mass of atom i. A diatomic molecule will have two rotational axes, one along the bond axis, the other perpendicular to it. Hence in a diatomic molecule, j quanta of energy may be distributed 2j+1 ways because there are two possibilities for every value of j except j = 0, for which there is only one possible way. The statistical weight factor is therefore 2j + 1. Hence:



250 12/4/13

qrot = (2 j +1)e j ( j+1)h2 /8π 2IkT∑ 8.29 Again the spacing between energy levels is relatively small (except for hydrogen) and 8.29 may be evaluated as an integral. For a diatomic molecule, the partition function for rotation is given by:

!

qrot =8" 2IkT#h2

8.30

where σ is the symmetry number and is equal to the number of equivalent ways the molecule can be oriented in space. It is 1 for a heteronuclear diatomic molecule (such as CO or 18O16O), and 2 for a homonuclear diatomic molecule such as 16O2 or a symmetric tri-atomic molecule such as 16O12C16O (more complex molecules will have higher symmetry numbers, e.g., 12 for 12C1H4). Equ. 8.30 also holds for linear polyatomic molecules with the symmetry factor equal to 2 if the molecule has a plane of symmetry (e.g., CO2) and 1 if it does not. For non-linear polyatomic molecules, the partition function is given by:

qrot =8π 2 (8π 3IAIBIC )

1/2 (kT )3/2

σh3 8.31

where IA, IB, and IC are the principal moments of inertia. In calculating the rotational contribution to the equilibrium constant, all terms cancel except for moment of inertia and the symmetry factor, and the contribution of rotational motion to isotope fractionation is also independent of temperature. For diatomic molecules, the equilibrium constant calculated from the ratios of partition functions re-duces to:

K rot =Iiσ i

⎛⎝⎜

⎞⎠⎟

ξi

i∏ 8.32

In general, bond lengths are largely independent of the isotope involved, so the moment of inertia term may be replaced by the reduced masses:

K rot ≅µi

σ i

⎛⎝⎜

⎞⎠⎟

ξi

i∏ 8.32a

8.3.1.3 Vibrational Partition Function We will simplify the calculation of the vibrational partition function by treating the diatomic mole-cule as a harmonic oscillator (as Fig. 8.1 suggests, this is a good approximation in most cases). In this case the quantum energy levels are given by:

Evib = n + 12

⎛⎝⎜

⎞⎠⎟ hν 8.33

where n is the vibrational quantum number and ν is vibrational frequency. Unlike rotational and vibra-tional energies, the spacing between vibrational energy levels is large at geologic temperatures, so the partition function cannot be integrated. Instead, it must be summed over all available energy levels. Fortunately, the sum has a simple form: for diatomic molecules; the summation is simply equal to:

qvib =e−hν /2kT

1− e−hν /kT 8.34

For molecules consisting of more than two atoms, there are many vibrational motions possible. In this case, the vibrational partition function is the product of the partition functions for each mode of motion, with the individual partition functions given by 8.34. For a non-linear polyatomic molecule consisting of i atoms and the product is performed over all vibrational modes, , the partition function is given by:



251 12/4/13

qvib =

e−hν /2kT

1− e−hν /kT

3−n

∏ 8.35

Where n is equal to 6 for non-linear polyatomic molecules and 5 for linear polyatomic molecules. The vibrational energy contribution to the equilibrium constant for diatomic molecules is thus:

Kvib =e−hνi /2kT

1− e−hνi /kT⎛⎝⎜

⎞⎠⎟i

∏ξi

8.36

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

!

qvib " e#h$ /2kT 8.37

Thus at low temperature, the vibrational contribution to the equilibrium constant approximates to:

Kvib ≅ e−ξhν /2kT

∏ 8.38

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

K = Mi

3/2 Iiσ i

⎡

⎣⎢

⎤

⎦⎥

e−hνi /2kT

1− e−hνi /kT⎛

⎝⎜⎞

⎠⎟i∏

ξi

8.39

This equation can be simplified through use of the Teller-Redlich spectroscopic theorem* to:

K = mi

3ri /2

σ i

hν i

kTe−hνi /2kT

1− e−hνi /kT⎛

⎝⎜⎞

⎠⎟i∏

ξi

8.40

where ri is the number of atoms being exchanged in the molecule. Returning to our reaction between CO2 and water, we can rearrange the equilibrium constant expres-sion (8.19) as:

K ==

q18Oq16O

⎛⎝⎜

⎞⎠⎟CO2

1/2

q16Oq18O

⎛⎝⎜

⎞⎠⎟H2O

8.41

Thus the equilibrium constant is the ratio of partition function ratios of the two isotopic versions of the two substances. We can see that the mass terms in equation 8.40 will cancel in the computation and consequently we can omit them. The partition function ratio with the mass terms omitted is sometimes referred to as the reduced partition function.

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

m2

m1

⎛⎝⎜

⎞⎠⎟

3/2I1I2

M1

M 2

⎛⎝⎜

⎞⎠⎟

3/2

= hν1 / kThν2 / kT

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



252 12/4/13

When theoretical calculations such as these are involved, isotope fractionation is also often expressed in terms of a β-factor defined as the ratio at equilibrium of the isotope ratio of the substance of interest to the isotope ratio of dissociated atoms.

βH2O

18O/16O ≡(18O / 16O)H2O

(18O / 16O)O 8.42

The denominator is the isotope ratio in a gas of monatomic oxygen and the numerator is the isotope ra-tio in water in equilibrium with it. The β factor for CO2 would also have the isotope ratio of dissociated atoms in the denominator and thus in our example of CO2-water exchange, the denominators cancel so that:

α =βCO2

βH2O

8.43

β-factors are closely related to the reduced partition function and are equal to it where the molecule of interest contains only one atom of the element of interest (this would be the case for water, for exam-ple, which contains only 1 oxygen atom). Where this is not the case, CO2 for example, the β-factor dif-fers from the reduced partition function by what Richet et al. (1987) termed an ‘excess factor’, arises from our interest in atomic isotopic ratios rather than of isotopic molecular abundances. The reason for this will become clear in the following example.

8.3.1.4 Example of fractionation factor calculated from partition functions To illustrate the use of partition functions in calculating theoretical fractionation factors, we will do the calculation for a very simple reaction: the exchange of 18O and 16O between O2 and CO: C16O + 18O16O ⇋ C18O + 16O2 8.44 The choice of diatomic molecules greatly simplifies the equations. Choosing even a slightly more com-plex model, such as CO2 would complicate the calculation because there are more vibrational modes possible. Richet et al. (1977) and Chacko et al. (2001) provide examples of the calculation for more com-plex molecules. Let’s first consider the relationship between the equilibrium constant and the fractionation factor for this reaction. The equilibrium constant for our reaction is:

K = [16O2 ][C18O]

[18O16O][C 16O] 8.45

where we are using the brackets in the unusual chemical sense to denote concentration. We can use concentrations rather than activities or fugacities because the activity coefficient of a phase is independ-ent of its isotopic composition. The fractionation factor, α, is defined as:

α = (18O / 16O)CO(18O / 16O)O2

8.46

We must also consider the exchange reaction: 18O18O + 16O16O ⇋ 216O18O for which we can write a second equilibrium constant, K2. It turns out that when both reactions are considered, α ≈ 2K. The reason for this is as follows. The isotope ratio in molecular oxygen is related to the concentration of the 2 molecular species as:

!

18O16O"

# $

%

& ' O2

=[18O16O]

[18O16O]+ 2[16O2 ] 8.47



253 12/4/13

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

!

18O16O"

# $

%

& ' CO

=[C18O][C16O]

8.48

Letting the isotope ratio equal R, we can solve 8.47 for [18O/16O]:

!

[18O /16O]= 2[16O2 ]RO21" RO2

8.49

and substitute it into 8.45: K =(1− RO2

)[C 18O]

2RO2[C 16O]

=(1− RO2

)RCO

2RO2

8.50

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

K ≅ RCO

2RO2

= α2

8.51

Relationships such as these are also why the β-factor can differ from the reduced partition function. Now let’s return to the problem of calculating K from the partition functions:

K =q16O2

qC 18O

q18O 16OqC 16O

8.52

where each partition function is the product of the translational, rotational, and vibrational partition functions. Since the reaction involves only diatomic molecules, we could simply use equation 8.40. However, it is informative to how the 3 separate modes of motion contribute to the overall equilibrium constant, so we will proceed by calculation the equilibrium constant for each mode of motion. The total equilibrium constant will then be the product of all three partial equilibrium constants. For translational motion we noted the ratio of partition functions reduces to the ratio of molecular masses raised to the 3/2 power. Hence:

K tr =q16O2

qC 18O

q18O 16OqC 16O

=M 16O2

MC 18O

M 18O 16OM

C 16O

⎛

⎝⎜⎞

⎠⎟

3/2

= 32 × 3034 × 28

⎛⎝⎜

⎞⎠⎟3/2

= 1.0126 8.53

We find that CO would be 12.6‰ richer in 18O if translational motions were the only modes of energy available. In the expression for the ratio of rotational partition functions, all terms cancel except the moment of inertia and the symmetry factors. The symmetry factor is 1 for all the molecules involved except 16O2. In this case, the terms for bond length also cancel, so the expression involves only the reduced masses. So the expression for the rotational equilibrium constant becomes:

