Isotopes - Notes Part I. What are isotopes and why might they be useful? nuclide - a nucleus with a specific number of protons+neutrons (nucleons) atomic number - number of protons (defines # of electrons, therefore chemistry) isobar - nuclides with the same atomic mass (p+n) eg 14C, 14N isotope - nuclides with the same atomic number but different numbers of neutrons (eg 14C, 12C, 13C). Isotopes behave in chemically similar ways (because they have the same number of electrons) but may be fractionated because they have different mass and therefore form bonds of different strengths Notation: XAZ , where X is the element symbon, A the number of total nucleons and Z the number

of protons (which is redundant with the element symbol) For example: 12C = 6 protons and 6 neutrons 13C = 6 protons and 7 neutrons 14N = 7 protons and 7 neutrons 15N = 7 protons and 8 neutrons 16O = 8 protons and 8 neutrons 18O = 8 protons and 10 neutrons Table 1. Some stable isotopes and their relative abundance

Element Isotope Relative abundance Hydrogen 1H 99.985% 2H (D) 0.0105% Carbon 12C 98.89%

13C 1.11%

Nitrogen 14N 99.34%

15N 0.37%

Oxygen 18O 99.76% 16O 0.1%

Sulfur 32S 95.0%

33S 0.76%

34S 4.22%

36S 0.014%

_______________________________________________________________________ The overall abundance of nucleons is related to nuclear, rather than chemical, properties. Nuclear Binding Energy, E = (Δ m) c2, where (Δ m) is the mass difference between the nucleus and the sum of the masses of individual protons and neutrons making up the nucleus. The most stable nuclide has the lowest binding energy per nucleon. Example: Conversion of Hydrogen to Helium (Fusion) Mass of 4 hydrogen atoms 6.696 x 10-24 g Mass of 1 helium atom - 6.648 x 10-24 g Δ m 0.048 x 10-24 g Using E = (Δ m) c2, this mass loss generates about 1 x 10-12 calories of energy per He atom formed. If 1 gram of H is converted to He, then about 1.5 x 1011 calories of energy are produced (enough to heat 2 million liters of water from room temperature to the boiling point). ______________________________________________________________________________________ Nucleons may be synthesized in 3 ways: 1) cosmological (the big bang) - produced H, He 2) stellar evolution - produces everything else

main sequence stars - burn H to He red giants - produce up to Fe (most stable nucleus is 56Fe) supernovae (s (slow) and r (rapid) process - produce elements heavier than Fe

3) in interstellar medium (including planetary atmospheres) induced by galactic cosmic rays (the cosmogenic nuclei –14C, 10Be, 36Cl, etc). Unstable nuclides undergo radioactive decay at a predictable and constant rate - this provides useful chronometers for earth and planetary sciences. (More later in discussion of radiocarbon). Properties of elemental abundance explained by stellar evolution 1) H, He are the most abundant elements in the universe 2) light elements are more abundant than heavy (except for anomalously low abundance of Li, B, Be) 3) even numbers of protons+neturons > odd numbers 4) elements with atomic number >30 show less variation in abundance than those with atomic number < 30 5) peak in abundance at Iron (Fe)

Figures are from W. S. Broecker, How to Build a Habitable Planet, Eldigio Press, NY

LIGHT STABLE ISOTOPES Nomenclature Ways of expressing differences in stable isotopes are based on our inability to measure absolute abundances as well as relative abundances of isotopes using mass spectrometric techniques (see below). Isotopic composition is expressed as the ratio of the rare isotope to the abundant isotope (R ), by convention:

HD

OO

NN

CC

lightheavy

abundantrareR ,,, 16

18

14

15

12

13===

Because we measure the ratio R in a sample compared to that of a known standard, and because the differences found in nature between sample and standard are relatively small, we express the isotopic signature as the deviation in parts per thousand (per mil) of R for the sample compared to that of the standard :

100011000tantan

tan xRR

xR

RR

dards

sample

dards

dardssample⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡ −=δ

The choice of standard varies with the element.

