Introduction to Stable Isotopes

Notation

Fractionation

Gear

Definitions

IsotopesAtoms of the same element (i.e., same number ofprotons and electrons) but different numbers ofneutrons.

Stable IsotopeDo not undergo radioactive decay, but they may beradiogenic (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)AZ

Number of nucleons in nucleusAMass numberNumber of neutrons in nucleusNNeutron numberNumber of protons in nucleusZAtomic number

DefinitionSymbolName

Same N, different A and ZIsotone

Same A, different Z and NIsobar

Same Z, different N and AIsotopeDefinitionNuclide

Nuclide classification

Koch 2007

What makes for a stable isotopesystem that shows large variation?

1) Low atomic mass

2) Relatively large mass differences between stableisotopes

3) Element tends to form highly covalent bonds

4) Element has more than one oxidation state orforms bonds with a variety of different elements

5) Rare isotopes aren’t in too low abundance to bemeasured accurately

Since natural variations in isotoperatios are small, we use δ notation

δHX = ((Rsample/Rstandard) -1) x 1000

where R = heavy/light isotope ratio forelement X and units are parts per thousand(or per mil, ‰)

e.g. δ18O (spoken “delta O 18)or δ34S (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: HF or LFdefined as H/(H+L) or L/(H+L)

2) Ratio of isotopes: usually denoted R, HR or H/LR,defined as H/L

3) Atom %: usually HAP or LAPdefined as 100*HF or 100*LF

Fry 2006

Over a 200‰ range, δHXvaries linearly with% heavy isotope, buthas easier toremember values.

Isotope Fractionation

1) Isotopes of an element have same numberand distribution of electrons, hence theyundergo the same chemical (and physical)reactions.

2) Differences in mass can, however, influencethe rate or extent of chemical or physicalreactions, or lead to partitioning of isotopesdifferentially among phases.

3) Isotopic sorting during chemical, physical, orbiological processes is called Fractionation.

Fractionation mechanisms

Equilibrium Isotope FractionationA quantum-mechanical phenomenon, drivenmainly by differences in the vibrational energiesof molecules and crystals containing atoms ofdiffering masses.

Kinetic Isotope FractionationOccur in unidirectional, incomplete, or branchingreactions due to differences in reaction rate ofmolecules or atoms containing different masses.

Fractionation terminologyFractionation factor: αA/B = HRA/HRB = (1000 + δHXA)/(1000 + δHXB)

Often kinetic fractionations, put light isotopeenriched substance in numerator

Biology

Often equilibrium fractionations, put heavyisotope enriched substance in numerator

Geochemistry

1000(αA/B -1)ΔA/BDiscrimination

1000 lnαA/BεA/BEnrichment

1000(αA/B -1)εA/BEnrichment

δA - δBΔA/BSeparation

FormulaSymbolTermDiscipline

Multiple Approximations1000 lnαA/B ≈ δA - δB ≈ 1000(αA/B-1)

Isotope Exchange Reactions

ALX + BHX ↔ AHX + BLX

Equilibrium Constant (K)

K = (AHX)(BLX)/(ALX)(BHX)K = (AHX/ALX)/(BHX/BLX)

K 1/n= αA/B = HRA/HRB = (1000 + δHXA)/ (1000 + δHXB)where HR = heavy/light isotope ratio in A or B

and n = number of exchange sites (1 in this case)

∆Grxn = -RT lnKwhere R is the gas constant and T is in kelvin

Hence, the isotopic distribution at equilibrium is a function of thefree energy of the reaction (bond strength) and temperature

H12CO3-(aq) + 13CO2(g) ⇔ H13CO3

-(aq) + 12CO2(g)

KHCO3/CO2 = 1.0092 (0°C) KHCO3/CO2 = 1.0068 (30°C)

Mass Dependent Fractionation

Bond energy (E) is quantized in discretestates. The minimum E state is not at thecurve minimum, but at some higher point.

Possible energy states are given by thefollowing equation:E = (n+½)hν, where n is a quantumnumber, h is Plank’s constant, and ν isthe vibrational frequency of the bond.

ν = (½π)√(k/µ), where k is the force constant of the bond (invariantamong isotopes), and µ is the reduced mass.