K rot =q16O2

qC 18O

q18O 16OqC 16O

=µ16O2

µC 18O

2µ18O 16Oµ

C 16O

⎛

⎝⎜⎞

⎠⎟= 12

16 ×1616 +16

× 12 ×1812 +18

18 ×1618 +16

× 12 ×1612 +16

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟= 0.9916

2 8.54

(we’ll ignore the 1/2, it will cancel out later when we calculate the fractionation factor). If rotation were the only mode of motion, CO would be 8‰ poorer in 18O. Notice that both the translational and rota-tional equilibrium constants do not depend on temperature. We’ll do the calculation for low temperature, which will allow us to use equation 8.38 to calculate the vibrational equilibrium constant:



254 12/4/13

Kvib =q16O2

qC 18O

q18O 16OqC 16O

= e−h(ν16O2

+νC 18O

−ν18O16O−ν

C 16O)

2kT 8.55

Further, since we expect the difference in vibrational frequencies to be quite small, we may make the approximation ex = x + 1. Hence:

Kvib = 1+−h2kT

ν 16O2+ν

C 18O{ }− ν 18O 16O+ν

C 16O{ }⎡⎣

⎤⎦ 8.56

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

ν = 12π

κµ

8.57

where κ is the forcing constant and depends on the nature of the bond, and hence will be independent of isotopic composition. In this case, we may write:

!C18O

= !C16O

µC16O

µC18O

= !C16O

6.8577.2

= 0.976!C16O

8.58

A similar expression may be written relating the vibrational frequencies of the oxygen molecule: ! O1816 O = 0.9718! O2

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

Kvib = 1+−h2kT

ν 16O21− 0.9718{ }−ν

C 16O1− 0.976{ }⎡

⎣⎤⎦

The measured vibrational frequencies of CO and O2 are 6.50 × 1013 sec-1 and 4.74 × 1013 sec-1. Substituting these values and values for the Planck and Boltzmann constants, we obtain:

Kvib = 1+5.544T

Kvib = 1+5.544T 8.59

At 300 K (room temperature), this evaluates to 1.0185. We may now write the total equilibrium constant expression as:

K = K trK rotKvib ≅M 16O2

MC 18O

M 18O 16OM

C 16O

⎛

⎝⎜⎞

⎠⎟

3/2 µ16O2µ

C 18O

2µ18O 16Oµ

C 16O

⎛

⎝⎜⎞

⎠⎟×

1+ h4π kT

κµ

C 16O

− κµ

C 18O

⎛

⎝⎜

⎞

⎠⎟ −

κµ16O2

− κµ18O 16O

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

8.60

Evaluating this at 300K we have:

K = 1.0126 ! 0.99162

!1.0185 = 1.0232

Since α = 2K, the fractionation factor is 1.023 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; at high temperature equilibrium constants depend on the inverse square of temperature). This temperature dependence is illustrated in Figure 8.2. Thus CO would be 23 permil richer in the



255 12/4/13

heavy isotope, 18O, than O2. This illustrates an important rule of stable isotope fractionations: the heavy isotope goes preferentially in the chemical compound in which the element is most strongly bound. Translational and rotational energy modes are, of course, not available to solids. Thus isotopic frac-tionations between solids are entirely controlled by the vibrational partition function. Fractionations be-tween coexisting solids can be calculated as we have done above. A few decades ago, the task was daunting because the varieties of vibrational modes available to atoms in a lattice make the task com-putationally complex. With the computer power available today, theoretically computed fractionation factors are becoming more common. The lattice may be treated as a large polyatomic molecule having 3N-6 vibrational modes, where N is the number of atoms in the unit cell. For large N, this approxi-mates to 3N. Vibrational frequency and heat capacity are closely related because thermal energy in a crystal is stored as vibrational energy of the atoms in the lattice. Einstein and Debye independently treated the problem by assuming the vibrations arise from independent harmonic oscillations. Their models can be used to predict heat capacities in solids. The vibrational motions available to a lattice may be divided into two types, the first of which is 'internal' or 'optical' vibrations between individual radicals or atomic groupings within the lattice such as CO3, and Si–O. The vibrational frequencies of these groups can be calculated from the Einstein function and can be measured by optical spectroscopy. The second type are vibrations of the lattice as a whole, called 'acoustical' vibrations, which can also be measured, but may also be calculated from the Debye function. From either calculated or observed vibrational fre-quencies, partition function ratios may be calculated, which in turn are directly related to the frac-tionation factor. Generally, the optical modes are the primary contribution to the partition function ra-tios. For example, for partitioning of 18O between water and quartz, the contribution of the acoustical modes is less than 10%. The ability to calculate fractionation factors is particularly important at low temperatures where reaction rates are quite slow, and experimental determination of fractionation therefore difficult. Figure 8.3 shows the calculated fractionation factor between quartz and water as a function of temperature. One the other hand, theoretically determined fractionation factors have the disadvantage that they generally assume perfect stoichiometry and crystallinity, whereas most natural crystals are chemically more complex. Thus experimental determination of fractionation factor remains important.

8.1.3.5 Isotopologues and Isotopic “Clumping” In the example we just considered, we were concerned only with how 18O was distributed between CO and O2. However, the CO and O2 will consist of a variety of molecules of distinct isotopic composition, or “isotopologues”. In-deed, there will be 12 such isotopologues: spe-cifically 12C16O, 12C17O, 12C18O, 13C16O, 13C17O, 13C18O, 16O2, 16O17O, 16O18O, 17O2, 17O18O, and 18O2. The statistical mechanical theory we just considered predicts that the distribution of iso-topes within a species will not be random but rather that some of these isotopologues will be thermodynamically favored, in particular the heavier isotopes of elements (13C and 18O) are more likely to be “clumped” together in a sin-gle molecule than to be randomly distributed among all molecules. The reason for this is that double heavy isotope substitution generally leads to a reduction in vibrational frequency

Figure 8.2. Fractionation factor, α= (18O/16O)CO/ (18O/16O)O2, calculated from partition functions as a function of temperature.



256 12/4/13

and energy greater than twice that of single substi-tution. Thus there is an energy advantage to bond-ing, or clumping, heavy isotopes together as op-posed to simply distributing them randomly (Eiler, 2013). As with the distribution of isotopes between chemical species, this clumping of isotopes within a species will be temperature dependent. We can use this as a geothermometer, and one that is inde-pendent of the isotopic composition of other phases. This, as we shall see in Chapter 11, is an important advantage. Let’s being by considering the distribution of iso-topes between the isotopologues of CO. There are 6 isotopologues and they can be related through the following two reactions: 12C16O + 13C17O ⇋ 13C16O + 12C17O 8.61 12C16O + 13C18O ⇋ 13C16O + 12C18O 8.62 (Since we can relate the 6 isotopologues through 2 reactions, we need only chose 4 of these iso-topologues as the components of our system.) The equilibrium constant for reaction 8.62 can be calcu-lated from:

K =q13C16O

q12C18O

q12C16Oq13C18O

8.63

A similar equation can be written for the equilibrium constant for 8.61. The individual partition func-tions can be calculated just as described in the previous section. Doing so, we find that the two heaviest species, 13C17O and 13C18O, will be more abundant that if isotopes were merely randomly distributed among the 6 isotopologues, i.e., the heavy isotopes tend to “clump”. Wang et al. (2004) introduced a delta notation to describe this effect:

Δi =Ri−e

Ri−r

−1⎛⎝⎜

⎞⎠⎟×1000 8.64

where Ri-e is that ratio of the observed or calculated equilibrium abundance of isotopologue i to the iso-topologue containing no rare isotopes and Ri-r is that same ratio if isotopes were distributed among iso-topologues randomly. Thus, for example, in the system above,

Δ 13C18O=

[13C18O][12C16O]

⎛⎝

⎞⎠e

[13C18O][12C16O]

⎛⎝

⎞⎠r

−1

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

×1000 8.65

Since Ri-r is the random distribution, it can be calculated directly as the probability of choosing isotopes randomly to form species. In the case of 13C18O, it is:

Figure 8.3. Calculated temperature dependencies of the fractionation of oxygen between water and quartz. After Kawabe (1978).



257 12/4/13

R13C18O−r= [13C18O]

[12C16O]

⎛⎝⎜

⎞⎠⎟r

= [13C][18O][12C][16O]

8.66

It gets a little more complex for molecules with more than 2 isotopes. In most cases, we are interested in combinations of isotopes rather than permutations, which is to say we don’t care about order. This will not be the case for highly asymmetric molecules such as nitrous oxide, N2O. The structure of this molecule is N-N-O and 14N15N16O will have different properties than 15N14N16O, so in that case, order does matter. The CO2 molecule is, however, symmetric and we cannot distinguish 16O12C18O from 18O12C16O. Its random abundance would be calculated as:

R13C16O18O−r= [13C16O18O]

[12C16O2 ]

⎛⎝⎜

⎞⎠⎟r

=

2[13C][16O][18O][12C][16O]2

= 2[13C][18O][12C][16O]

8.67

The factor of 2 is in the denominator to take account of both 12C16O18O and 12C18O16O. As Wang et al. (2004) showed, value of ∆ as defined in equation 8.65 is related to the equilibrium constant (equation 8.63) for the exchange reaction (equation 8.61) as:

Δ ≈ −1000 ln KK r

8.68

Since Kr refers to the case of random distribution of isotopes, it is equal to 1 and since K will have a value close to 1, we may use the approximation ln x ≈ 1 – x, so that 8.68 reduces to: Δ ≈ (K −1)×1000 8.69 Figure 8.4 shows the ∆ values calculated for 3 isotopologues of CO as a function of temperature by Wang et al. (2004). ∆ values vary with the inverse of temperature, a direct consequence of the inverse temperature dependence of the vibrational equilibrium constant expressed in equation 8.59. The calcu-lations predict that at room temperature 13C18O and 13C18O concentrations will be 1 to 2 per mil above a purely random distribution of isotopes. Both are quite rare compared to the most abundant iso-topologue, 12C16O; the concentration of 13C18O will be less than 0.003%, that of 13C17O about .0004%. Very high precision and sensitivity is required to analyze the abundance of these isotopes and detect the small enrichment resulting from clumping. CO is a relatively rare species in nature (on Earth anyway) and it is of limited geochemical interest, at least compared to CO2 and its related forms such as carbonate and bicarbonate ions and, particularly, carbonate minerals. The latter are among the most common sedimentary minerals. Most form by pre-cipitation from water (often biologically mediated), and the oxygen isotopic fractionation resulting from this precipitation reaction is temperature dependent. Thus the isotopic composition of carbonates has proved to be a useful geothermometer and a very useful paleoclimatic tool. However, the isotopic composition of carbonate precipitated from water depends both on temperature and the isotopic com-

Figure 8.4. Predicted enrichment of isotopologues in CO2 gas based on statistical mechanical calculations. After Wang et al. (2004).



