Isotope Geochemistry. Stable Isotope Geochemistry I: Theory

Transcription

1 Stable Isotope Geochemistry I: Theory INTRODUCTION Stable isotope geochemistry is concerned with variations of the isotopic compositions of light elements 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 compounds (O), or are important constituents of naturally-occurring solids and fluids. The abundance of the rare isotope is sufficiently high (generally at least tenths of a percent) to facilitate analysis. It was once thought that elements not meeting these criteria would not show measurable variation in isotopic composition. However, as new techniques offering greater sensitivity and higher precision have become available (particularly use of the MC-ICP-MS), geochemists have begun to explore isotopic variations of metals such as Mg, Ca, Ti, Cr, Fe, Zn, Cu, Ge, Mo, Ti, and Tl. The isotopic variations observed in these metals have generally been quite small, except in materials affected or produced by biologically processes, where fractionations are a little larger, but still smaller than the former group of elements. Nevertheless, some geologically useful information has been obtained from isotopic study of these metals and exploration of their isotope geochemistry continues. Stable isotopes can be applied to a variety of problems. One of the most common is geothermometry. This use derives from the extent of isotopic fractionation varying inversely with temperature: fractionations are large at low temperature and small at high temperature. Another application is process identification. For instance, plants that produce C 4 hydrocarbon chains (that is, hydrocarbon chains 4 carbons long) as their primary photosynthetic products fractionate carbon differently than to plants that produce C 3 chains. This fractionation is retained up the food chain. This allows us to draw some inferences about the diet of fossil mammals from the stable isotope ratios in their bones. Sometimes stable isotopes are used as 'tracers' much as radiogenic isotopes are. So, for example, we can use oxygen isotope ratios in igneous rocks to determine whether they have assimilated crustal material. The δ Notation NOTATION AND DEFINITIONS Variations in stable isotope ratios are typically in the parts per thousand range and hence are generally reported as permil variations, δ, from some standard. Oxygen isotope fractionations are generally reported in permil deviations from SMOW (standard mean ocean water): #! 18 O = (18 O/ 16 O) sam "( 18 O/ 16 O) SMOW % $ ( 18 O/ 16 O) SMOW & ' The same formula is used to report other stable isotope ratios. Hydrogen isotope ratios, δd, are reported 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 Canyon Diablo iron meteorite. Cl isotopes are also reported relative to seawater; Li and B are reported relative to NBS (which has now become NIST: National Institute of Standards and Technology) standards /22/12

2 Table 8.1. Isotope Ratios of Stable Isotopes Element Notation Ratio Standard Absolute Ratio Hydrogen δd D/H ( 2 H/ 1 H) SMOW Lithium δ 6 Li 6 li/ 7 Li NBS L-SVEC Boron δ 11 B 11 B/ 10 B NBS Carbon δ 13 C 13 C/ 12 C PDB Nitrogen δ 15 N 15 N/ 14 N atmosphere Oxygen δ 18 O 18 O/ 16 O SMOW, PDB δ 17 O 17 O/ 16 O SMOW Chlorine δ 37 Cl 37 Cl/ 35 Cl seawater ~ Sulfur δ 34 S 34 S/ 32 S CDT Unfortunately, a dual standard has developed for reporting O isotopes. Isotope ratios of carbonates are reported relative to the PDB carbonate standard. This value is related to SMOW by: δ 18 O PDB = δ 18 O SMOW Table 19.1 lists the values for standards used in stable isotope analysis. The Fractionation Factor An important parameter in stable isotope geochemistry is the fractionation factor, α. It is defined as: " A# B $ R A R B 8.03 where R A and R B are the isotope ratios of two phases, A and B. The fractionation of isotopes between two phases is often also reported as A-B = δ A δ B. The relationship between and α is: (α - 1)10 3 or 10 3 ln α 8.04 We derive it as follows. Rearranging equ. 8.01, we have: R A = (δ A )R STD / where R denotes an isotope ratio. Thus α may be expressed as: " = (# $ +103 )R STD /10 3 (# % )R STD /10 3 = (# $ +103 ) (# % ) Subtracting 1 from each side and rearranging, and since δ is generally << 10 3, we obtain: " #1 = ($ % # $ ) & ($ & ) ' ($ % # $ & ) = ( )10 # The second equation in 8.04 results from the approximation that for x 1, ln x 1 x. As we will see, α is related to the equilibrium constant of thermodynamics by α A-B = (K/K ) 1/n 8.08 where n is the number of atoms exchanged, K is the equilibrium constant at infinite temperature, and K is the equilibrium constant is written in the usual way (except that concentrations are used rather than activities because the ratios of the activity coefficients are equal to 1, i.e., there are no isotopic effects on the activity coefficient). THEORY OF MASS DEPENDENT ISOTOPIC FRACTIONATIONS Isotope fractionation can originate from both kinetic effects and equilibrium effects. The former might be intuitively expected (since for example, we can readily understand that a lighter isotope will diffuse 204 8/22/12

