Introduction to Stable Isotope Geochemistry Lecture 32

Beginnings Harold Urey Stable isotope geochemistry is concerned with variations in the isotopic compositions of elements arising small differences in the chemical behavior of different isotopes of an element. These can provide a very large amount of useful information about chemical (both geochemical and biochemical) processes. Harold Urey and his associates Jacob Bigeleisen and Maria Mayer essentially initiated this field, predicting that different isotopes would have slightly different behavior (the first success of which was the discovery of deuterium, for which he won the Nobel Prize). The story really begins in 1947 when he and colleagues predicted the oxygen isotopic composition of carbonates would vary with temperature. What has been learned in the 70 years since those papers were published would undoubtedly astonish even Urey. Urey was also instrumental in founding the modern fields of geochronology and cosmochemistry. Harold Urey

Stable Isotopes Traditionally, the principal elements of interest in stable isotope geochemistry were H, C, N, O, and S. Over the last two decades, Li and B have also become “staples” of isotope geochemistry. These elements have several common characteristics: They have low atomic mass. The relative mass difference between their 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 have become available (particularly MC-ICP-MS), geochemists have begun to explore isotopic variations of many more elements, including Mg, Si, Cl, Ca, Ti, Cr, Fe, Zn, Cu, Ge, Sr, Se, Mo, Tl, and U, among others. Isotopic variations are much smaller in these elements.

Delta Notation Stable isotope ratios variations are generally small, so they are reported as per mil (‰) deviations from a standard. For example, oxygen isotope ratios (except in carbonates) are reported relative to Standard Mean Ocean Water (SMOW) as: Carbonates are reported relative to the Pee Dee Belemite (PDB). They are related as: δ18OPDB = 1.03086δ18OSMOW + 30.86

‘Traditional’ Stable Isotopes Element Notation Ratio Standard Absolute ratio Hydrogen δD D/H (2H/1H) SMOW 1.557 ×10-4 Lithium δ7Li 7Li/6Li NIST 8545 (L-SVEC) 12.285 Boron δ11B 11B/10B NIST 951 4.044 Carbon δ13C 13C/12C PDB 1.122 × 10-2 Nitrogen δ15N 15N/14N Atmosphere 3.613 × 10-3 Oxygen δ18O 18O/16O SMOW, PDB 2.0052 × 10-3   δ17O 17O/16O 3.76 × 10-4 Sulfur δ34S 34S/32S CDT 4.416 × 10-2 δ33S 33S/32S 7.877 × 10-3

Fractionation Factor The fractionation factor, α, is the ratio of isotope ratios in two phases predicted or found to occur: Fractionation factors can also be expressed as the difference in delta values: ∆A-B = δA - δB Because α is usually not very different from 1, the two are related as ∆ ≈ (α - 1) × 103 or ∆ ≈ 103 ln α

Theory of Stable Isotope Fractionations By fractionation, we mean an enrichment or depletion of one isotope over another, that is a change in the isotope ratio, as a consequence of some process.

Fractionations Isotope fractionation can originate from both kinetic effects and equilibrium effects. Equilibrium Fractionations: Quantum mechanics predicts that the mass of an atom affects its vibrational motion, and therefore the strength of chemical bonds. It also affects rotational and translational motions. From an understanding of these effects of atomic mass, it is possible to predict the small differences in the chemical properties of isotopes quite accurately. Kinetic Fractionations Lighter isotopes form weaker bonds and therefore react faster. They also diffuse more rapidly. These also lead to isotopic differences.

Equilibrium Fractionations Equilibrium fractionations arise from translational, rotational and vibrational motions of molecules in gases and liquids and atoms in crystals. Isotopes will be distributed so as to minimize the vibrational, rotational, and translational energy of a system. All these motions are quantized (but quantum steps in translation are very small). Of the three types of energies, vibrational energy makes by far the most important contribution to isotopic fractionation. Vibrational motion is the only mode of motion available to atoms in a solid. These effects are small. For example, the equilibrium constant for the reaction: ½C16O2 + H218O ⇄ ½C18O2 + H216O is only about 1.04 at 25°C and the ∆G of the reaction, given by -RT ln K, is only -100 J/mol.

Predicting Fractionations The Boltzmann Distribution Law states that the probability of a molecule having internal energy Ei is: g is a weighting factor to account for degenerate states (different states with same energy). The denominator is the partition function: From partition functions we can calculate the equilibrium constant: We can divide the partition function into three parts: Qtotal = QtransQrotQvib The separate partition functions can be calculated separately.

Note error in book: remove 2 in exponential term in denominator Partition Functions The following applies to diatomic molecules, for which the sums in the partition function fortunately have simple solutions. Principles are the same for multiatomic molecules and crystals, but the equations are more complex because there are many possible vibrations and rotations. Vibrational Partition Function: where h is Planck’s constant (converting frequency to energy) and ν is the vibrational frequency of the bond. Rotational Partition Function where I is the moment of inertia I = µr2; i.e., the reduced mass (µ) of atoms times bond length. σ is a symmetry factor; σ= 1 for a non-symmetric molecule (18O16O) and 2 for a symmetric one (16O16O) Translational partition function for molecule of mass m is (derived from Schrödinger’s equation for particle in a box): Note error in book: remove 2 in exponential term in denominator

Partition Function Ratios Since what we really want is the ratios of partition functions for isotope exchange reactions. Most terms cancel (including bond length). This ratio for two isotopic species (isotopologues) of the same diatomic molecules, e.g., 16O18O and 16O16O, will be: We see the partition function ratio is temperature dependent (which arises only from the vibrational contribution: temperature canceled in other modes). We also see that we can predict fractionations from measured vibrational frequencies and atomic and molecular masses.