258 12/4/13

position of the water; calculating temperature thus requires knowing the isotopic composition of the water. In contrast, the relative abundance of carbonate isotopologues depends only on temperature and the isotopic composition of the carbonate. Since the latter is readily measured, temperature can be cal-culated from the abundance of the isotopologues, provided those can be measured. The difficulty is two-fold. First, the abundance carbonate ion isotopologues containing two or more rare isotopes is very small; the most abundant will be 13C18O16O2 with a relative abundance of about 67 ppm. Second, no analytical technique can directly measure the abundance of these rare isotopologues in carbonates. Carbonates are analyzed by first digesting the material in anhydrous phosphoric acid, which generates CO2 gas. Carbon and oxygen isotopic compositions are then determined by analyzing the CO2 as the CO2

+ ion in a gas source mass spectrometer. Ghosh et al. (2006) showed that the abun-dance of 13C18O16O isotopologue of analyzed CO2 gas was proportional to the abundance of 13C-18O bonds in the carbonate. They found that the ∆ value of the 13C18O16O isotopologue in the CO2 gas, re-ferred to as ∆47

*, was about 0.2‰ higher than that of the carbonate, but this fractionation appears to be nearly constant. Analyzing both natural carbonate from corals known to have grown at different tem-peratures and carbonates precipitated from laboratory solutions, Ghosh et al. (2006) showed ∆47 was in-versely proportional to temperature as predicted by statistical mechanics (Figure 8.5). Ghosh et al. (2006) concluded that ‘clumped’ isotopic analysis could be used as a geothermometer with a precision of about ±2˚C. To further improve application of the clumping technique, workers from four laboratories (Dennis et al., 2011) proposed a method for standardizing and reporting clumped isotopic species to an absolute reference frame to avoid interlaboratory biases that result, among other things, from differences in frac-tionation that occur during acid digestion and ionization in the mass spectrometer. The approach in-

* Two other CO2 isotopologues also have mass 47, but their concentration is much, much lower than that of 13C18O16O.

Figure 8.5. Data for ∆47 measured in experimental and natural carbonates by Ghosh et al. (2006) as a function of inverse temperature. Both 1/T (solid line) and a 1/T2 (dashed line) are fitted to the data.



259 12/4/13

volves standardization against prepared CO2 gases whose ∆47 can be theoretically predicted. These in-clude gases that vary in C and O isotopic composition that are equilibrated to a common temperature (typically 1000˚C), and hence should have a common (and theoretically predicted) ∆47, and a series of gases with the same isotopic composition equilibrated with water at a known temperature, and hence should also have theoretically predicted ∆47 values. The gases are then analyzed and the measured val-ues regressed against theoretical ones to provide calibration functions. We’ll consider specific examples of applications of the clumping technique in Chapter 11.

8.3.2 Kinetic Fractionation Kinetic effects are normally associated with fast, incomplete, or unidirectional processes like evapora-tion, diffusion and dissociation reactions. As an example, recall that temperature is related to the aver-age kinetic energy. In an ideal gas, the average kinetic energy of all molecules is the same. The kinetic energy is given by:

! !

E =12mv2 8.70

Consider two molecules of carbon dioxide, 12C16O2 and 13C16O2, in such a gas. If their energies are equal, the ratio of their velocities is (45/44)1/2, or 1.011. Thus 12C16O2 can diffuse 1.1% further in a given amount of time at a given temperature than 13C16O2. This result, however, is largely limited to ideal gases, i.e., low pressures where collisions between molecules are infrequent and intermolecular forces negligible. For the case of air, where molecular collisions are important, the ratio of the diffusion coeffi-cients of the two CO2 species is the ratio of the square roots of the reduced masses of CO2 and air (mean molecular weight 28.8):

!

D12CO2

D13CO2

=µ 13CO2

µ 12CO2

=17.56117.406

=1.0044 8.71

Hence we predict that gaseous diffusion would lead to only a 4.4‰ fractionation. In addition, molecules containing the heavy isotope are more stable and have higher dissociation en-ergies than those containing the light isotope. This can be readily seen in Figure 8.1. The energy re-quired to raise the D2 molecule to the energy where the atoms dissociate is 441.6 kJ/mole, whereas the energy required to dissociate the H2 molecule is 431.8 kJ/mole. Therefore it is easier to break bonds such as H-H and C-H than D-D and C-D. Where reactions go to completion, this difference in bonding energy plays no role: isotopic fractionations will be governed by the considerations of equilibrium dis-cussed in the previous section. Where reactions do not achieve equilibrium the lighter isotope will be preferen-tially concentrated in the reaction products, because of this effect of the bonds involving light isotopes in the reactants being more easily broken. Large kinetic effects are associated with biologically mediated reactions (e.g., bacterial reduction), because such reactions generally do not achieve equilibrium. Thus 12C is enriched in the products of photosynthesis in plants (hydrocarbons) relative to atmospheric CO2, and 32S is enriched in H2S produced by bacterial reduction of sulfate. We can express this in a more quantitative sense. The rate at which reactions occur is given by:

!

R = Ae"Eb /kT 8.72 where A is a constant called the frequency factor and Eb is the barrier energy. Referring to Figure 8.1, the barrier energy is the difference between the dissociation energy, ε, and the zero-point energy. The constant A is independent of isotopic composition, thus the ratio of reaction rates between the HD molecule and the H2 molecule is:

!

RDRH

=e"(#"1 2h$D ) /kT

e"(#"1 2h$H ) /kT 8.73



260 12/4/13

or

!

RDRH

= e("H #"D )h /2kT 8.74

Substituting for the various constants, and using the wavenumbers given in the caption to Figure 8.1 (remembering that ω = cν where c is the speed of light) the ratio is calculated as 0.24; in other words we expect the H2 molecule to react four times faster than the HD molecule, a very large difference. For heavier elements, the rate differences are smaller. For example, the same ratio calculated for 16O2 and 18O16O shows that the 16O will react about 15% faster than the 18O16O molecule. The greater translational velocities of lighter molecules also allow them to break through a liquid sur-face more readily and hence evaporate more quickly than a heavy molecule of the same composition. The transition from liquid to gas in the case of water also involves breaking hydrogen bonds that form between the hydrogen of one molecule and an oxygen of another. This bond is weaker if 16O is involved rather than 18O, and thus is broken more easily, meaning H2

16O is more readily available to transform into the gas phase than H2

18O. Thus water vapor above the ocean typically has δ18O around -13 per mil, whereas at equilibrium the vapor should only be about 9 per mil lighter than the liquid. Let's explore this example a bit further. Kinetic fractionation of O isotopes between water and water vapor is an example of Rayleigh distillation (or condensation), as is fractional crystallization. Let A be the amount of the species containing the major isotope, H2

16O, and B be the amount of the species con-taining the minor isotope, H2

18O. The rate at which these species evaporate is proportional to the amount present:

dA=kAA 8.75a and dB=kBB 8.75b Since the isotopic composition affects the reaction, or evaporation, rate, kA ≠ kB. We'll call this ratio of the rate constants α. Then

!

dBdA

="BA

8.76

Rearranging and integrating, we have

!

ln BB˚

=" ln AA˚

or

!

BB˚

=AA˚"

# $

%

& ' (

8.77

where A° and B° are the amount of A and B originally present. Dividing both sides by A/A°

!

B /AB˚/A˚

=AA˚"

# $

%

& ' ()1

878

Since the amount of B makes up only a trace of the total amount of H2O present, A is essentially equal to the total water present, and A/A° is essentially identical to ƒ, the fraction of the original water re-maining. Hence:

!

B /AB˚/A˚

= ƒ"#1 8.79

Subtracting 1 from both sides, we have

!

B /A" B˚/A˚B˚/A˚

= ƒ#"1 "1 8.80

Comparing the left side of the equation to equation 8.1, we see the permil fractionation is given by:

! = 1000( f " #1 #1) 8.81 Of course, the same principle applies when water condenses from vapor. Assuming a value of α of 1.01, δ will vary with ƒ, the fraction of vapor remaining, as shown in Figure 8.6.



261 12/4/13

Even if the vapor and liquid remain in equilibrium throughout the conden-sation process, the isotopic composi-tion of the remaining vapor will change continuously. The relevant equation is:

!