3 faster than a heavier one), but the latter may be somewhat surprising. After all, we have been taught 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 mechanical effects predict the small differences in the chemical properties of isotopes quite accurately. We shall now consider the manner in which isotopic fractionations arise. The electronic structures of all isotopes of an element are identical and since the electronic 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 molecular vibrations. The energy of a molecule can be described Figure Energy-level diagram for the hydrogen atom. Fundamental vibration frequencies are 4405 cm -1 for H 2, 3817 cm -1 for HD, and 3119 cm -1 for D 2. The zero-point energy of H 2 is greater than that for HD which is greater than that for D 2. After O'Neil (1986). in terms of several components: electronic, nuclear spin, translational, rotational and vibrational. 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. Translational and rotational motion can be described by classical mechanics, but an adequate description 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: 1 2 C16 O 2 + H 2 18 O = 1 2 C18 O 2 + H 2 16 O is only about 1.04 at 25 C. Figure 8.01 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 /22/12

4 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 attracted to each other by the net charge on atoms. At intermediate distances, the potential energy is at a minimum. The energy and the distance over which the atoms vibrate 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 tells us that even at absolute 0 the atoms will vibrate at a ground frequency ν 0. The system will have energy of 1 / 2 hν 0, where h is Planck's constant. This energy level is called the Zero Point Energy (ZPE). Its energy 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 energy level for a given quantum number will be lower for a bond involving a heavier isotope of an element, as suggested in Figure Thus bonds involving heavier isotopes 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 participate more readily in a given chemical reaction. If the reaction fails to go to completion, which is often the case, this tendency gives rise to kinetic fractionations of isotopes. There are other causes of kinetic fractionations as well, and will consider them in due course. We will now consider in greater detail the basis for equilibrium fractionation, and see that they can be predicted from statistical mechanics. Equilibrium Fractionations Urey (1947) and Bigeleisen and Mayer (1947) pointed out the possibility of calculating the equilibrium 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 available to an atom or molecule. At equilibrium, the ratio of the number of molecules having internal energy E i to the number having the zero point energy E 0 is: n i n 0 = g i e "E i / kt where n 0 is the number of molecules with ground-state or zero point energy, n i is the number of molecules with energy E i 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 E i ). 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 number of particles in that state divided by the number of particles in those states: E =! i! i n i E i n i! g i E i e " E i /kt =! g i e " E i /kt The partition function, q, is the denominator of this equation: * The energy level is said to be 'degenerate' if two or more states have the same energy level E i /22/12

5 q = g i e "E i / kt # 8.11 Substituting 8.11 into 8.10, we can rewrite 8.10 in terms of the partial derivatives of q: " lnq E = kt 2 "T 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 8.12 S = U T Rlnq 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 = "Rlnq 8.14 And for the entropy and energy changes of a reaction, we have: #!U " T!S = "Rln$ q n 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 = "Rln$ q n 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 = " q n 8.18 For example, in the reaction involving exchange of 18 O between H 2 O and CO 2, the equilibrium constant is simply: 1/2 K = q C 16 O 2 1/2 q q H2 18 O q C H2 O O2 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 can be written as: q total = q tr q vib q rot 8.20 i.e., the product of the translational, rotational and vibrational partition functions. It is convenient to treat these three modes of motion separately. Let's now do so Entropy is defined in the second law of thermodynamics, which states: ds = dq rev T where Q rev is heat gained by a system in a reversible process. Entropy can be thought of as a measure of the randomness of a system /22/12

6 Translational Partition Function Writing a version of equation for translational energy, q trans is expressed as: q trans = g tr,i e "E tr,i # / kt 8.21 i Now all that remains is to find and 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, so we can use a classical mechanical approach to calculating the energy. The quantum translational energy of a particle in a cubical box is given by: E trans = n 2 h 2 8md 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 into and integrating: ( )1/ 2 # q trans = e "n 2 h 2 8md $ 2 kt dn = 2%mkT 0 h d 8.23 gives an expression for q trans for each dimension. The total three-dimensional translational partition function is then: q trans = ( 2"mkT )3/2 V 8.24 h where V is volume and is equal to d 3. (It may seem odd that the volume should enter into the calculation, 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 K tr as: " K tr = # q tr 8.25 K tr reduces to the product of the molecular masses raised to the stoichiometric coefficient times threehalves: " K tr = M i 3/2 # i 8.26 i where we have replace m with M, the molecular mass. Thus the translational contribution to the partition function and fractionation factor is independent of temperature. Rotational Partition Function The allowed quantum rotational energy states are: E rot = j( j + 1)h2 8! 2 I where j is the rotational quantum number and I is the moment of inertia. For a diatomic molecule, I= µd 2, where d is the bond length, m i is the atomic mass of atom i, and µ is reduced mass: 8.27 µ = m 1 m 2 m 1 + m /22/12

