Introduction to Isotopic Fractionation Reading: Fritz and Clark, Chapter 1, excluding parts on radionuclides Other resources for more information: Tom Johnson s Isotope Geochem Class Notes: http://classes.geology.illinois.edu/12fallclass/geo562/notes.html Mass-dependent isotope fractionation and Kinetic Isotope Effects Bill White s 2007 isotope geochem. lecture notes: http://www.geo.cornell.edu/geology/classes/geo656/656home.html Equilibrium Isotope Fractionation #27 Kinetic Isotope Fractionation #28 Faure and Mensing, Isotopes: Principles and Applications R. Criss, Principles of Stable Isotope Distribution Motivation: Different isotopes of the same element are nearly the same chemically- same e - config. However, slight chemical differences exist between isotopes, for a few different reasons. Thus, chemical reactions may fractionate the isotopes. Fractionate= to partition unevenly, i.e., light isotopes somewhat enriched in one part of the system, while heavier isotopes enriched somewhere else It turns out this effect can be useful in several ways: 1. Fractionation may be temperature dependent- provides potential paleothermometers 2. Different sources of an element or compound may have distinct isotope ratios, and we can thus use isotope abundances as indicators of source or mass balance 3. Different mechanisms for reactions may have differing fractionations. In these cases, occurrences of certain reaction mechanisms might be inferred from isotope ratios. Background you need to use stable isotopes: 1) Understanding of fractionation processes. 2) Knowledge of terminology, notation and mathematical tools 3) Introduction to the behavior of each element. 4) Knowledge of measurement capabilities How does mass dependent isotopic fractionation occur? In basic chemistry classes, one treats all the stable isotopes of an element identically. You are told to assume absolutely no difference in chemical properties between isotopes of an element. You re told the extra neutrons in the nucleus have no effect on chemical properties. This is approximately true: Chemical bonds are formed by electrons and, to a close approximation, those electrons interact with the positive charges (protons) of the nucleus but do not interact with neutrons at all. From a classical physics point of view, bond strengths are determined by electrons and protons only. Extra neutrons doesn t affect bond strengths or the thermodynamic energies (e.g., ΔG s) of molecules. But here s a simple example that contradicts that view: Consider Gaseous diffusion: H 2 vs. D 2 At any given temperature: average kinetic energies are equal for H 2 and D 2 Recall that Kinetic energy = 1/2 mv 2 Because D 2 is double the mass, it s velocity is less by a factor of 2 Thus, it diffuses more slowly.
If we let a mixture of H 2 and D 2 diffuse through a narrow tube, the H 2 molecules tend to come out the far end earlier and the and D 2 molecules later This partial separation of isotopes is call isotopic fractionation This example doesn t involve chemical bonding. Here s another example that does: Example #2: Evaporation H 2 16 O versus H 2 18 O Evaporation: H 2 O s break hydrogen bonds and break free of the liquid water H 2 16 O breaks hydrogen bonds with other water molecules more easily than H 2 18 O This can be observed, but it is a small effect Vapor entering the air is enriched in H 2 16 O relative to the water By about 1% (more details on this later) This is observed experimentally, but it does not make sense from a classical physics perspective. His raises the question: Why do lighter isotopes break their bonds more easily? Quantum effects cause bonds involving lighter isotopes to be more easily broken than those involving heavier isotopes. - In a classical sense, bond enthalpies/strengths are the same: They are determined by electronic configurations only, and all isotopes of an element have the same number of protons and thus the same electronic configuration. BUT, effectively, bond energies are slightly different because of quantum effects: A central principle in quantum mechanics: Chemical bonds have certain discreet energy levels, and can t exist with energies other than those levels. This is related to the wave nature of matter. It is somewhat similar to harmonics on a guitar string. Furthermore, even at absolute zero, there is some fundamental vibrational energy, called the zero point energy; the bond does not settle into the lowest energy point on the curve E This figure depicts the energy well of a chemical bond. In the classical model, the bond is like a spring and it can have any energy, depending on separation distance (i.e., how it is compressed or stretched). However, we know from observing real molecules that this is not true. They have discreet energy levels. Theory and experiment tell us the lowest available energy is not zero. This lowest energy is called the zero point energy or ZPE. In this diagram, the ZPE the distance between the lowest point on the classical energy function and the horizontal line representing the actual energy of a ground state molecule. Distance Zero point energies are greater for bonds involving light isotopes (dashed line) than for heavier isotopes (solid line). Note: I believe the Clark and Fritz version of this diagram is incorrect. They show two lines for the potential energy, one for heavier isotopes, one for lighter. But the classical bond energy model depends on electron interactions only and NOT on mass, so there should be only one line, with two different ZPE s. Zero- point energies of bonds involving lighter isotopes are greater! Lighter isotopes have slightly higher zero-point energies than heavier ones! This can be proven with the tools of quantum mechanics and statistical mechanics- standard fare for a P-chem course (see brief description below). This makes some sense from a classical point of view: 1) Given the same bond strength, bonds with lighter isotopes vibrate faster. Think of the bond as a spring, and think about how oscillation would be different for a lead weight versus a wood one (same spring). 2) Faster oscillation means higher vibrational energy