" = 1# 1(1# ƒ) /$ + f

%

& '

(

) * +1000 8.82

The effect of equilibrium condensation is also shown in Figure 8.6. It is important to bear in mind that kinetic fractionations often occur in combination with equilibrium frac-tionation. Although the above example of Rayleigh condensation illustrates how kinetic effects can amplify equilib-rium fractionations, kinetic effects can also counteract and partially or whole nullify equilibrium fractionations.

8.4 MASS INDEPENDENT FRACTIONATION Most isotopic studies focus on the fractionation between the two most abundant isotopes of an ele-ment, for example, 16O and 18O. Some elements, however, have 3 or more isotopes. For example, O consists of 17O, as well as 16O and 18O, although 17O is an order of magnitude less abundant than 18O (which is two orders of magnitude less abundant than 16O). The reason for this focus is that, based on the theory we have just reviewed, mass fractionation should depend on mass difference. This is referred to as mass-dependent fractionation. The mass difference between 17O and 16O is half the dif-ference between 18O and 16O, hence we expect the fractionation between 17O and 16O to be about half that between 18O and 16O. In the example of fractionation between CO and O2 in the previous section, it is easy to show from equation 8.56 that through the range of temperatures we expect near the surface of the Earth (or Mars), that the ratio of fractionation factors ∆17O/∆18O should be ≈ 0.53. In the limit of in-finite temperature, ∆17O/∆18O ≈ 0.52. The empirically observed ratio for terrestrial fractionation (and also within classes of meteorites) is ∆17O/∆18O ≈ 0.52. Because the fractionation between 17O and 16O bears a simple relationship to that between 18O and 16O, the 17O/16O ratio is rarely measured. However, as we saw in Chapter 5, not all O isotope variation in solar system materials follows the expected mass-dependent fractionation. Furthermore, we saw that there is laboratory evidence that mass-independent fractionation can occur. Mass independent fractionation has subsequently been demonstrated to occur in nature, and indeed may provide important clues to Earth and Solar System processes and history, and we will return to this topic later. Mass-independent fractionation is nevertheless rare. First observed in meteorites (Chapter 5), it has subsequently been observed in oxygen isotope ratios of atmospheric gases, most dramatically in strato-spheric ozone (Figure 8.7), and more recently in sulfur isotope ratios of Archean sediments and modern sulfur-bearing aerosols in ice. The causes of mass-independent fractionation are incompletely under-stood and it seems likely there may be more than one cause. Only a brief discussion is given here, a fuller discussion of the causes of mass independent fractionation can be found in Thiemens (2006).

Figure 8.6. Fractionation of isotope ratios during Rayleigh and equilibrium condensation. δ is the per mil difference be-tween the isotopic composition of original vapor and the iso-topic composition as a function of ƒ, the fraction of vapor remaining.



262 12/4/13

There is at least a partial theoretical explanation in the case of ozone (Heidenreich and Thiemens, 1986, Gao and Marcus, 2001). Their theory can be roughly explained as follows. Formation of ozone in the stratosphere typically involves the energetic collision of monatomic and molecular oxygen, i.e.: O + O2 → O3* The ozone molecule thus formed is in a vibrationally excited state (designated by the asterisk) and, con-sequently, subject to dissociation if it cannot loose this excess energy. The excess vibrational energy can be lost either by collisions with other molecules, or by partitioning to rotational energy. In the strato-sphere, collisions are comparatively infrequent hence repartitioning of vibrational energy represents an important pathway to stability. Because there are more possible energy transitions for asymmetric spe-cies such as 16O16O18O and 16O16O17O than symmetric ones such as 16O16O16O, the former can repartition its excess energy and form a stable molecule. At higher pressures, such as prevail in the troposphere, the symmetric molecule can readily lose energy through collisions, lessening the importance of the vi-brational to rotational energy conversion. Gao and Marcus (2001) were able to closely match observed experimental fractionations, but their approach was in part empirical because a fully quantum me-chanical treatment is not yet possible. Theoretical understanding of mass-independent sulfur isotope fractionations is less advanced. Mass independent fractionations similar to those observed in Archean rocks (discussed in Chapter 11) have been produced in the laboratory by photo-dissociation (photolysis) of SO2 and SO using deep ultravio-let radiation (wavelengths <220 nm). Photolysis at longer wavelengths does not produce mass inde-pendent fractionations. Current explanations therefore focus on ultraviolet photolysis. However, there as yet is no theoretical explanation of this effect and alternative explanations, including ones that in-volve the role in symmetry in a manner analogous to ozone, cannot be entirely ruled out.

Figure 8.07. Oxygen isotopic composition in the stratosphere and troposphere show the effects of mass independent fractionation. A few other atmospheric trace gases show similar effects. Essentially all other material from the Earth and Moon plot on the terrestrial fractionation line. Af-ter Johnson et al. (2001).



263 12/4/13

8.5 HYDROGEN AND OXYGEN ISOTOPE RATIOS IN THE HYDROLOGIC SYSTEM We noted above that isotopically light water has a higher vapor pressure, and hence lower boiling point than isotopically heavy water. Let's consider this in a bit more detail. Raoult's law states that the partial pressure, p, of a species above a solution is equal to its molar concentration in the solution times the standard state partial pressure, p°, where the standard state is the pure solution. So for example:

!

pH216O = pH2

16Oo H2

16O[ ] 8.83a and

!

pH218O = pH2

18Oo H2

18O[ ] 8.83b

Since the partial pressure of a species is proportional to the number of atoms of that species in a gas, we can define α, the fractionation factor between liquid water and vapor in the usual way:

!

"l /v =pH2

18O/ p

H216O

[H218O]/[H2

16O] 8.84

By solving 8.80a and 8.80b for [H216O] and [H2

18O] and substituting into 8.81 we arrive at the rela-tionship:

!

"l /v =pH2

18Oo

pH2

16Oo 8.85

Interestingly enough, the fractionation factor for oxygen between water vapor and liquid turns out to be just the ratio of the standard state partial pressures. The next question is how the partial pressures vary with temperature. According to classical thermodynamics, the temperature dependence of the partial pressure of a species may be expressed as:

!

d lnPdT

="HRT 2 8.86

where T is temperature, ∆H is the enthalpy or latent heat of evaporation, and R is the gas constant. Over a sufficiently small range of temperature, we can assume that ∆H is independ-ent of temperature. Rearranging and integrating, we obtain:

ln p = !"HRT

+ const 8.87

We can write two such equa-tions, one for [H2

16O] and one for [H2

18O]. Dividing one by the other we obtain:

!

lnpH2

18Oo

pH2

16Oo = A" B

RT 8.88

where A and B are constants. This can be rewritten as: α = ae-B/RT 8.89

Figure 8.8. Temperature dependence of fractionation factors between vapor and water (solid lines) and vapor and ice (dashed lines) for various species of water.



264 12/4/13

(where a = eA). Over a larger range of temperature, ∆H is not constant. The fractionation factor in that case depends on the inverse square of temperature, so that the tempera-ture dependence of the fractiona-tion factor can be represented as:

!

ln" = A# BT 2 8.90

Figure 8.8 shows water-vapor and ice-vapor fractionation factors for oxygen. Over a temperature range relevant to the Earth's surface, the fractionation factor for oxygen shows an approximately inverse dependence on temperature. Hy-drogen isotope fractionation is clearly non-linear over a large range of temperature. Given the fractionation between wa-ter and vapor, we might predict that there will be considerable variation in the isotopic composition of water in the hydrologic cycle, and indeed there is. Furthermore, these variations form the basis of estimates of pa-leotemperatures and past ice volumes. Let's now consider the question of iso-topic fraction in the hydrosphere in greater detail. As water vapor condenses, the droplets and vapor do not remain in equilibrium if the precipitation occurs and the droplets fall out of the atmos-phere. So the most accurate description of the condensation process is Rayleigh distillation, which we discussed above. To a first approximation, condensation of water vapor will be a function of temperature. As air rises, it cools. You may have noticed the base elevation of clouds is quite uniform on a given day in a given locality. This elevation rep-resents the isotherm where condensa-tion begins. At that height, the air has become supersaturated, and condensation begins, forming clouds. Water continues to condense until equilibrium is again achieved. Further condensation will only occur if there is further cooling, which generally occurs as air rises. The point is that the parameter ƒ, the fraction of vapor remaining, can be approximately represented as a function of temperature. To

Figure 8.10. Variation of δ18O in precipitation as a function of mean annual temperature. This relationship is approxi-mately described as δ18Ο = 0.69Τ ˚C − 13.6. After Dansgaard (1964).

Figure 8.9. Calculated dependence of δ18O on temperature based on equ. 8.90. We assume the water vapor starts out 10 per mil de-pleted in δ18O.



265 12/4/13

explore what happens when water vapor condenses, lets construct a simple hypothetical model of con-densation and represent ƒ as hypo-thetical function of temperature such as:

!

