Introduction to Stable Isotopes Notation Fractionation Gear

Definitions Isotopes Atoms of the same element (i.e., same number of protons and electrons) but different numbers of neutrons. Stable Isotope Do not undergo radioactive decay, but they may be radiogenic (i.e., produced by radioactive decay). Usually the number of protons and neutrons is similar, and the less abundant isotopes are often “heavy”, i.e., they have an extra neutron or two.

Nomenclature ( X) AZAZ NameSymbolDefinition Atomic numberZNumber of protons in nucleus Neutron numberNNumber of neutrons in nucleus Mass numberANumber of nucleons in nucleus NuclideDefinition IsotopeSame Z, different N and A IsobarSame A, different Z and N IsotoneSame N, different A and Z Nuclide classification

Koch 2007

What makes for a stable isotope system that shows large variation? 1)Low atomic mass 2)Relatively large mass differences between stable isotopes 3)Element tends to form highly covalent bonds 4)Element has more than one oxidation state or forms bonds with a variety of different elements 5)Rare isotopes aren’t in too low abundance to be measured accurately

Since natural variations in isotope ratios are small, we use δ notation δ H X = ((R sample /R standard ) -1) x 1000 where R = heavy/light isotope ratio for element X and units are parts per thousand (or per mil, ‰) e.g. δ 18 O (spoken “delta O 18) or δ 34 S (spoken “delta S 34) Don’t ever say “del”. Don’t ever say “parts per mil”. It makes you sound like a knuckle-head.

Other notation used in ecology 1)Fraction of isotope in total mixture: H F or L F defined as H/(H+L) or L/(H+L) 2) Ratio of isotopes: usually denoted R, H R or H/L R, defined as H/L 3) Atom %: usually H AP or L AP defined as 100* H F or 100* L F Fry 2006 Over a 200‰ range, δ H X varies linearly with % heavy isotope, but has easier to remember values.

Isotope Fractionation 1)Isotopes of an element have same number and distribution of electrons, hence they undergo the same chemical (and physical) reactions. 2)Differences in mass can, however, influence the rate or extent of chemical or physical reactions, or lead to partitioning of isotopes differentially among phases. 3) Isotopic sorting during chemical, physical, or biological processes is called Fractionation.

Fractionation mechanisms Equilibrium Isotope Fractionation A quantum-mechanical phenomenon, driven mainly by differences in the vibrational energies of molecules and crystals containing atoms of differing masses. Kinetic Isotope Fractionation Occur in unidirectional, incomplete, or branching reactions due to differences in reaction rate of molecules or atoms containing different masses.

Fractionation terminology Fractionation factor: α A/B = H R A / H R B = ( δ H X A )/( δ H X B ) DisciplineTermSymbolFormula GeochemistryOften equilibrium fractionations, put heavy isotope enriched substance in numerator SeparationΔ A/B δ A - δ B Enrichmentε A/B 1000( α A/B -1) BiologyOften kinetic fractionations, put light isotope enriched substance in numerator DiscriminationΔ A/B 1000( α A/B -1) Enrichmentε A/B 1000 ln α A/B Multiple Approximations 1000 ln α A/B ≈ δ A - δ B ≈ Δ A/B ≈ ε A/B

Isotope Exchange Reactions A L X + B H X ↔ A H X + B L X Equilibrium Constant (K) K = (A H X)(B L X)/(A L X)(B H X) K = (A H X/A L X)/(B H X/B L X) K 1/n = α A/B = H R A / H R B = ( δ H X A )/ ( δ H X B ) where H R = heavy/light isotope ratio in A or B and n = number of exchange sites (1 in this case) ∆G rxn = -RT lnK where R is the gas constant and T is in kelvin Hence, the isotopic distribution at equilibrium is a function of the free energy of the reaction (bond strength) and temperature H 12 CO 3 - (aq) + 13 CO 2 (g) ⇔ H 13 CO 3 - (aq) + 12 CO 2 (g) K HCO3/CO2 = (0°C) K HCO3/CO2 = (30°C)

Mass Dependent Fractionation Bond energy (E) is quantized in discrete states. The minimum E state is not at the curve minimum, but at some higher point. Possible energy states are given by the following equation: E = (n+½)h, where n is a quantum number, h is Plank’s constant, and  is the vibrational frequency of the bond.  = (½)√(k/), where k is the force constant of the bond (invariant among isotopes), and  is the reduced mass.  = (m 1 m 2 )/(m 1 +m 2 ) As bond energy is a function of , E is lower when a a heavier isotope is substituted for a lighter one. Zero Point Energy Difference Consider a C-O bond where we substitute 18 O for 16 O. m 1 =12, m 2 = 16 or 18, 16  = 12*16/28 = , 18  = 12*18/30 = For E, most terms dropout, so 18 / 16  = √( 16 / 18 ) = So, 18 E < 16 E