7 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: q rot = (2 j +1)e j( j+1)h 2 /8" 2 I i kt # 8.29 i Again the spacing between energy levels is relatively small (except for hydrogen) and may be evaluated as an integral. For a diatomic molecule, the partition function for rotation is given by: q rot = 8" 2 IkT #h 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 18 O 16 O), and 2 for a homonuclear diatomic molecule such as 16 O 2 or a symmetric triatomic molecule such as 16 O 12 C 16 O (more complex molecules will have higher symmetry numbers). Equ also holds for linear polyatomic molecules with the symmetry factor equal to 2 if the molecule has a plane of symmetry (e.g., CO 2 ) and 1 if it does not. For non-linear polyatomic molecules, the partition function is given by: q rot = 8" 2 (8" 2 ABC) 1/ 2 (kt) 3 / 2 #h where A, B, and C are the principal moments of inertia of the molecule and σ is equal to the number of equivalent ways of orienting the molecule in space (e.g., 2 for H 2 O, 12 for CH 4 ). In calculating the rotational contribution to the partition function and 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 we may write: " I K rot = i % )# $ & ' 8.32 i! i In general, bond lengths are also independent of the isotope involved, so the moment of inertia term may be replaced by the reduced masses. Vibrational Partition Function We will simplify the calculation of the vibrational partition function by treating the diatomic molecule 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: ( " E vib = n + 1 % $ ' h( 8.33 # 2& where n is the vibrational quantum number and ν is vibrational frequency. Unlike rotational and vibrational 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: q vib = /2kT e hν 8.34 hν /kt 1 e 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, 209 8/22/12

8 with the individual partition functions given by 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: 3 n e hν /2kT q vib = e hν /kt Where n is equal to 6 for non-linear polyatomic molecules and 5 for linear polyatomic molecules (there 3-6 vibrational modes for non-linear polyatomic molecules and 3-5 modes of motion for linear ones). At room temperature, the exponential term in the denominator approximates to 0, and the denominator therefore approximates to 1, so the relation simplifies to: q vib " e #h$ /2kT 8.36 Thus at low temperature, the vibrational contribution to the equilibrium constant approximates to: K vib = e "#! h$! % /2kT 8.37! which has an exponential temperature dependence. The full expression for the equilibrium constant calculated from partition functions for diatomic molecules is then: " - K = q tr i q rot vib ( i q i ) " % i 3/2 I # = M i ( i ' * e+"h, i /2kT 0 i #/ i & $ i ) 1+ e +"h, i. /2kT i 1 By use of the Teller-Redlich spectroscopic theorem*, this equation simplifies to: * $ #U /2 1 3/2 e ' K = + & m i ) 8.39 %" i 1# e #U ( where m is the mass of the isotope exchanged and U is defined as: U = h" kt = hc# kt and ω is the vibrational wave number and c the speed of light. Example of fractionation factor calculated from partition functions i 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 18 O and 16 O between O 2 and CO: C 16 O + 18 O 16 O = C 18 O + 16 O The choice of diatomic molecules greatly simplifies the equations. Choosing even a slightly more complex model, such as CO 2 would complicate the calculation because there are more vibrational modes possible. Chacko et al. (2001) provide an example of the calculation for more complex molecules such as CO 2. The equilibrium constant for our reaction is: 8.40 * 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: " $ # m 2 m 1 % ' & 3/2 I1 " M 1 % $ ' I 2 # & M 2 3/2 = U 1 U 2 where m is the isotope mass and M is the molecular mass. We need not concern ourselves with its details /22/12

9 K = [16 O 2 ][C 18 O] [ 18 O 16 O][C 16 O] 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 independent of its isotopic compositions. The fractionation factor, α, is defined as: 8.42 " = (18 O / 16 O) CO ( 18 O / 16 O) O O 18 O + 16 O 16 O = 2 16 O 18 O We must also consider the exchange reaction: for which we can write a second equilibrium constant, K 2. 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: " $ # % ' & 18 O = 16 O O 2 [ 18 O 16 O] [ 18 O 16 O]+ 2[ 16 O 2 ] ( 16 O 2 has 2 16 O atoms, so it must be counted twice) whereas the ratio in CO is simply: " $ # % ' & 18 O 16 O CO = [C18 O] [C 16 O] Letting the isotope ratio equal R, we can solve 8.44 for [ 18 O 16 O]: and substitute it into 8.42: K = (1" R O 2 )[C 18 O] 2R O2 [C 16 O] [ 18 O / 16 O] = 2 [16 O 2 ]R O2 1" R O Since the isotope ratio is a small number, the term (1 R) 1, so that: = 2 (1" R O 2 )R CO 2R O K " 2 R CO 2R O2 = # 2 We can calculate K from the partition functions as: 8.48 K = q 16 O 2 q C 18 O q 18 O 16 O q C 16 O where each partition function is the product of the translational, rotational, and vibrational partition functions. However, we will proceed by calculating an 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 = q 16 O 2 q C 18 O q 18 O 16 O q C 16 O " = M M 16 O 2 C % 18 O $ M M ' # 18 O 16 O C 16 O & 3/2 " 32 ( 30 % = $ ' # 34 ( 28 & 3/ = We find that CO would be 12.6 richer in 18 O 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 16 O /22/12