ƒ =T " 22350

8.91

Since T is in kelvins, this equation means that ƒ will be 1 at 273 K (0°C) and will be 0 at 223 K (–50° C). In other words, we suppose condensa-tion begins at 0° C and is complete at –50°C. Now we also want to include temperature dependent fractionation in our model, so we will use equa-tion 8.86. Realistic values for the constant a and B are 0.9822 and -66.057J/mole respectively, so that 8.90 becomes:

α = 0.9822 e66.057/RT 8.92 Substituting 8.91, 8.92, and R=8.314J/mol-K into equation 8.81, our model is:

δ 18O = 103 × T − 22350

⎛⎝⎜

⎞⎠⎟0.9822e7.9448T −1

−1⎡

⎣⎢⎢

⎤

⎦⎥⎥

8.93

So we predict that the isotopic composition of water vapor should be a function of temperature. We can, of course, write a similar equation for equilibrium condensation. Figure 8.9 shows the temperature dependence we predict for water vapor in the atmosphere as a function of temperature (we have as-sumed that the vapor begins with δ18O of -10 before condensation begins). Of course, ours is not a particular sophisticated model; we have included none of the complexities of the real atmosphere. It is interesting to now look at some actual observations to compare with our model. Figure 8.10 shows the global variation in mean annual δ18O in precipitation, which should be somewhat heavier than vapor, as a function of mean annual air temperature. The actual observations show a linear dependence on temperature and a somewhat greater range of δ18O than our prediction. This reflects both the ad hoc nature of our model and the complexities of the real system. We did not, for example, consider that some precipitation is snow and some rain, nor did we consider the variations that evaporation at various temperatures might introduce. It is also worth noting that monthly averages show considerable deviation from the trend of mean annual data shown in Figure 8.10. Along with these factors, distance from the ocean also appears to be an important variable in the iso-topic composition of precipitation. The further air moves from the site of evaporation (the ocean), the more water is likely to have condensed and fallen as rain, and therefore, the smaller the value of ƒ. To-pography also plays an important role in the climate, rainfall, and therefore in the isotopic composition of precipitation. Mountains force air up, causing it to cool and the water vapor to condense. Thus the water vapor in air that has passed over a mountain range will be isotopically lighter than air on the ocean side of a mountain range. These factors are illustrated in the cartoon in Figure 8.11. Hydrogen as well as oxygen isotopes will be fractionated in the hydrologic cycle. Indeed, δ18O and δD are well correlated in precipitation, as is shown in Figure 8.12. The fractionation of hydrogen isotopes, however, is greater because the mass difference is greater. The correlation shown in Figure 8.12 is known as the Meteoric Water Line (or sometimes the Global Meteoric Water Line because local correlations

Figure 8.11. Cartoon illustrating the process of Rayleigh frac-tionation and the increasing fractionation of oxygen isotopes in rain as it moves inland.



266 12/4/13

between δD and δ18O often define somewhat shallower slopes than the global trend shown in Figure 8.12). The equation for the Meteoric Water Line is δD = δ18O + 10. As is ap-parent from both this equation and Figure 8.12, seawater (which by definition has δD = δ18O = 0) does not lie on this line. This reflects the role played by kinetic frac-tionation in evaporation, as we discussed in the section 8.3.2. Figure 8.13 shows the variation in oxygen iso-topic composition of me-teoric surface waters in the North America. The distribution is clearly not purely a function of mean annual tempera-ture, and this illustrates the role of the factors dis-cussed above. We will re-turn to the topic of the hydrologic system in Chapter 10 when we dis-cuss paleoclimatology.

8.6 ISOTOPE FRACTIONATION IN

THE BIOSPHERE Biological processes often involve large iso-topic fractionations, as we noted earlier. Indeed, biological processes are the most important cause of variations in the iso-tope composition of car-bon, nitrogen, and sulfur. For the most part, the largest fractionations occur during the initial

Figure 8.12. Northern hemisphere variation in mean annual δD and δ18O in precipitation and meteoric waters. The relationship between δD and δ18O is approximately δD = 8δ18O + 10. After Dansgaard (1964).

Figure 8.13. Variation of δ18O in precipitation in the United States (computed from analysis of tap water). δ18O depends on orographic effects, mean an-nual temperature, and distance from the sources of water vapor. After Bowen et al. (2007).



267 12/4/13

production of organic matter by the so-called primary producers, or autotrophs. These include all plants and many kinds of bacteria. The most important means of production of organic matter is photosynthe-sis, but organic matter may also be produced by chemosynthesis, for example at mid-ocean ridge hy-drothermal vents. Large fractions of both carbon and nitrogen occur during primary production. Addi-tional fractionations also occur in subsequent reactions and up through the food chain as hetrotrophs consume primary producers, but these are generally smaller.

8.6.1 Carbon Isotope Fractionation During Photosynthesis The most important of process producing isotopic fractionation of carbon is photosynthesis. Photo-synthetic fractionation of carbon isotopes is primarily kinetic. The early work of Park and Epstein (1960) suggested fractionation occurred in several steps. Subsequent work has elucidated the fraction-ations involved in these steps, which we will consider in more detail here. For terrestrial plants (those utilizing atmospheric CO2), the first step is diffusion of CO2 into the boundary layer surrounding the leaf, through the stomata, and finally within the leaf. The average δ13C of various species of plants has been correlated with the stomatal conductance (Delucia et al., 1988), in-dicating that diffusion into the plant is indeed important in fractionating carbon isotopes. On theoreti-cal grounds, a 4.4‰ difference in the diffusion coefficients is predicted (12CO2 will diffuse more rapidly; equation 8.71) so a fractionation of –4.4‰ is expected. Marine algae and aquatic plants can utilize either dissolved CO2 or HCO3

– for photosynthesis:

CO2(g) → CO2(aq) + H2O → H2CO3 → Η+ +HCO–3

An equilibrium fractionation of +0.9 per mil is associated with dissolution (13CO2 will dissolve more readily), and an equilibrium +7 to +12‰ fractionation (depending on temperature) occurs during hy-dration and dissociation of CO2. Thus, we expect dissolved HCO–

3 to be about 8 to 12 per mil heavier than atmospheric CO2. At this point, there is a divergence in the chemical pathways. Most plants use an enzyme called ribu-lose bisphosphate carboxylase oxygenase (RUBISCO) to catalyze a reaction in which ribulose bisphosphate re-acts with one molecule of CO2 to produce 2 molecules of 3-phosphoglyceric acid, a compound contain-ing 3 carbon atoms, in a process called carboxylation (Figure 8.14). Energy to drive this reaction is provided by another reaction, called photophosphorylation, in which electromagnetic energy is used to dissociate water, producing oxygen. The carbon is subsequently re-duced, carbohydrate formed, and the ribulose bisphosphate regenerated. Such plants are called C3 plants and this process is called the Benson-Calvin, or Calvin, cycle. C3 plants constitute about 90% of all plants and comprise the majority of cultivated plants, including wheat, rice, and nuts. Algae and autot-rophic bacteria also produce a 3-carbon chain as the initial photosynthetic product. There is a kinetic fractionation associated with carboxylation of ribulose bisphosphate that has been determined by sev-eral methods to be -29.4‰ in higher terrestrial plants. Bacterial carbolaxylation has different reaction mechanisms and a smaller frac-tionation of about –20‰. Thus for terrestrial plants a fractionation of about –34‰ is expected from the sum of the fraction. The actual ob-served total fractionation is in the range of –20 to –30‰. The disparity between the ob-served total fractionation and that expected from the sum of the steps presented something of a conun-drum. The solution appears to be a model that assumes the amount of

Figure 8.14. Ribulose bisphosphate (RuBP) carboxylation, the reaction by which C3 plants fix carbon during photosynthesis.



268 12/4/13

carbon isotope fractionation expressed in the tis-sues of plants depends on ratio the concentration of CO2 inside plants to that in the external en-vironment (e.g., Farquhar et al., 1982). In its sim-plest version, the model may be described by the equation:

!

" = a+ (ci /ca )(b # a) 8.94 where a is the isotopic fractionation due to diffu-sion into the plant, ci is the interior CO2 concen-tration, ca is the ambient or exterior CO2 con-centration, and b is the fractionation occur-ring during carboxylation. According to this model, where an unlimited amount of CO2 is available (i.e., when ci/ca ≈ 1), carboxylation alone causes fractionation. At the other ex-treme, if the concentration of CO2 in the cell is limiting (i.e., when ci/ca ≈ 0), essentially all carbon in the cell will be fixed and there-fore there will be little fractionation during this step and the total fractionation is essen-tially just that due to diffusion alone. Both laboratory experiments and field observations provide strong support for this model. More recent studies have shown that Rubisco enzyme exists in at least 2 different forms and that these two different forms fractionate carbon isotopes to differing degrees. Form I, which is by far the most common, typically produces the fractionation mentioned above; fractionation produced by Form II, which appears to be restricted to a few autotrophic bacteria and some dinoflagellates, can be as small as 17.8‰. The other photosynthetic pathway is the Hatch-Slack cycle, used by the C4 plants, which include hot-region grasses and related crops such as maize and sugarcane. These plants use phosphoenol pyruvate carboxylase (PEP) to initially fix the carbon and form oxaloacetate, a compound that contains 4 carbons (Fig. 8.15). A much smaller fractionation, about -2.0 to -2.5‰ occurs during this step. In phospho-phoenol pyruvate carboxylation, the CO2 is fixed in outer mesophyll cells as oxaloacetate and carried as part of a C4 acid, either malate or asparatate, to inner bundle sheath cells where it is decarboxylated and refixed by RuBP (Fig. 8.16). The environment in the bundle sheath cells is almost a closed system, so that virtually all the carbon carried there is refixed by RuBP. Consequently there is little fractionation during this step. C4 plants have average δ13C of -13‰. As in the case of RuBP photosynthesis, the frac-tionation appears to depend on the ambient concentration of CO2. This dependence was modeled by Farquhar et al. (1983) as:

!