Temperature Dependence The temperature dependence is: At low-T (~surface T and below), the exponential term is small and the denominator approximates to 1. Hence and the fractionation factor can be expressed as: At higher temperature, however, this approximation no longer holds and α varies (~) with the inverse square of T: The temperature dependence leads to important applications in geothermometry & paleoclimatology. O isotope fractionation between CO2 and H2O

Kinetic Fractionations: Reaction Rates Looking again at the hydrogen molecular bond, we see it takes less energy to break if it is H-H rather than D-H. This effectively means the activation energy is lower and the rate constant, k, will be higher: DH will react faster than H2. We can calculate a kinetic fractionation factor from the ratio of rate constants: This will make no difference if the reaction goes to completion, but will make a difference for incomplete reactions (good example is photosynthesis, which does not convert all CO2 to organic carbon).

Kinetic Fractionation: Diffusion Lighter isotopic species will diffuse more rapidly. Energy is equally partitioned in a gas (or liquid). The translational kinetic energy is simply E = ½mv2. Consider two molecules of carbon dioxide, 13C16O2 and 12C16O2, in 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. But this applies to ideal gases (i.e., low pressures where collisions between molecules are infrequent. For the case of air, where molecular collisions are important, the ratio of the diffusion coefficients 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): leading to a 4.4‰ fractionation (actually observed).

Rayleigh Distillation/Condensation Different isotopologues of water evaporate at different rates and have different condensation temperatures. We can imagine two ways in which condensation occurs: droplets remain in isotopic equilibrium with vapor droplets do not remain in equilibrium: fractional condensation If the fractionation between vapor and liquid is α, for fractional condensation, the fractionation, ∆, varies with fraction of vapor remaining, ƒ, as: ∆ = 1000(ƒα-1-1) For equilibrium condensation it is: Fractional condensation can lead to quite extreme compositions of remaining vapor. Isotopic composition of vapor when the fraction of original vapor, ƒ, remains.

Isotope Fractionation Summary As a rule, heavy isotopes partition preferentially into phases in which they are most strongly bound (because this results in the greatest reduction in system energy). Covalent bonds, and bonds to heavier atoms, are generally strongest and hence will most often incorporate the heavy isotope. Largest fractionations will occur where the atomic environment or bond energy differences are greatest So, for example, fractionation of O between silicates are not large, because the O is mainly bound to Si. Fractionations tend to be large between different oxidation states of an element (e.g., for C, N, S). Lighter isotopes are likely to be enriched in the products of incomplete reactions and also reactions where diffusion is important.

Mass Dependent Fractionation if a 4‰ fractionation of δ18O is observed in a particular sample, what value of δ17O do we predict? We might guess it would ½ as much. Mass occurs in a variety of ways in the partition function, as m3/2, as reduced mass, and in the exponential term. Consequently, the ratio of fractionation of 17O/16O to that of 18O/16O in most cases is about 0.52. Nevertheless, the fractionation between isotopes predicted by this equation is proportional to the difference in mass – this is referred to as mass-dependent fractionation. There are some exceptions where the ratio of fractionation of 17O/16O to that of 18O/16O is ≈1. Since the extent of fractionation in these cases seems independent of the mass difference, this is called mass-independent fractionation. Examples Oxygen in meteorites Sulfur in Archean sulfides Oxygen in stratospheric gases

Mass Independent Fractionation Most examples of ‘MIF’ seem to be related to photochemical reactions. Formation of ozone in the stratosphere involves the energetic collision of monatomic and molecular oxygen: O + O2 → O3 The ozone molecule is in a vibrationally excited state and subject to dissociation if it cannot lose 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 infrequent, hence repartitioning of vibrational energy represents an important pathway to stability. Because there are more possible energy transitions for asymmetric species such as 16O16O18O and 16O16O17O than symmetric ones such as 16O16O16O, the former can more readily repartition its excess energy and form a stable molecule. In the troposphere, collisions more frequent, reducing this effect.

Isotope Geothermometry One of the principal uses of stable isotopes is geothermometry. Stable isotope geothermometers are based on the temperature dependence of the fractionation factor or equilibrium constant, which can generally be expressed as: (at low temperatures, the form of changes to α ∝ 1/T). Temperature dependence can be theoretically calculated or experimentally measured. Measuring the isotopic composition of two phases allows us to calculate the temperature at which they equilibrated (assuming, of course, that they did equilibrate).

Isotope “Clumping” Consider the distribution of 18O between CO and O2 (Example 9.1). CO and O2 will consist actually of 12 isotopically distinct molecules or “isotopologues”, such as 12C16O, 12C17O, 13C18O, 16O17O, etc. The distribution of isotopes between these species will not be random but rather some of these isotopologues will be thermodynamically favored. Essentially, grouping the heavy isotopes in one molecule, e.g., 13C18O, reduces bond energy by a bit more than twice the reduction of putting one heavy isotope in the molecule. Thus “clumping” of heavy isotopes reduces system energy. This ‘clumping’ depends on temperature (greater at low T). By analyzing the various isotopologues of the species, one can calculate equilibrium temperatures. The advantage is that we need analyze just one phase involved in the reaction, for example, carbonate precipitated from water. In addition to calculating temperature, one can also calculate the isotopic composition of the water. This is a very new field, but holds great promise in isotope geothermometry.