10 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 = q 16 O 2 q C 18 O q 18 O 16 O q C 16 O! 16 ' ' 18 $! I 16 I O = 2 C $ 18 O # " 2I I & 18 O 16 O C 16 O % = 1 ' # & 2 # 18 ' ' 16 & ' " # % & = (ignore the 1/2, it will cancel out later). If rotation were the only mode of motion, CO would be 8 poorer in 18 O. The vibrational equilibrium constant may be expressed as: K vib = q q 16 O 2 C 18 O q q = e 18 O 16 O C 16 O "h(# 16 O2 +# C 18 O "# C 16 O "# 18 O 16 O ) KT 8.52 Since we expect the difference in vibrational frequencies to be quite small, we may make the approximation e x = x + 1. Hence: K vib "1+ h 2KT [{ # $# C 16 O C 18 O}$ {# 16 $# O 18 2 O 16 O} ] 8.53 Let's make the simplification that the vibration frequencies are related to reduced mass as in a simple Hooke's Law harmonic oscillator: " = 1 2# k µ where k is the forcing constant, and depends on the nature of the bond, and will be the same for all isotopes of an element. In this case, we may write:! C 18 O =! C 16 O µ C 16 O µ C 18 O =! C 16 O = 0.976! C 16 O 8.55 A similar expression may be written relating the vibrational frequencies of the oxygen molecule:! 16 O 18 O = ! 16 O2 Substituting these expressions in the equilibrium constant expression, we have: ( ) K vib =1+ h 2kT " [1# 0.976]#" [1# ] C 16 O 16 O 2 The measured vibrational frequencies of CO and O 2 are sec -1 and sec -1. Substituting these values and values for the Planck and Boltzmann constants, we obtain: K vib = T At 300 K (room temperature), this evaluates to We may now write the total equilibrium constant expression as: " M 16 M O K = K tr K rot K vib! 2 C % 18 O $ M M ' # 18 O 16 O C 16 O & 1 3 h 21+ 4)kT 43 + " -$, -# k µ C 16 O 3/2 * k % µ ' C 18 O & * " $ # " µ 16 µ O 2 C % 18 O $ 2µ µ ' # 18 O 16 O C 16 O & ( k * k %. 5 3 ' 06 µ 16 O 2 µ 18 O 16 O & / /22/12

11 Evaluating this at 300 K we have: K = ! ! = Figure Fractionation factor, α= ( 18 O/ 16 O) CO / ( 18 O/ 16 O) O2, calculated from partition functions as a function of temperature. Figure Calculated temperature dependencies of the fractionation of oxygen between water and quartz. After Kawabe (1978). Since α = 2K, the fractionation factor is 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 temperature). This temperature dependence is illustrated in Figure Thus CO would be 23 permil richer in the heavy isotope, 18 O, than O 2. 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 fractionations between solids are entirely controlled by the vibrational partition function. In principle, fractionations between coexisting solids could be calculated as we have done above. The task is considerably complicated by the variety of vibrational modes available to a lattice. 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 approximates 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 'internal' or 'optical' vibrations between individual radicals or atomic groupings within the lattice such as CO 3, and Si O. The vibrational frequencies of these groups can be calculated from the Einstein function and can be measured by optical spectroscopy. In addition, there are vibrations of the lattice as a whole, called 'acoustical' vibrations, which can also be measured, but may be calculated from the Debye function. From either calculated or observed vibrational frequencies, partition function ratios may be calculated, which in turn are directly related to the fractionation factor. Generally, the optical modes are the primary contribution to the partition 213 8/22/12