" = a+ (b4 +b3# $ a)(ci /ca ) 8.95 where a is the fractionation due to diffusion of CO2 into the plant as above, b4 is the fractionation during transport into bundle-sheath cells, b3 is the fractionation during carboxylation (~ –3‰), φ is the fraction CO2 leaked from the plant. A third group of plants, the CAM plants, have a unique metabolism called the ‘Crassulacean acid me-tabolism’. These plants generally use the C4 pathway, but can use the C3 pathway under certain condi-tions. They are generally adapted to arid environments and include pineapple and many cacti, they have δ13C intermediate between C3 and C4 plants. Terrestrial plants, which utilize CO2 from the atmosphere, generally produce greater fractionations than marine and aquatic autotrophs, which utilize dissolved CO2 and HCO 3

– , together referred to as

Figure 8.15. Phosphoenolpyruvate carboxylation, the reaction by which C4 plants fix CO2 during photosynthesis.

Figure 8.16. Chemical pathways in C4 photosynthesis. From Fogel and Fuentes (1993).



269 12/4/13

dissolved inorganic carbon or DIC. As we noted above, there is about a +8‰ equilibrium fractionation be-tween dissolved CO2 and HCO 3

– . Since HCO3– is

about 2 orders of magnitude more abundant in sea-water than dissolved CO2, many marine algae utilize this species, and hence tend to show a lower net frac-tionation during photosynthesis. Diffusion is slower in water than in air, so diffusion is often the rate-limiting step. Most aquatic plants have some mem-brane-bound mechanism to pump DIC, which can be turned on when DIC is low (Sharkey and Berry, 1985). When DIC concentrations are high, fractiona-tion in aquatic and marine plants is generally similar to that in terrestrial plants. When it is low and the plants are actively pumping DIC, the fractionation is less because most of the carbon pumped into cells is fixed. Thus carbon isotope fractionations can be as low as 5‰ in algae. The model describing this frac-tionation is: ! = d + b3(F3 / F1) 8.96 where d is the equilibrium effect between CO2 and HCO-

3 , b3 is the fractionation associated with car-boxylation, and (F3/F1) is the fraction of CO2 leaked out of the cell. Though the net fractionation varies between species and depends on factors such as light intensity and moisture stress, higher C3 plants have average bulk δ13C values of -27‰; algae and lichens are typically -12 to 23‰. In aquatic systems where the pH is lower than in seawater, CO2 becomes a more important species and some algae can utilize this rather than HCO 3

– . In those cases, the total fractionation will be greater. An interesting illustration of this, and the effect of the CO2 concentration on net fractionation is shown in Figure 8.17, which shows data on the isotopic composition of algae and bacteria in Yellowstone hot springs. Some fractionation is also associated with respiration (the oxidation of carbohydrate to CO2), but the net effect of this is uncertain. Not surprisingly, the carbon isotope fractionation in C fixation is also temperature dependent. Thus higher fractionations are observed in cold-water phytoplankton than in warm water species. However, this observation also reflects a kinetic effect: there is generally less dissolved CO2 available in warm wa-ters because of the decreasing solubility at higher temperature (indeed, as we will see in Chapter 10, the effect of CO2 concentration on fractionation in marine algae can be used to estimate ancient CO2 levels). As a result, a larger fraction of the CO2 is utilized and there is consequently less fractionation. Surface waters of the ocean are generally enriched in 13C because of uptake of 12C during photosynthesis (Figure 8.18). The degree of enrichment depends on the productivity: biologically productive areas show greater enrichment. Deep water, on the other hand, is depleted in 13C (perhaps it would be more accu-rate to say it is enriched in 12C). Organic matter falls through the water column and is decomposed and "remineralized", i.e., converted to inorganic carbon, by the action of bacteria, enriching deep water in 12C. Thus biological activity acts to "pump" carbon, and particularly 12C from surface to deep waters.

Figure 8.17. Dependence of δ13C of algae and bacterial on CO2 concentration from hydro-thermal springs in Yellowstone National Park. Carbon isotope fractionation also depends on the pH of the water, because this determines the species of carbon used in photosynthesis. Data from Estep (1984), after Fogel and Fuen-tes (1993).



270 12/4/13

Nearly all organic matter originates through photo-synthesis. Subsequent reactions convert the photo-synthetically produced carbohydrates to the variety of other organic compounds utilized by organisms. Fur-ther fractionations occur in these reactions as discussed in Hayes (2001). Lipids (fats, waxes, etc.) in plants tend to be isotopically lighter (i.e., have lower δ13C) than other components (such as cellulose). The effect is usu-ally small, a few per mil, compared to the fractionation during photosynthesis. These fractionations are thought to be kinetic in origin and may partly arise from organic C-H bonds being enriched in 12C while organic C-O bonds are enriched in 13C. 12C is preferen-tially consumed in respiration (again, because bonds are weaker and it reacts faster), which would tend to enrich residual organic matter in 13C. Thus the carbon isotopic composition of organisms becomes more posi-tive moving up the food chain. Interestingly, although the energy source for che-mosynthesis is dramatically different than for photo-synthesis, the carbon-fixation process is similar and still involves the Calvin cycle. Not surprisingly then, carbon fractionation during chemosynthesis is similar to that during photosynthesis. Some chemosynthetic bacteria, notably some of the symbionts of hydrothermal vent

organisms, have Rubisco Form II, and hence show smaller fractionations. 8.6.2 Nitrogen Isotope Fractionation in Biological Processes

Nitrogen is another important element in biological processes, being an essential component of all amino acids and proteins. As in the case of carbon, most of terrestrial nitrogen isotopic variation results from biological processes. These processes, however, are considerably more complex because nitrogen exists in more forms and more oxidation states. There are five important forms of inorganic nitrogen: molecular nitrogen (N2), nitrate (NO3

+), nitrite (NO2+), ammonia (NH3), and ammonium (NH4

-). Equilib-rium isotope fractionations that can be quite large occur between these five forms. Except for the ammonium–ammonia reaction, the reactions between these forms are all redox reactions and they are predominantly biologically mediated. Significant kinetic fractionations occur during these biological mediated reactions. Heterotrophs, such as you and I, get their nitrogen from what they eat and there is only a slight dif-ference, generally around 1 or 2‰, between animals and what they eat. Autotrophs, including algae, plants and bacteria, must assimilate nitrogen from the environment. Plants and algae cannot assimilate and utilize N2; they must used some type of fixed nitrogen, which can be any of the remaining 4 forms listed above. The dirty work of converting nitrogen to ammonia (and from there to other forms of ni-trogen), a process called fixation, is done by bacteria, including photosynthetic ones. This involves only a small fractionation of -3 to +1‰. Reduced nitrogen (e.g., ammonia) is the form of nitrogen that is ultimately incorporated into organic matter, as NH2 amine groups, by autotrophs. There is a fractionation of up to -20‰ when autotrophs uptake of ammonium (14N is taken up preferentially), the extent depending on the ammonium abun-dance. When ammonium is abundant, fractionation tends to be large, when it is not, most available ammonium is taken up and there is little fractionation (Figure 8.19). Most plants, including many ma-

Figure 8.18. Depth profile of total dissolved inorganic carbon and δ13C in the North At-lantic. Data from Kroopnick et al. (1972).



271 12/4/13

rine algae, can utilize oxidized nitrogen, NO3

– and NO 2– , as well as reduced nitrogen.

In these cases, nitrogen must first be reduced by the action of reductase enzymes. δ15N frac-tionations of 0 to –24‰ have been measured for the assimilation of NO3

– . There is a small fractionation when that ammonium is subsequently incorporated into organic molecules. There are two principle reactions by which ammonia is incorporated into organic matter: formation of glutamate from α-ketoglutarate via the glutamate ded-hydrogenase reaction, and formation of glu-tamine from glutamate via the enzyme glu-tamine synthetase. A positive fractionation (i.e., the product is enriched in 15N) of +2 to +4 has been measured for the glutamate ded-hydrogenase reaction, and the fractionation

for the glutamine synthetase reaction is also expected to be positive, because N is bound more strongly in the product than in ammonia. The net result of these various frac-tionations is that organic nitrogen is usually heavier than atmospheric ni-trogen. The isotopic compositions of marine particulate nitrogen and non-nitrogen-fixing plankton are typically –3‰ to +12‰ δ15N. Terrestrial plants unaffected by artificial fertilizers gen-erally have a narrower range of +6‰ to

+13 per mil. Legumes (and a few other kinds of plants) are a special case. While they cannot fix nitro-gen, they have symbiotic bacteria in their root nodules that can. As a result, legumes have distinctly lower δ15N than other terrestrial, in the range of –2 to +4‰. Marine blue-green algae range from -4 to +2, with most in the range of -4 to -2‰. A caveat to all this is that most fixed nitrogen in modern ecosystems now derived, directly or indi-rectly, from artificial fertilizers. These fertilizers contain ammonia derived from atmospheric N2 through the Haber process, in which there is little isotopic fractionation. Consequently, modern plants, particularly those raised on artificial fertilizers, have lower δ15N.