From vibrational frequency calculations, we can determine values for α (= K 1/n ). The derivation is time consuming, so here are some general rules. 1)Equilibrium fractionation usually decrease as T increases, roughly proportional to 1/T 2. 2)All else being equal, isotopic fractionations are largest for light elements and for isotopes with very different masses, roughly scaling as . 3)At equilibrium, heavy isotopes will tend to concentrate in phases with the stiffest bonds. The magnitude of fractionation will be roughly proportional to the difference in bond stiffness between the equilibrated substances. Stiffness is greatest for short, strong chemical bonds, which correlates with: a) high oxidation state in the element b) high oxidation state in its bonding partner c) bonds for elements near the top of the periodic table d) highly covalent bonds Equilibrium Isotope Fractionation

Temperature dependence of α Often expressed in equations of the form: 1000 ln α = a(10 6 /T 2 ) + b(10 3 /T) + c where T is in kelvin, and a, b, and c are constants.

In a closed systems, fractionation between 2 phases must obey conservation of mass (i.e., the total number of atoms of each isotope in the system is fixed). In a system with 2 phases (A and B), if f B is the fraction of B then 1-f B is the fraction of A. δ ∑ = δ A (1-f B ) + δ B f B δ A = δ ∑ + f B (δ A -δ B ) δ A = δ ∑ + f B ε A/B δ B = δ ∑ -(1- f B )ε A/B Equilibrium Fractionation, Closed System ε A/B ≈ 50‰ A B

Equilibrium Fractionation, Open System Consider a systems where a transformation takes place at equilibrium between phase A and B, but then phase B is immediately removed. This is the case of Rayleigh Fractionation/Distillation. I won’t present the derivation here, but the equations that describes phenomenon are: R A = R A0 f ( α A/B -1) δ A = (( δ A0 )f ( α A/B -1) )-1000 δ B = (( δ A )/ α A/B )-1000 where R A is the isotopic of ratio of phase A after a certain amount of distillation, R A0 is the initial isotopic ratio of phase A, f is the fraction of phase A remaining (A/A 0 = fraction of phase A remaining), and α A/B is the equilibrium fractionation factor between phase A and B.

Equilibrium Fractionation, Open System Hoefs 2004

Equilibrium Fractionation, Open System ε A/B ≈ 10‰ A = pond B = evaporate C = ∑ evaporate

Kinetic Isotope Fractionation Consider a unidirectional reaction where the reactant must pass over an energy barrier in the form of an activated complex to form the product. The energy required to surmount this barrier, E a, is the difference in energy between the activated complex and the reactant. We know from the earlier discussion that a reactant molecule with a heavier isotope will have a lower zero point energy than a reactant with a lighter isotope. The activated complex is pretty unstable, so isotopic substitutions don’t affect its energy state. As a consequence, the heavier isotope has a higher E a than the lighter isotope and reacts slower, leading to a heavy isotope depletion in the product. EaEa

Kinetic Fractionation, Closed System We can define a kinetic rate constant for the reaction above: k = Ae -Ea/RT, where A is the amount of reactant If H refers to the reactant with the heavy isotope, for R → P: dA/dt = kA and d H A/dt = H k H A If β Ξ H k/k, then d H A/A = βdA/A Integrating this equation between A 0 and A for both isotopes, we get: ln( H A/ H A 0 ) = βln(A/A 0 ) or H A/ H A 0 = (A/A 0 ) β If we divide by A/A 0, we get: ( H A/ H A 0 )/(A/A 0 ) = (A/A 0 ) β-1 Recall that R A ≈ H A/A and that R A0 = H A 0 /A 0, therefore R A /R A0 = (A/A 0 ) β-1 If f Ξ A/A 0, we get: R A = R A0 f β-1 This is just the Rayleigh Equation again. So, a closed system kinetic fractionation behaves just like an open system equilibrium fractionation. Not surprising, the unidirectional nature of the reaction is, in effect, removing the reactant from the product.

Kinetic Fractionation, Closed System

Kinetic Fractionation, Open System Consider a system with 1 input and 2 outputs (i.e., a branching system). At steady state, the amount of R entering the system equals the amounts of products leaving: R = P + Q. A similar relationship holds for isotopes: δ R = δ P f P + (1-f P )δ Q. Again, this should look familiar; it is identical to closed system, equilibrium behavior, with exactly the same equations: δ P = δ R + (1-f P )ε P/Q δ Q = δ R - f P ε P/Q

Open Systems at Steady State

Open system approaching steady state

Nier-type mass Spectrometer Ion Source Gas molecules ionized to + ions by e - impact Accelerated towards flight tube with k.e.: 0.5mv 2 = e + V where e + is charge, m is mass, v is velocity, and V is voltage Magnetic analyzer Ions travel with radius: r = (1/H)*(2mV/e + ) 0.5 where H is the magnetic field higher mass > r Counting electronics

Dual Inlet sample and reference analyzed alternately 6 to 10 x viscous flow through capillary change-over valve 1 to 100 μmole of gas required highest precision Continuous Flow sample injected into He stream cleanup and separation by GC high pumping rate 1 to 100 nanomoles gas reference gas not regularly altered with samples loss of precision