12 function ratios. For example, for partitioning of 18 O 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 8.03 shows the calculated fractionation factor between quartz and water as a function of temperature. Isotopologues and Isotopic Clumping In the example we just considered, we were concerned only with how 18 O was distributed between CO and O 2. However, the CO and O 2 will consist of a variety of molecules of distinct isotopic composition, or isotopologues. Indeed, there will be 12 such isotopologues: specifically 12 C 16 O, 12 C 17 O, 12 C 18 O, 13 C 16 O, 13 C 17 O, 13 C 18 O, 16 O 2, 16 O 17 O, 16 O 18 O, 17 O 2, 17 O 18 O, and 18 O 2. The statistical mechanical theory we just considered predicts that the distribution of isotopes with a species will not be random but rather that some of these isotopologues will be thermodynamically favored. As with the distribution of isotopes between chemical species, the distribution of isotopes with a species will be temperature dependent. Thus, the relative abundance of isotopologues within a species, called clumping, can be used as a geothermometer, and one that is independent of the isotopic composition of other phases. This, as we shall see, is an important advantage. Let s begin by considering the distribution of isotopes between the isotopologues of CO. There are 6 isotopologues and they can be related through the following two reactions: 12 C 16 O + 13 C 17 O 13 C 16 O + 12 C 17 O C 16 O + 13 C 18 O 13 C 16 O + 12 C 18 O 8.58 (Since we can relate the 6 isotopologues through 2 reactions, we need only chose 4 of these isotopologues as the components of our system.) The equilibrium constant for reaction 8.57 can be calculated from: K = q 13 C 16 O q 12 C 17 O q 12 C 16 O q 13 C 17 O A similar equation can be written for the equilibrium constant for The individual partition functions can be calculated just as described in the previous section. Doing so, we find that the two heaviest species, 13 C 17 O and 13 C 18 O, 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: 8.59 Δ i = R i e 1 R i r where R i-e is that ratio of the observed or calculated equilibrium abundance of isotopologue i to the isotopologue containing no rare isotopes and R i-r is that same ratio if isotopes were distributed among isotopologues randomly. Thus, for example, in the system above, [ 13 C 18 O] Δ = [ 12 C 16 O] e 13 C 18 O [ 13 C 18 O] [ 12 C 16 O] r Since R i-r is the random distribution, it can be calculated directly as the probability of choosing isotopes randomly to form species. In the case of 13 C 18 O, it is: 214 8/22/12

13 R = [13 C 18 O] 13 C 18 O r [ 12 C 16 O] r = [13 C][ 18 O] [ 12 C][ 16 O] 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, N 2 O. The structure of this molecule is N-N-O and 14 N 15 N 16 O will have different properties than 15 N 14 N 16 O, so in that case, order does matter. The CO 2 molecule is, however, symmetric and we cannot distinguish 12 C 16 O 18 O from 12 C 18 O 16 O. Its random abundance would be calculated as: R = [13 C 16 O 18 O] 13 C 16 O 18 O r [ 12 C 16 O 2 ] r = 2[13 C][ 16 O][ 18 O] = 2[13 C][ 18 O] [ 12 C][ 16 O] 2 [ 12 C][ 16 O] The factor of 2 is in the denominator to take account of both 12 C 16 O 18 O and 12 C 18 O 16 O. Kinetic Fractionation Kinetic effects are normally associated with fast, incomplete, or unidirectional processes like evaporation, diffusion and dissociation reactions. As an example, recall that temperature is related to the average kinetic energy. In an ideal gas, the average kinetic energy of all molecules is the same. The kinetic energy is given by: E = 1 2 mv Consider two molecules of carbon dioxide, 12 C 16 O 2 and 13 C 16 O 2, in such a gas. If their energies are equal, the ratio of their velocities is (45/44) 1/2, or Thus 12 C 16 O 2 can diffuse 1.1% further in a given amount of time at a given temperature than 13 C 16 O 2. 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 coefficients of the two CO 2 species is the ratio of the square roots of the reduced masses of CO 2 and air (mean molecular weight 28.8): D 12 CO = µ 13 CO 2 = = D 13 CO 2 µ CO 2 Hence we would predict that gaseous diffusion will lead to only a 4.4 fractionation. In addition, molecules containing the heavy isotope are more stable and have higher dissociation energies than those containing the light isotope. This can be readily seen in Figure The energy required to raise the D 2 molecule to the energy where the atoms dissociate is kj/mole, whereas the energy required to dissociate the H 2 molecule is kj/mole. Therefore it is easier to break bonds such as C-H than 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 discussed in the previous section. Where reactions do not achieve equilibrium the lighter isotope will be preferentially 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 12 C is enriched in the products of photosynthesis in plants (hydrocarbons) relative to atmospheric CO 2, and 32 S is enriched in H 2 S 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 "E b /kt /22/12