8.6.3 Oxygen and Hydrogen Isotope Fractionation by Plants Oxygen is incorporated into biological material from CO2, H2O, and O2. However, both CO2 and O2 are in oxygen isotopic equilibrium with water during photosynthesis, and water is the dominant form. Therefore, the isotopic composition of plant water determines the oxygen isotopic composition of plant material. The oxygen isotopic composition of plant material seems to be controlled by exchange reac-tions between water and carbonyl oxygens (oxygens doubly bound to carbon):

C !16O +H218O C ! 18O"H2

16O

Table 8.2 Nitrogen Isotopic Fractionation in the Exogenic Nitrogen Cycle. Reaction Fractionation N2 fixation N2 → 2NH3 <2‰ NH3 → NH4

+ +19 to 33‰ Nitrification NH4

+→ NO3- -15 to -35‰

Denitirification NO3- → N2 -5 to -30‰

Nitrate reduction NO3- → NH4

+ -20 to -35‰

Figure 8.19. Dependence of nitrogen isotope frac-tionation by bacteria and diatoms on dissolved ammo-nium concentration. From Fogel and Fuentes (1993).



272 12/4/13

Fractionations of +16 to +27‰ (i.e., the organically bound oxygen is heavier) have been measured for these reac-tions. Consistent with this, cellulose from most plants has δ18O of +27±3‰. Other factors, however, play a role in the oxygen isotopic composition of plant material. First, the isotopic com-position of water varies from δ18O ≈ -55‰ in Arctic regions to δ18O ≈ 0‰ in the oceans. Second, less than complete equilibrium may be achieved if photo-synthesis is occurring at a rapid pace, resulting in less fractionation. Finally, some fractionation of water may occur during transpiration, with residual wa-ter in the plant becoming heavier. Hydrogen incorporated into organic matter during photosynthesis is de-rived meteoritic water, whose isotopic composition varies with distance from source and temperature (e.g., Figure 8.12). Hydrogen isotope fractionation during photosynthesis occurs such that the light iso-tope is strongly enriched in organic material. In marine algae, isotope fractionations of –100 to –150‰ have been observed, which is similar to that observed in terrestrial plants: –86 to -120‰. Among terres-trial plants, there appears to be a difference between C3 and C4 plants. The former fractionations of –117 to –121‰, while fractionations -86 to –109‰ have been observed in C4 plants. As a consequence, hy-drogen isotopic compositions of organic matter show wide variations. As is the case for oxygen, variations in the isotopic composition of available water and fractionation during transpiration are important in controlling the hydrogen isotopic composition of plants. This is il-lustrated in Figure 8.20.

8.6.4 Carbon and Hydrogen Isotopic Composition of Organic Matter in Sediments Organic matter buried in sediment undergoes a series of diagenetic changes, some of which are bio-logically mediated (by bacteria) and some of which are abiotic. A complex, refractory† substance called kerogen is the product the early stages of diagenesis. Sediments typically contain about a half a percent of such organic material. Little change in carbon and hydrogen isotopic composition occurs in early di-agenesis; consequently, the δ13C and δD of kerogen are similar to the isotopic compositions of its source with δ13C typically in the range of –20 to -32‰ and δD in the range of -75 to -150‰. Subsequent heat and pressure transforms kerogen. This process produces coal from Type III kerogen, which is derived from cellulose-rich terrestrial organic matter accumulated in swamps and peat bogs. Lipid-rich material derived from aquatic and marine algae and non-woody parts of terrestrial plants form Type I and II kerogens, which ultimately transforms to petroleum and natural gas. The transformations to coal and petroleum involve the progressive loss of hydrogen and oxygen, and hence decrease in H:C and O:C ratios of the material. Most of this likely occurs through generation and loss of H2O and CO2, but gen-eration low molecular weight hydrocarbons that compose natural gas (methane, CH4; ethane, C2H6; propane, C3H8; and butane, C4H10‡) can also occur. Fractionations tend to be small during this phase and

† “Refractory” in this context means resistant to oxidation or bacterial decomposition. ‡ These compounds, in which the carbons are chained together, are called alkanes and are referred to by the number of carbons in the chain, e.g., C2 = ethane.

Figure 8.20. Isotopic Fractionations of hydrogen during primary production in terrestrial plants. After White (1988).



273 12/4/13

petroleum liquids typically have δ13C within 1‰ or of the kerogen that gave rise to them. Hydrogen isotopic fractionations also tend to be relatively small, although as Figure 8.21 shows, the δD of petroleum is skewed to lighter isotopic composition relative to kerogen. This is likely due mainly to the lipids, which tend to isotopically lighter contributing dispropor-tionately to petroleum generation. Similarly small isotopic changes occur in the generation of coal. These changes in isotopic composition are similar to or smaller than the isotopic heterogeneity of the kerogen itself. Thus carbon and hydrogen isotopic composition is used extensively in the petroleum as a tracer to determine the source rock of a petroleum reservoir. As temperature increases beyond 100˚ C or so hydrocarbons composing petroleum begin to break down, or ‘crack’, into lower molecular weight compounds and ultimately into natural gas. Kinetic fractionations can become important during this phase as 13C forms stronger bonds. Consequently, gases generated, particularly methane, will be isotopically lighter than the precursor material. Clayton (1991) concluded that the effect of cracking is to increase δ13C of oil by about 1.5‰ by 50% cracking. The isotopic composition of methane generated during the thermal cracking phase can have consid-erably lighter carbon and hydrogen than the oil. As Figure 8.22 shows, this thermogenic methane may have δ13C as low as -60‰ and δD as low as -300‰. Methane may also be produced by microbial processes. Some of these occur in anoxic sediments and soils during early diagenesis. Indeed, it is not uncommon to see bubbles of methane rising to the surface in small ponds from anoxic sediment accumulating beneath them. Methane is also generated in guts of animals, particularly ruminants, such as cattle (but also humans). In all cases, the organisms actually producing the methane are a group of archaea called methanogens. They produce methane through two dominant pathways: CO2 reduction by reaction with hydrogen, and fermenta-tion. In the latter process, acetic acid (CH3COOH) is broken down to produce CH4 plus CO2. Chemi-cally, this biogenic methane is distinct from thermogenic methane discussed above by the near absence of C2 and higher alkanes in the latter. They are also isotopically distinct, as may be seen in Figure 8.22. Equilibrium fractionation between CO2 and CH4 is quite large. Richet et al. (1977) derived the following equation:

1000 lnαCH4−CO2= 29.2

1000T

⎛⎝⎜

⎞⎠⎟ − 29.6 8.97

Figure 8.21. Hydrogen isotopic composi-tions of lipids, phytoplankton, and sedi-mentary organic matter. After Schoell (1984).



274 12/4/13

According to this equation, α should rise from 1.05 to 1.08 as temperature falls from 100˚C to 0˚C. Thus at the ocean floor (~0˚C), methane generated in equilibrium with seawater CO2 would be expected to have δ13C of around -80‰. Methane in marine sediments is most often somewhat heavier than this (Whitcar et al., 1986). Experiments by Bolz et al. (1996) suggest why. In cultural experiments they found that fractionation between CO2 and CH4 was close to equilibrium values when microbial populations were stable and only a small fraction of the CO2 was being converted to CO2. Fractionation was less than equilibrium values when populations were growing and large fractions of available CO2 (up to 30 or 40%) was converted to CH4. Thus as is the case in photosynthesis, kinetic factors may limit fractiona-tion during methanogenesis. As Figure 8.22 shows, carbon isotope fractionation is generally smaller when methane is produced by fermentation, but hydrogen fractionation tends to be greater.

8.6.5 Biological Fractionation of Sulfur isotopes Sulfur is a critical but minor component of living tissue (C:S atomic ratio is about 200). Plants take up sulfur as sulfate and subsequently reduce it to sulfide and incorporate into cysteine (an amino acid with the composition HO2CCH(NH2)CH2SH). There is apparently no fractionation of sulfur isotopes in transport across cell membranes and incorporation, but there is a fractionation of +0.5 to –4.5‰ in re-duction process, referred to as assimilatory sulfate reduction. This is substantially less than the expected fractionation of about -20‰, suggesting that nearly all the sulfur taken up by primary producers is re-duced and incorporated into tissue. Sulfur, however, plays two other important roles in biological processes. First, sulfur in the form of sulfate can act as an electron acceptor or oxidant, and is utilized as such by sulfur-reducing bacteria.

Figure 8.22. Carbon and hydrogen isotopic composition of methane. “Dry” refers to methane not associated with petroleum. “A” is the isotopic composition of atmospheric methane, “E” and “Z” are the isotopic compositions of abiotic methane from the East Pacific Rise (Whelan, 1988), and “the Zimbales peridotite in the Philippines (Abrajano et al., 1988), respectively. These methanes have isotopic compositions close to that of mantle carbon. After Schoell (1988) and Whitcar et al. (1986).



275 12/4/13

This process, in which H2S is liberated, is called dissimilatory sulfate reduction and plays an important role in biogeochemical cycles, both as a sink for sulfur and source for atmospheric oxygen (because the sulfide is usually then buried in sediments). A large fractionation of +5 to –46‰ is associated with this process. This process produces by far the most significant fractionation of sulfur isotopes, and thus governs the isotopic composition of sulfur in the exogene. Sedimentary sulfate typically has δ34S of about +17, which is similar to the isotopic composition of sulfate in the modern oceans (+20), while sedimentary sulfide has a δ34S of –18. The living biomass has a δ34S of ≈ 0. The final important role of sulfur is a reductant. Sulfide is an electron acceptor used by some types of photosynthetic bacteria as well as other bacteria in the reduction of CO2 to organic carbon. Most unique among these perhaps are the chemosynthetic bacteria of submarine hydrothermal vents. They utilize H2S emanating from the vents as an energy source and form the base of the food chain in these unique ecosystems. A fractionation of +2 to – 18‰ is associated with this process.

