1. Stable Isotope Geochemistry Theory
Lecture 34

2. 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 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 65 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

3. 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.
t 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, Se, Mo, Tl, and U, also isotopic variations are much smaller in these elements.

4. 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 = δ18OSMOW

5. Common 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
× 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

6. 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
The two are related as
∆ ≈ (α - 1) × 103 or ∆ ≈ 103 ln α

7. 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.

8. 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

9. Predicting Fractionations
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.
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.

10. Partition Functions
The following applies to diatomic molecules. Principles are the same for multiatomic molecules and crystals, but the equations are more complex because there are many possible vibrations and rotationns.
Vibrational Partition Function:
where h is planks 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; reduced mass 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 (derived from Schrödinger’s equation for particle in a box):

11. 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., H218O and H216O, 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.

12. 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

13. 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 equilibrium constant from the ratio of rate constants:
This will make no difference is 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).

14. 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, 12C16O2 and 13C16O2, in a gas. If their energies are equal, the ratio of their velocities is (45/44)1/2, or 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).