14 where A is a constant called the frequency factor and E b is the barrier energy. Referring to Figure 8.01, 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 H 2 molecule is: or R D = e"(#"1 2h$ D )/kt R H e "(#"1 2h$ H R D R H = e (" H #" D )h /2kT )/kt 8.60 Substituting for the various constants, and using the wavenumbers given in the caption to Figure 8.01 (remembering that ω = cν where c is the speed of light) the ratio is calculated as 0.24; in other words we expect the H 2 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 16 O 2 and 18 O 16 O shows that the 16 O will react about 15% faster than the 18 O 16 O molecule. The greater translational velocities of lighter molecules also allows them to break through a liquid surface 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 16 O is involved rather than 18 O, and thus is broken more easily, meaning H 16 2 O is more readily available to transform into the gas phase than H 18 2 O. Thus water vapor above the ocean typically has δ 18 O 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. An interesting example of a kinetic effect is the fractionation of O isotopes between water and water vapor. This is another example of Rayleigh distillation (or condensation), as is fractional crystallization. Let A be the amount of the species containing the major isotope, H 16 2 O, and B be the amount of the species containing the minor isotope, H 18 2 O. The rate at which these species evaporate is proportional to the amount present: da=k A A 8.62a and db=k B B 8.62b Since the isotopic composition affects the reaction, or evaporation, rate, k A k B. We'll call this ratio of the rate constants α. Then db da = " B 8.63 A Rearranging and integrating, we have 8.61 ln B B = " ln A A or B B = " A % $ ' # A & ( 8.64 where A and B are the amount of A and B originally present. Dividing both sides by A/A B / A B / A = " A % $ ' # A & ()1 865 Since the amount of B makes up only a Figure Fractionation of isotope ratios during Rayleigh and equilibrium condensation. δ is the per mil difference between the isotopic composition of original vapor and the isotopic composition as a function of ƒ, the fraction of vapor remaining /22/12

15 trace of the total amount of H 2 O present, A is essentially equal to the total water present, and A/A is essentially identical to ƒ, the fraction of the original water remaining. Hence: B / A B / A = ƒ"#1 Subtracting 1 from both sides, we have B / A " B / A B / A 8.66 = ƒ #"1 " Comparing the left side of the equation to 26.1, we see the permil fractionation is given by:! = 1000( f " #1 # 1) 8.68 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 Even if the vapor and liquid remain in equilibrium throughout the condensation process, the isotopic composition of the remaining vapor will change continuously. The relevant equation is: % " = ' 1# & 1 (1# ƒ)/$ + f The effect of equilibrium condensation is also shown in Figure 8.4. FRACTIONATION OF SEVERAL ISOTOPES ( * ) In the example in the previous section we considered only the fractionation between 18 O and 16 O, and indeed almost all research on oxygen isotope fraction focuses on just these two isotopes. However, a third isotope of oxygen, 17 O, exists, although it is an order of magnitude less abundant than 18 O (which is two orders of magnitude less abundant than 16 O). 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 17 O and 16 O is half the difference between 18 O and 16 O, hence we expect the fractionation between 17 O and 16 O to be half that between 18 O and 16 O. In the example of fractionation between CO and O 2 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 17 O/ 18 O should be In the limit of infinite temperature, 17 O/ 18 O The empirically observed ratio for terrestrial fractionation (and also within classes of meteorites) is 17 O/ 18 O Because the fractionation between 17 O and 16 O bears a simple relationship to that between 18 O and 16 O, the 17 O/ 16 O 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 of Oxygen and Sulfur Nearly all observed isotopic fractionation is mass dependent fractionation. There are, however, some exceptions where the ratio of fractionation of 17 O/ 16 O to that of 18 O/ 16 O is close to 1. Since the extent of fractionation in these cases seems independent of the mass difference, this is called mass independent fractionation. Mass independent fractionation is rare. It was first observed oxygen isotope ratios in meteorites (Chapter 5) and has subsequently been observed in oxygen isotope ratios of atmospheric gases, most dramatically in stratospheric ozone (Figure 8.05), and most recently in sulfur isotope ratios of Archean sediments and modern sulfur-bearing aerosols in ice. The causes of mass independent fractionation are poorly understood and it seems likely there may be more than one cause. Only a brief 217 8/22/12

16 Figure 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. After Johnson et al. (2001). discussion is given here, a fuller discussion of the causes of mass independent fractionation can be found in Thiemens (2006). 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 + O 2 O 3 * The ozone molecule thus formed is in a vibrationally excited state (designated by the asterisk) and, consequently, 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 stratosphere, collisions are comparatively infrequent hence repartitioning of vibrational energy represents an important pathway to stability. Because there are more possible energy transitions for asymmetric species such as 16 O 16 O 18 O and 16 O 16 O 17 O than symmetric ones such as 16 O 16 O 16 O, 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 vibrational 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 mechanical 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 a subsequent chapter) have been produced in the laboratory by photo-dissociation (photolysis) of SO 2 and SO using deep ultraviolet radiation (wavelengths <220 nm). Photolysis at longer wavelengths does not produce mass independent fractionations. Current explanations therefore focus on ultraviolet photolysis. However, 218 8/22/12