REFERENCES AND SUGGESTIONS FOR FURTHER READING Abrajano, T. A., N. C. Sturchio, J. K. Bohlke, G. L. Lyon, R. J. Poreda and C. M. Stevens 1988, Methane-

hydrogen gas seeps, Zambales Ophiolite, Philippines: Deep or shallow origin? Chemical Geology, 71: 211-222, 10.1016/0009-2541(88)90116-7.

Bigeleisen, J. and M. G. Mayer 1947. Calculation of equilibrium constants for isotopic exchange reac-tions, Journal of Chemical Physics, 15: 261-267.

Botz, R., H.-D. Pokojski, M. Schmitt and M. Thomm, Carbon isotope fractionation during bacterial methanogenesis by CO2 reduction, Organic Geochemistry., 25: 255-262, 10.1016/s0146-6380(96)00129-5 1996.

Bowen, R. 1988 Isotopes in the Earth Sciences, Elsevier, Essex, 647p. Bowen, G. J. 2010. Isoscapes: Spatial pattern in isotopic biogeochemistry, Annual Review of Earth and

Planetary Sciences, 38: 161-187, doi: doi:10.1146/annurev-earth-040809-152429. Bowen, G. J., J. R. Ehleringer, L. A. Chesson, E. Stange and T. E. Cerling, Stable isotope ratios of tap wa-

ter in the contiguous United States, Water Resour. Res., 43: W03419, doi: 10.1029/2006wr005186, 2007. Broecker, W. and Oversby, V. 1971. Chemical Equilibria in the Earth, McGraw-Hill, New York, 318 pp. Chacko, T., Cole, D. & Horita, J. 2001. Equilibrium oxygen, hydrogen and carbon isotope fractionation

factyors applicable to geologic systems. In: Valley, J. W. & Cole, D. (eds.) Reviews in Mineralogy and Geochemistry v. 43: Stable Isotope Geochemistry.

Dansgaard, W. 1964. Stable isotopes in precipitation, Tellus, 16, 436-463. Dennis, K. J., Affek, H. P., Passey, B. H., Schrag, D. P. & Eiler, J. M. 2011. Defining an absolute reference

frame for "clumped‚" isotope studies of CO2. Geochimica et Cosmochimica Acta, 75, 7117-7131. Eiler, J. M. 2013. The Isotopic anatomies of molecules and minerals. Annual Review of Earth and Planetary

Sciences, 41, 411-441, doi: doi:10.1146/annurev-earth-042711-105348. Estep, M. L. F. 1984. Carbon and hydrogen isotopic compositions of algae and bacteria from hydro-

thermal environments, Yellowstone National Park. Geochimica et Cosmochimica Acta, 48, 591-599, doi: 10.1016/0016-7037(84)90287-4.

Farquhar, G. D. 1983. On the Nature of Carbon Isotope Discrimination in C4 Species. Functional Plant Bi-ology, 10, 205-226, doi: 0.1071/PP9830205.

Farquhar, G. D., O'Leary, M. H. & Berry, J. A. 1982. On the Relationship Between Carbon Isotope Dis-crimination and the Intercellular Carbon Dioxide Concentration in Leaves. Functional Plant Biology, 9, 121-137, doi: 10.1071/PP9820121.

Faure, G. 1986. Principles of Isotope Geology, 2nd ed., J. Wiley & Sons, New York, 589 p. Ferronsky, V. I., and V. A. Polyakov 1982. Environmental Isotopes in the Hydrosphere, ed., 466 pp., John

Wiley and Sons, Chichester.



276 12/4/13

Fogel, M. L., and M. L. Cifuentes 1993. Isotope Fractionation during Primary Production, in Organic Geochemistry: Principles and Applications edited by M. H. Engel and S. A. Macko, p. 73-98, Plenum, New York.

Gao, Y. Q. and R. A. Marcus 2001. Strange and unconventional isotope effects in ozone formation, Sci-ence, 293:259-263.

Ghosh, P., J. Adkins, H. Affek, B. Balta, W. Guo, E. A. Schauble, D. Schrag and J. M. Eiler, 13C-18O bonds in carbonate minerals: A new kind of paleothermometer, Geochimica et Cosmochimica Acta, 70: 1439-1456,10.1016/j.gca.2005.11.014 2006.

Hayes, J. M. 2001. Fractionation of carbon and hydrogen isotopes in biosynthetic processes. In: VALLEY, J. W. & COLE, D. (eds.) Reviews in Mineralogy and Geochemistry v. 43: Stable Isotope Geochem-istry. Washington: Mineralogical Society of America.

Hiedenreich, J. E. and M. Thiemens 1986. A non‐mass‐dependent oxygen isotope effect in the produc-tion of ozone from molecular oxygen: The role of molecular symmetry in isotope chemistry. Journal of Chemical Physics, 86: 2129-2136.

Hoefs, J. 1988. Stable Isotope Geochemistry, 3rd ed., Springer-Verlag, Berlin, 241p. Johnson, D. G., K. W. Jucks, W. A. Traub and K. V. 2001. Chance, Isotopic composition of stratospheric

ozone, J. Geohys. Res., 105:9025-9031. Kawabe, I., 1979. Calculation of oxygen isotopic fractionation in quartz-water system with special refer-

ence to the low temperature fractionation, Geochimica et Cosmochimica Acta 42: 613-621. O'Leary, M. H. 1981. Carbon isotope fractionation in plants, Phytochemistry, 20, 553-567. O'Leary, M. H. 1988. Carbon isotopes in photosynthesis. BioScience, 38, 328-336, doi: 10.2307/1310735. O'Neil, J. R., Theoretical and experimental aspects of isotopic fractionation, in Stable Isotopes in High

Temperature Geologic Processes, Reviews in Mineralogy Volume 16, J. W. Valley, et al., eds., Mineralogical Society of America, Washington, pp. 1-40, 1986.

Park, R. and S. Epstein, 1960. Carbon isotope fractionation during photosynthesis, Geochim. Cosmochim. Acta, 21, 110-126.

Richet, P., Bottinga, Y. & Javoy, M. 1977. A Review of Hydrogen, Carbon, Nitrogen, Oxygen, Sulphur, and Chlorine Stable Isotope Fractionation Among Gaseous Molecules. Annual Review of Earth and Planetary Sciences, 5, 65-110, doi :10.1146/annurev.ea.05.050177.000433.

Schoell, M. 1988 Multiple origins of methane in the Earth, Chemical Geology, 71:1-10, doi: 10.1016/0009-2541(88)90101-5.

Schoell, M. 1984. Recent advances in petroleum isotope geochemistry, Organic Geochemistry, 6:645-663. Sharkey, T. D. & Berry, J. A. 1985. Carbon isotope fractionation of algae as influenced by an inducible

CO2 concentrating mechanism. In: LUCAS, W. J. & BERRY, J. A. (eds.) Inorganic Carbon Uptake by Aquatic Photosynthetic Organisms. Rockville, MD: American Society of Plant Physiology.

Sharp, Z. D. 2007. Principles of Stable Isotope Geochemistry, Upper Saddle River, NJ, Pearson Prentice Hall. Urey, H. 1947. The thermodynamic properties of isotopic substances, J. Chem. Soc. (London), 1947: 562-

581. Wang, Z., E. A. Schauble and J. M. Eiler 2004. Equilibrium thermodynamics of multiply substituted iso-

topologues of molecular gases, Geochimica et Cosmochimica Acta, 68: 4779-4797, doi: 10.1016/j.gca.2004.05.039.

Welhan, J. A., Origins of methane in hydrothermal systems, Chemical Geology, 71:183-198, 10.1016/0009-2541(88)90114-3 1988.

White, J. W. C. 1988. Stable hydrogen isotope fractionation in plants: a review of current theory and some potnetial applications. In: RUNDEL, P. W., EHLERINGER, J. R. & NAGY, K. A. (eds.) Stable Iso-topes in Ecological Research. New York: Springer Verlag.

Whiticar, M. J., E. Faber and M. Schoell 1986. Biogenic methane formation in marine and freshwater en-vironments: CO2 reduction vs. acetate fermentation--Isotope evidence, Geochimica et Cosmochimica Acta, 50: 693-709,10.1016/0016-7037(86) 90346-7.



277 12/4/13

PROBLEMS 1. Using statistical mechanical approach outlined in this chapter, calculate the fractionation factor ∆CO-O2

for the 17O/16O ratio at 273 K. What is the expected ratio of fractionation of 17O/16O to that of 18O/16O at this temperature?

2. Show that at 300K, equilibrium fractionation between O2 and CO should lead to ∆17O/∆18O ≈ 0.52.

3. Derive a temperature dependent expression for the equilibrium constant in equation 8.62:

K =q13C16O

q12C17O

q12C16Oq13C17O

What is the value at 300 K? What is the value at 1000K?

4. Calculate the ratio [13C16O18O]/[12C16O2] in CO2 gas with a bulk isotopic composition of δ13CPDB = -6‰ and δ18OSMOW = 0. Assume that the standards have the isotopic compositions listed in Table 8.1.

5. Calculate the δ18O of raindrops forming in a cloud after 80% of the original vapor has already con-densed assuming (1.) the water initial evaporated from the ocean with δ18O = 0, (2.) the liquid-vapor fractionation factor, α = 1.0092.

Documents

Chapter 8. Theory of Stable Isotope Fractionations - EAS