µ = (m1m2 )/(m1+m2)As bond energy is a function of µ, E is lower when aa heavier isotope is substituted for a lighter one.Zero Point Energy Difference

Consider a C-O bond where we substitute 18O for 16O.m1=12, m2 = 16 or 18,16µ = 12*16/28 = 6.85714, 18µ = 12*18/30 = 7.2000For E, most terms dropout, so 18 ν/16ν = √(16µ/ 18µ) = 0.9759.So, 18E < 16E

From vibrational frequency calculations, we can determine values forα (= K1/n). The derivation is time consuming, so here are some

general rules.

1) Equilibrium fractionation usually decrease as T increases, roughlyproportional to 1/T2.

2) All else being equal, isotopic fractionations are largest for lightelements and for isotopes with very different masses, roughlyscaling as µ.

3) At equilibrium, heavy isotopes will tend to concentrate in phaseswith the stiffest bonds. The magnitude of fractionation will beroughly proportional to the difference in bond stiffness betweenthe equilibrated substances. Stiffness is greatest for short, strongchemical bonds, which correlates with:a) high oxidation state in the elementb) high oxidation state in its bonding partnerc) bonds for elements near the top of the periodic tabled) highly covalent bonds

Equilibrium Isotope Fractionation

Temperature dependence of α

Often expressed in equations of the form:1000 lnα = a(106/T2) + b(103/T) + c where T is in kelvin, and a, b, and c are constants.

In a closed systems, fractionation between 2 phases must obeyconservation of mass (i.e., the total number of atoms of each isotope inthe system is fixed). In a system with 2 phases (A and B), if fB is thefraction of B then 1-fB is the fraction of A.

δ∑ = δA (1-fB) + δBfB

δA = δ∑ + fB (δA-δB)

δA = δ∑ + fBεA/B

δB = δ∑ -(1- fB )εA/B

Equilibrium Fractionation, Closed System

εA/B ≈ 50‰ A

B

Equilibrium Fractionation, Open System

Consider a systems where a transformation takes place at equilibriumbetween phase A and B, but then phase B is immediately removed. This isthe case of Rayleigh Fractionation/Distillation. I won’t present thederivation here, but the equations that describes phenomenon are:

RA = RA0f(αA/B-1)

δA = ((1000 + δA0)f(αA/B-1))-1000

δB = ((1000 + δA)/αA/B)-1000

where RA is the isotopic of ratio of phase A after a certain amount ofdistillation, RA0 is the initial isotopic ratio of phase A, f is the fraction ofphase A remaining (A/A0 = fraction of phase A remaining), and αA/B is theequilibrium fractionation factor between phase A and B.

Equilibrium Fractionation, Open System

Hoefs 2004

Equilibrium Fractionation, Open System

εA/B ≈ 10‰

A = pondB = evaporateC = ∑ evaporate

Kinetic Isotope Fractionation

Consider a unidirectional reaction where the reactant must pass over anenergy barrier in the form of an activated complex to form the product.The energy required to surmount this barrier, Ea, is the difference inenergy between the activated complex and the reactant. We know fromthe earlier discussion that a reactant molecule with a heavier isotope willhave a lower zero point energy than a reactant with a lighter isotope. Theactivated complex is pretty unstable, so isotopic substitutions don’t affectits energy state. As a consequence, the heavier isotope has a higher Eathan the lighter isotope and reacts slower, leading to a heavy isotopedepletion in the product.

Ea

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 dHA/dt =HkHA

If β Ξ Hk/k, then dHA/HA = βdA/A

Integrating this equation between A0 and A for both isotopes, we get:ln(HA/HA0) = βln(A/A0) or HA/HA0 = (A/A0)β

If we divide by A/A0, we get:(HA/HA0)/(A/A0) = (A/A0)β-1

Recall that RA ≈ HA/A and that RA0 = HA0/A0,therefore RA/RA0 = (A/A0)β-1

If f Ξ A/A0, we get:

RA = RA0fβ-1

This is just the Rayleigh Equation again.So, a closed system kinetic fractionation behaves just like an open systemequilibrium fractionation. Not surprising, the unidirectional nature of thereaction 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 branchingsystem). At steady state, the amount of R entering the systemequals the amounts of products leaving: R = P + Q. A similarrelationship holds for isotopes: δR = δPfP + (1-fP)δQ. Again, thisshould look familiar; it is identical to closed system, equilibriumbehavior, with exactly the same equations:δP = δR + (1-fP)εP/QδQ = δR - fPεP/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.: 0.5mv2 = 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