17 there as yet is no theoretical explanation of this effect and alternative explanations, including ones that involve the role in symmetry in a manner analogous to ozone, cannot be entirely ruled out. 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: p = p o H H 16 2 O H 16 2 O[ 16 2 O] 8.70a and p = p o H H 18 2 O H 18 2 O[ 18 2 O] 8.70b 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 = p H 2 18 O / p H 2 16 O [H 2 18 O]/[H 2 16 O] By solving 8.70a and 8.70b for [H 16 2 O] and [H 18 2 O] and substituting into 8.71 we arrive at the relationship: 8.71 " l /v = p o H 18 2 O 8.72 o p H 16 2 O 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: 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 independent of temperature. Rearranging and integrating, we obtain: We can write two such equations, one for [H 16 2 O] and one for [H 18 2 O]. Dividing one by the other we obtain: o d ln P dt = "H RT ln p =! "H RT + const 8.74 ln p H 18 2 O = A " B o RT 8.75 Figure Temperature dependence of fractionation factors between vapor and water (solid lines) and vapor and ice (dashed lines) p H 2 16 O for various species of water /22/12

18 where A and B are constants. This can be rewritten as: α = ae -B/RT 8.76 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 temperature dependence of the fractionation factor can be represented as: ln" = A # B T Figure 8.06 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 Figure Calculated dependence of δ 18 O on temperature based on equ We assume the water vapor starts out 10 per mil depleted in δ 18 O. shows an approximately inverse dependence on temperature. Hydrogen isotope fractionation is clearly non-linear over a large range of temperature. Given the fractionation between water 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 paleotemperatures and past ice volumes. Let's now consider the question of isotopic 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 atmosphere. 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 represents the isotherm where condensation 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 explore what happens when water vapor condenses, lets construct a hypothetical model of condensation and represent ƒ as hypothetical function of temperature such as: ƒ = T " 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 condensation 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 equation Realistic values for the constant a and B are and J/mole respectively, so that 8.76 becomes: 8.78 α = e /RT 8.79 Substituting 8.78, 8.79 and R=8.314J/mol-K into equation 8.69, our model is: *, $ " 18 O v =1000 & T # 223 ' + ) -, % 50 ( e /T #1., #1/ 0, /22/12

19 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 8.07 shows the temperature dependence we predict for water vapor in the atmosphere as a function of temperature (we have assumed that the vapor begins with δ 18 O 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 8.08 shows the global variation in δ 18 O 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 δ 18 O 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. Along with these factors, distance from the ocean also appears to be an important variable in the isotopic 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. Topography 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 Hydrogen as well as oxygen isotopes will be fractionated in the hy- Figure Variation of δ 18 O in precipitation as a function of mean annual temperature. Figure Cartoon illustrating the process of Rayleigh fractionation and the increasing fractionation of oxygen isotopes in rain as it moves inland /22/12

20 Figure Northern hemisphere variation in δd and δ 18 O in precipitation and meteoric waters. The relationship between δd and δ 18 O is approximately δd = 8δ 18 O After Dansgaard (1964). drologic cycle. Indeed, δ 18 O and δd are reasonably well correlated in precipitation, as is shown in Figure The fractionation of hydrogen isotopes, however, is greater because the mass difference is greater. Figure 8.10 shows the variation in oxygen isotopic composition of meteoric surface waters in the North America. The distribution is clearly not purely a function of mean annual temperature, and this illustrates the role of the factors discussed above. We will return to the topic of the hydrologic system in a future lecture when we discuss paleoclimatology. ISOTOPE FRACTIONATION IN THE BIOSPHERE As we noted, biological processes often involve large isotopic fractionations. Indeed, biological processes are the most important cause of variations in the isotope composition of carbon, nitrogen, and sulfur. For the most part, the largest fractionations occur during the initial production of organic matter by the so-called primary producers, or autotrophs. These include all plants and many kinds of bacteria. The Figure Variation of average δ 18 O in precipitation most important means of production of organic matter is photosynthesis, but organic over North America. δ 18 O depends on orographic effects, mean annual temperature, and distance from the matter may also be produced by chemosynthesis, for example at mid-ocean ridge hy- sources of water vapor. drothermal vents. Large fractions of both carbon and nitrogen occur during primary production. Additional fractionations also occur in subsequent reactions and up through the food chain as hetrotrophs consume primary producers, but these are generally smaller /22/12