3) Greater vibrational energy means the bond for a lighter isotope has greater vibrational energy and the additional energy needed to break the bond is less 4) Thus, the bond with the lighter isotope is effectively weaker. This is echoed in quantum mechanics, which provides a much better model for bonds. Allowable energy levels are related to vibrational frequencies of photons that can be accepted by or emitted from a bond: ΔE = hν This is the energy jump or gap between allowed energy states. where h is Planck s constant and ν is the photon s frequency. So the spacing between energy level is given by hν The zero point energy is given by 1/2hν (long story). We observe that lighter isotopes give off higher frequency/shorter wavelength photons So once again, the result is the same: Bonds involving lighter isotopes have greater energy and therefore can overcome the bond energy more easily to beak the bond Isotopic fractionation occurring when bonds are broken or rearranged is known as a kinetic isotope effect, or KIE. Temperature effects: As temperature increases above absolute zero, the bond energy, averaged over all molecules present, increases. This occurs by the promotion of some molecules to higher energy levels. The greater the temperature, the greater the proportion of molecules in the higher energy levels. As a result of this, the average energy difference between bonds with lighter isotopes and those with heavier isotopes diminishes as T increases. Another Example of a KIE: Microbial reduction of SO 4 = - sulfate reduction at room T is mediated by bacteria- e.g., desulfovibrio - the reaction begins by the breakage of one S-O bond - reaction rate slightly greater for lighter isotopes - products enriched in lighter isotopes - reactants become increasingly enriched in heavier isotopes as the reaction proceeds Equilibrium Isotope Effects The bond breakage effects described above deal with reaction rates (a.k.a. kinetics). What happens if a reaction approaches equilibrium? Do zero point energy effects still cause isotopic fractionation? Yes. At equilibrium, the energies of all the various compounds involved depend on which isotopes are present. Example: If we have equilibrium between the oxygen isotopes in water and CO 2, we could write: C 16 O 2 + H 2 18 O = C 18 O 16 O + H 2 16 O - This chemical reaction is merely exchange of an 18 O from H 2 O to CO 2, balanced by a complementary movement of 16 O the other way - Any isotope exchange reaction can be written A 1 + B 2 = A 2 + B 1 where 1 = light isotope, 2 = heavy isotope The zero point energies of all four molecules in this reaction are different; each has its own vibrational frequency. The reaction products in this case happen to have slightly lower energy (ΔG) than the reactants, because of the ZPE differences. This means the product side is slightly favored in the equilibrium. In other words, as equilibrium is approached, 18 O will be slightly favored in CO 2, whereas 16 O will be slightly favored in the H 2 O. This is an equilibrium isotopic fractionation. General rule to predict which compound or bond has a preference for the heavier isotope:
The compound with the stronger (higher frequency; higher energy) bonds generally ends up enriched in the heavier isotopes relative to that with weaker bonds. Example: Equilibrium isotopic fractionation between water and CaCO 3 precipitating from solution - dissolved CO 3 2- in solution is in isotopic equilibrium with O atoms with water - dissolved CO 3 2- also in isotopic equilibrium with O atoms at surface of CaCO 3 crystal - therefore, the crystal is in equilibrium with the H 2 O - CaCO 3 crystal has stronger bonds than H 2 O prefers heavier oxygen isotopes- chemical difference - There is a predictable partitioning of isotopes in this case. Equilibrium isotopic fractionation is thermodynamically fixed; it is a function of temperature, but does not depend on how the reaction happens. If we measure equilibrium fractionation once for a given equilibrium at a given T, we know the fractionation will always be the same at that temperature. So you can go to a reference a look up a relevant equilibrium fractionation factor. Isotopic fractionation between water and calcite as a function of T. Isotopic fracitonation during biotic calcite precipitation is thought to be very close to equilibrium. Biological precipitation may involve a small deviation from equilibrium. Equilibrium fractionation decreases with T and generally reaches near-zero values at magmatic temperatures. There s no significant pressure dependence. Kinetic isotopic fractionation DOES depend on the reaction mechanisms, rates, etc. Most biogeochemical reactions have multiple reaction steps. Often there are different reaction mechanisms by which a reaction can occur. In these cases, the kinetic isotope effect may vary with reaction mechanism, even though the reactants and products may have well-defined energies. Thus, if we measure the magnitude of a kinetic isotope effect in one setting (e.g., the laboratory), we are likely to find it is different in another setting (e.g., the field). Partition Functions We won t go there. Ignore the section in Clark and Fritz, unless you are prepared to dive deeper into quantum mechanics. They are a useful mathematical construct that elegantly cuts to the core of energy and isotopic fractionation. Many isotope geochemistry texts present how they are used but don t explain why they work very well. Notation we use δ notation (why?): o because 18 O/ 16 O = 0.0020002, and typical differences are 0.1%, or even smaller: o thus. 18 O/ 16 O = 0.0020004 vs. 0.0020000- it s annoying to write out all those decimal places! o We could work with percent differences (for above example, difference is 0.02%; often we can measure ratios to this level of precision or better). o The community uses per mil differences, 10 times more sensitive than percent differences. This is delta notation:
δ sample = R sample R std R std 1,000 all samples compared to international standard People often use big delta for the difference between two samples: Δ sample1 sample2 δ sample1 δ sample2 When we develop theoretical models for isotope fractionation, we use alpha: α 1 2 R 1 this is the ratio of two isotope ratios we are comparing R 2 Very important: α gives the magnitude of isotopic fractionation for a process or equilibrium Because we normally work in delta notation, we need an easy conversion between alpha and delta. For a system where phases 1 and 2 are in isotopic equilibrium: Δ 1 2 1000lnα (a very close approximation) OR Δ 1 2 1000( α 1) (a very close approximation) For kinetic isotope effects, you will often see the magnitude of isotopic fractionation given in terms of ε: ε = (α -1) x 1000 ε δ product - δ reactant where product means the flux of reaction product (i.e., not the total pool of product accumulated; the product produced at one instant in time) ε gives a convenient representation of alpha in per mil units; this is convenient to use routinely because we use per mil units for all our analyses and everyday discussions Rayleigh distillation: Model that describes isotope ratios of reactant and reaction product as a reaction proceeds. Example: During evaporation, perhaps we can use the isotopes in the remaining water (compared to the starting value) to find the amount of evaporation. If we leave a pan of water out to evaporate, how does the delta value change as a function of the amount evaporated? Note this is like a distillation process: Lighter isotopes are preferentially evaporated, and over time, the remaining water becomes enriched in heavier isotopes. R = R 0 f α 1 this gives the isotope ratio of the remaining water as the process proceeds, where R = the isotope ratio (D/H or 18 O/ 16 O in the evaporation example); R 0 is initial value f = the fraction of reactant (water in this case) remaining α is the fractionation factor for the process or equilibrium The above equation is awkward because we use δ notation for everyday work. Converted to δ notation it is: δ = (δ 0+1000) f α-1 1000
you may see δ 1000(f α-1 1) in some places; this is an approximation that works well when δ is fairly small A very useful approximation is δ = δ 0+ ε [ln(f)] The reaction product created at any instant in time is of course tied to the reaction product by: R prod = αr reactant but this is closely approximated by δ prod δ reactant + ε If the reaction product accumulates in a single pool somewhere (often not the case), its isotope ratio is given by: R prod R 0 = 1 f α 1 f OR δ prod = ( δ 0 +1000) 1 f α 1000 1 f Delta 15.00 10.00 5.00 0.00-5.00-10.00 Rayleigh model for isotopic fractionation during rainfall: Solid line is the delta value of the remaining water vapor in the air. Dashed line is the rain falling at one point in time. Dotted line is the value of the cumulative rainfall, combined. -15.00-20.00-25.00 1.00 0.80 0.60 0.40 Fraction Remaining 0.20 0.00 Other types of isotopic fractionation Magnetic isotope effect, MIE: Nuclei of isotopes with odd mass number (e.g., 199 Hg and 201 Hg, but not 200 Hg) are magnetic; under certain circumstances they can alter reaction kinetics by altering the spins (magnetic polarization) of electrons. This happens only under special circumstances where radicals are involved. As of now, photochemical (sunlight-driven) reactions are the only ones that show MIE effects. Gas-phase symmetry effects: Gaseous molecules have more ways to gain or lose energy than liquids and solids. Gas molecules spin rapidly and move freely through air. Symmetry is important for the spinning energy, and chemical reactions are altered when symmetry is broken. For example, 16 O 3, the most abundant ozone isotopologue, is symmetric, but 17 O 16 O 2 and 18 O 16 O 2 are not. As a result, isotopic fractionation occurs, and it has a distinctive pattern. With mass-dependent fractionation, 17 O fractionates relative to 16 O about half as strongly as 18 O does. The symmetry effect is very different. Thus, when 17 O abundance departs from the mass-dependent fractionation trend, we know a gas-phase reaction was involved. Self-shielding effects:
Ultraviolet rays cause chemical reactions in the atmosphere. These reactions get weaker as one moves from the top of the atmosphere downward, because many of the UV rays are blocked. But the blockage of the various wavelengths is not equal. Wavelengths absorbed by common isotopologues (e.g., 32 S-bearing molecules) are blocked more than those absorbed by rare ones (e.g., 33 S-bearing molecules). So reaction rates are not equal and this causes isotopic fractionation. Once again, a distinctive pattern occurs, and we can detect atmospheric reactions by looking at tyree isotopes of an element instead of just two (e.g., measure BOTH 34 S/ 32 S and 33 S/ 32 S; not just 34 S/ 32 S as is commonly done).