Element Standard R

Carbon Pee Dee Belemnite (calcium

carbonate, original, has been

replaced but still expressed)

13C/12C = 0.0112372 18O/16O = 0.002671

Nitrogen Atmospheric N2 0.0036765

Hydrogen,

Oxygen

VSMOW (standard mean

ocean water)

D/H = 0.00015576 18O/16O = 0.00200520

By definition, the standards have δ = 0‰, though the standard needs to be specified for oxygen.

For example, a leaf with δ13C value of –28 ‰ has an isotope ratio ⎥⎦

⎤⎢⎣

⎡

dards

sampleRR

tan of

(-28/1000) + 1, or 0.972.

Isotopes have traditionally been measured using mass spectrometry: A mass spectrometer has four main parts: 1.- Ion source (in stable isotope mass specs this is a source of positive ions) 2.- Analyzing magnet 3.- Ion collector (Faraday cup) 4. Detector 1. Ionization Atoms and molecules are ionized by bombardment with high energy electrons (~80eV) 2: Acceleration Ions are accelerated through a voltage and focused into a beam using electrostatic deflection by small charged plates with small holes in the middle (electrostatic lenses) 3: Isotope separation by ion deflection Charged ions moving through a magnetic field are deflected – heavier ions get deflected less than lighter ions. 4: Detection Ions are collected in Faraday cups which are placed precisely to intersect the mass of interest as it exits the magnet. The kinetic energy of an ion of mass m and charge e moving through a voltage potential V is given as:

2

21 mveVE == (8)

where v is velocity. Ions exiting the ion source should all have the same charge and the same kinetic energy becuase they have been accelerated through the same voltage potential In that case, isotopes of different mass will have slightly different velocities: (since they have the same E)

meVv 2

= (9)

When the ions enter a magnetic field (one that is perpendicular to the ion trajectory), they are deflected into circular paths that obey the equation

2

rvmBev = (10)

where B is the magnetic field strength and r is the radius of the circular trajectory taken by the ions. Combining equations (8) and (10), we obtain:

22

2r

VrB

em

×⎟⎟⎠

⎞⎜⎜⎝

⎛= ,

For each gas analyzed B and V are set in the instrument such that KVrB

=⎟⎟⎠

⎞⎜⎜⎝

⎛

2

2 (a constant)

In this way, 2Krem=

New instruments have been designed to measure the isotopic signature of gases using laser spectroscopy – this takes advantage of the different absorption in the infra-red of the molecules with different isotopic makeup (e.g. 12CO2 versus 13CO2). [See next section for why this is so]. Laser measurements come in various forms – tunable diode lasers and cavity ring down. The technology is advancing very rapidly, and it is likely that in the future mass spectrometry will not be the most common method for stable isotope measurement.

Narrow band, idealizedtransmission spectra

generated from HITRANhttp://www.cfa.harvard.edu/hitran/

4.54 µm 4.17 µm

12CO2 4.332333 µm

13CO2 4.332435 µm

Bowling et al. (2003) AFM

II. Fractionation of Stable Isotopes in Nature The energy of molecular bonds are quantized and the energy of the total molecule depends on the degree of activation of its rotational, translation, and vibrational modes. The most important factor that varies at room temperature for different isotopes is the energy associated with vibration. The vibrational energy in a molecular bond (Ev) in the ground state (zero point energy) is defined as:

ABv hE ν21

= , where h is Plank’s constant, and νΑΒ vibrational frequency for one atom (A)

in relation to the other (B). νΑΒ is defined as:

ABAB

kµπ

ν21

= , where k is the elastic constant in the bond A-B and µAB the reduced mass:

BA

BAAB mm

mm+

×=µ , where mA and mB are the atomic masses of A and B respectively.

Bonds between different isotopes of species A and B will have different vibrational energies at ground state – molecules with lighter isotopes have higher vibrational frequencies than those with heavy isotopes. For example, for H2 versus HD µΗ2 =0.50391261 and µHD=0.67171137.

This means that H2 has a lower dissociation energy (less energy required to break the bond) than HD.