21 Carbon Isotope Fractionation During Photosynthesis The most important of process producing isotopic fractionation of carbon is photosynthesis. As we earlier noted, photosynthetic 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 fractionations involved in these steps, which we will consider in more detail here. For terrestrial plants (those utilizing atmospheric CO 2 ), the first step is diffusion of CO 2 into the boundary layer surrounding the leaf, through the stomata, and internally in the leaf. The average δ 13 C of various species of plants has been correlated with the stomatal conductance (Delucia et al., 1988), indicating that diffusion into the plant is indeed important in fractionating carbon isotopes. On theoretical grounds, a 4.4 difference in the diffusion coefficients is predicted ( 12 CO 2 will diffuse more rapidly; see Lecture 27) so a fractionation of 4.4 is expected. Marine algae and aquatic plants can utilize either dissolved CO 2 or HCO 3 for photosynthesis: CO 2(g) CO 2(aq) + H 2 O H 2 CO 3 Η + +HCO 3 An equilibrium fractionation of +0.9 per mil is associated with dissolution ( 13 CO 2 will dissolve more readily), and an equilibrium +7 to +12 fractionation (depending on temperature) occurs during hydration and dissociation of CO 2. Thus, we expect dissolved HCO 3 to be about 8 to 12 per mil heavier than atmospheric CO 2. At this point, there is a divergence in the chemical pathways. Most plants use an enzyme called ribulose bisphosphate carboxylase oxygenase (RUBISCO) to catalyze a reaction in which ribulose bisphosphate reacts with one molecule of CO 2 to produce 2 molecules of 3-phosphoglyceric acid, a compound containing 3 carbon atoms, in a process called carboxylation (Figure 8.12). 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 reduced, carbohydrate formed, and the ribulose bisphosphate regenerated. Such plants are called C 3 plants, and this process is called the Benson-Calvin, or Calvin, cycle. C 3 plants constitute about 90% of all plants and include algae and autotrophic bacteria and comprise the majority of cultivated plants, including wheat, rice, and nuts. There is a kinetic fractionation associated with carboxylation of ribulose bisphosphate that has been determined by several methods to be in higher terrestrial plants. Bacterial carbolaxylation has different reaction mechanisms and a smaller fractionation of about 20. Thus for terrestrial plants a fractionation of about 34 is expected from the sum of the fraction. The actual observed total fractionation is in the range of 20 to 30. The disparity between the observed total fractionation and that expected from the sum of the steps presented something of a conundrum. The solution appears to be a model that assumes the amount of carbon isotope fractionation expressed in the tissues of plants depends on ratio the concentration of CO 2 inside plants to that in the external environment. The model may be described by the equation: " = a + (c i /c a )(b # a) 8.81 where a is the isotopic fractionation due to diffusion into the plant, c i is the interior CO 2 concentration, c a is the ambient or exterior CO 2 concentration, and b is the fractionation occurring during carboxylation. According to this model, where an unlimited amount of CO 2 is available (i.e., Figure Ribulose bisphosphate (RuBP) carboxylation, the reaction by which C 3 plants fix carbon during photosynthesis /22/12

22 when c i /c a 1), carboxylation alone causes fractionation. At the other extreme, if the concentration of CO 2 in the cell is limiting (i.e., when c i /c a 0), essentially all carbon in the cell will be fixed and therefore there will be little fractionation during this step and the total fractionation is essentially 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 The other photosynthetic pathway is the Figure Phosphoenolpyruvate carboxylation, the reaction by which C 4 plants fix CO 2 during photosynthesis. Figure Chemical pathways in C 4 photosynthesis. Hatch-Slack cycle, used by the C 4 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. 8.13). A much smaller fractionation, about -2.0 to -2.5 occurs during this step. In phosphophoenol pyruvate carboxylation, the CO 2 is fixed in outer mesophyll cells as oxaloacetate and carried as part of a C 4 acid, either malate or asparatate, to inner bundle sheath cells where it is decarboxylated and refixed by RuBP (Fig. 8.14). The environment in the bundle sheath cells is almost a closed system, so that virtually all the carbon carried there is refixed by RuBP, so there is little fractionation during this step. C 4 plants have average δ 13 C of As in the case of RuBP photosynthesis, the fractionation appears to depend on the ambient concentration of CO 2. This dependence can be modeled as: " = a + (b 4 + b 3 # $ a)(c i /c a ) 8.82 where a is the fractionation due to diffusion of CO 2 into the plant as above, b 4 is the fractionation during transport into bundle-sheath cells, b 3 is the fractionation during carboxylation (~ 3 ), φ is the fraction CO 2 leaked from the plant. A third group of plants, the CAM plants, have a unique metabolism called the Crassulacean acid metabolism. These plates generally use the C 4 pathway, but can use the C 3 pathway under certain conditions. These plants are generally adapted to arid environments and include pineapple and many cacti, they have δ 13 C intermediate between C 3 and C 4 plants. Terrestrial plants, which utilize CO 2 from the atmosphere, generally produce greater fractionations than marine and aquatic plants, which utilize dissolved CO 2 and HCO 3, together referred to as dissolved inorganic carbon or DIC. As we noted above, there is about a +8 equilibrium fractionation between dissolved CO 2 and HCO 3. Since HCO 3 is about 2 orders of magnitude more abundant in seawater than dissolved CO 2, many marine algae utilize this species, and hence tend to show a lower net fractionation during photosynthesis. Diffusion is slower in water than in air, so diffusion is often the rate-limiting step. Most aquatic plants have some membrane-bound mechanism to pump DIC, which can be turned on when DIC is low. When DIC concentrations are high, fractionation 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 fractionation is: 224 8/22/12