Thus there will be different physical properties for molecules with different isotopic abundances – these are most pronounced where mass differences are largest. Example: Characteristic constants of H2O and D2O (From Hoefs, 1973, Stable isotope geochemistry) Property H2O D2O Density (20 C) 0.9982 1.1050 Temperature of greatest density 4.0 11.6 Mole volume (20C) cm3/mole 18.049 18.124 Melting point (760 torr, C) 0.00 3.82 Boiling point (760 torr, C) 100.00 101.42 Vapor pressure (at 100C in torr) 760.00 721.60 Viscosity (@20.2 C in centipoises) 1.00 1.26 Ionic product a 20C 1x10-14 0.16 x 10-14 There are two basic kinds of fractionation: Equilibrium, and kinetic (nonequilibrium). The degree of separation of isotopes between two phases or reactant versus products in a chemical reaction is called the Isotopic fractionation factor (α):

For a reaction A à B (which could represent reactants and products in a chemical

reaction, or two phases, e.g. liquid and vapor), α is defined as:

B

ABA R

R=−α (2)

where RA and RB are the ratios ⎥⎦

⎤⎢⎣

⎡

dards

sampleRR

tan in A and B, respectively. [Note, we

sometimes get sloppy in our notation, and forget the Rstandard, since it always cancels out of subsequent relations]

Equilibrium Exchange Reactions [H2

16O]vapor + [H218O]liquid ↔ [H2

16O]vapor + [H218O]liquid

The equilibrium constant for this reaction (remember your basic chemistry!) is

[ ][ ]

[ ][ ]

[ ][ ]

[ ][ ]

liquidvapor

liquidOH

liquidOH

vaporOH

vaporOH

liquidOH

liquidOH

vaporOH

vaporOHeq RRlvK /),(

162

182

162

182

182

162

162

182

===

In this case Keq = the fractionation factor, α (more generally, α = Κ (1/n), where n is the

maximum number of exchange positions that are equivalent). The first innovative use of stable isotopes – paleothermometry For isotope exchange reactions in general, there is a small change in enthalpy (ΔHR). This means that the fractionation factor will change with temperature. The Clausius-Clapeyron relation says: D(lnKeq)/d((1/T)) = ΔHR/R (where R is the gas constant) So, for example, in the isotope exchange reaction: CaC16O16O16O + H2

18O = CaC16O16O18O + H216O

(CaCO3 is calcium carbonate, or limestone)

α = RCaCO3/RH2O [ ][ ][ ][ ] 3/1

31618

2

3/13

18162

OCaCOH

OCaCOH=

is temperature dependent. Experiments have determined the relationship for calcium carbonate in equilibrium with water:

T (Centigrade) = 16.9 – 4.2 (δ18Ocarbonate – δ18Owater) + 0.13(δ18Ocarbonate – δ18Owater)2

δ18Owater can be measured – and if we assume it is constant, then we can get temperature of equilibration from δ18Ocarbonate.

Spherical chamber form of O. universa

Cesare Emiliani - δ18O of planktonic foraminifera can record temperature changes in the past

But – how much is temperature and how much is δ18Owater on glacial-interglacial timescales?

(Bemis et al (1998); Spero et al., in prep.)

5

10

15

20

25

30

-3-2-1012

O. universa HL (So. Cal. Bight& Puerto Rico combined)

Catalina 2000Bemis et al. (1998)Puerto Rico 1999

5

10

15

20

25

30

Tem

pera

ture

(o C)

δ18Oc - δ18O

w (‰ vs PDB)

T = 15.0 - 4.36(δshell

-δw) + 0.35(δ

shell-δ

w)2

Kinetic Effects Example – Photosynthesis CO2 + H2O + light è CH2O + O2

Let us assume that δ13CO2 (substrate) equals –7.4‰ (Rs= 0.9926) and the δ13C of the plant material (product) equals -27.6‰ (Rp= 0.9724). The value of α is given by:

02077.19724.09926.0

===p

s

RR

α

Usually α is expressed as an “enrichment factor” (ε) : 1000)1( ×−= αε ,

in geochemical and atmospheric chemistry applications. In this case ε = 20.77‰. In other words, the isotopic signature of the plant carbon is ~21‰ “lighter” than the substrate CO2. If we express ε as a function of delta (δ) values, we obtain:

ABBA Δ≈−≈ δδαln103 , where ΔΑΒ approximates ε

plantaCOproductosubstrato δδδδ −=−=Δ2

, or Δ = -7.4‰– (-27.6 ‰)=20.2‰. NOTE: For biologists, ε = 103ln α, and Δ = (α−1) x 1000 (the same as ε for geochemists). Although the differences are generally small, this confusion of nomenclature should be noted. Diffusion Fick’s Law (from Stefan-Bolzman relation):

2

2

1121⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+=

baab mm

kTn

Dππσ

where Dab is the rate of diffusion for molecule a in b (for example, CO2 in air) σ is the collision diameter between unlike molecules (a and b) n is the number of moles k is the Bolzman constant T is temperature ma is the mass of species a, and mb is the mass of species b For different isotopes the collisition diameter is the same – there the diffusion coefficients are related by:

2/1

213

212

213

212

213

212 *

*)()(

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛ +=

COair

COair

COair

COair

mmmm

mmmm

CODCOD

= 1.0044

(mair = 28, m12CO2 = 44, m13CO2 = 45) Isotope Mixing Isotopes are useful for determining the contribution of different sources to a mixture. We use the isotope mass balance. For example, if A is the proportion of the first source (with isotopic signature δA) , and B is the proportion of the second source (with isotopic signature δB)) Then the overall mass balance equation is A+B =1 (3) and the isotope mass balance is:

PBA BA δδδ =×+× )()( (4) Substituting eq(3) into eq(4) we obtain

BA

BPAδδδδ

−

−= (5)

This model requires that: 1) There are only two sourcess 2) The two sources are different in their isotopic signature Example: Combinations of cane sugar (C4 photosynthesis; –12‰; δB) and tree-flower sugar (C3 photosynthesis pathway; –27‰; δA) to a jar of honey, assuming the honey has δ13C value of –21‰ (δP). If we use equation (3) :

6.0)12(27)12(21=

−−−

−−−=A

Isotopes and the water cycle Rayleigh Distillation (example evaporation)

Rayleigh distillation [Rayleigh (1896)] is used to described processes where two

components equilibrate, followed by the emoval of one phase or component after equilibration. An example is the continuous removal of liquid from vapor as rain. In the case of water vapor in the atmosphere, from which liquid rain condenses (in equilibrium with the vapor) and falls (ie is removed from the system), the isotopic composition of the remaining vapor will be enriched in the lighter isotope. (This is because the lighter isotopes will tend to remain in the vapor, while the heavier isotopes will partition into the liquid). The Rayleigh equation describing the evolution of the vapor in this case is:

)1( −= αfRR

ov

v (6)

where Rvo is the initial vapor composition and Rv the composition of vapor (v) remaining at and given instant thereafter, f is the fraction of water vapor remaining (ie the amount at any given instant divided by the amount at time 0) and α is the fractionation factor (Rl/Rv (l=liquid)). Expressed in delta notation:

)1(

10001000 −=+

+ α

δ

δf

ov

v (7)

The graph below follows the isotopic evolution of vapor and the liquid condensing from it as as function of f (assuming α= 1.0092 (for fractionation of 18O in H2O) and an initial δ18O of vapor of -9.2‰.

Raleigh Distillation . Rayleigh distillation. Isotopic fractionation of oxygen isotopes in weater during condensation of water from vapor, with constant removal of the liquid as it forms (Faure, 1986).

The first liquid to condense is in equilibrium with the initial vapor (δ18O=-9.7 ‰) and has δ18O=0‰. If this condensate is removed (ie as a raindrop) without being re-evaporated, the remaining vapor will have less of the heavy isotope. Any water that condenses from this vapor will in turn have ‘lighter’ isotope values (ie more depleted in the heavy isotope)

Bowen and Revenaugh, 2003