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GE 140a 2019 Lecture 3 NOMENCLATURE OF ISOTOPIC DISTRIBUTIONS

GE 140a 2019 Lecture 3 NOMENCLATURE OF ISOTOPIC DISTRIBUTIONS. Language. Isotopes : Nuclides that share a number of protons but differ in number of neutrons Stable Isotopes : Isotopes that are neither the product of nor undergo radioactive decay.

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GE 140a 2019 Lecture 3 NOMENCLATURE OF ISOTOPIC DISTRIBUTIONS

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  1. GE 140a 2019 Lecture 3 NOMENCLATURE OF ISOTOPIC DISTRIBUTIONS

  2. Language Isotopes: Nuclides that share a number of protons but differ in number of neutrons Stable Isotopes: Isotopes that are neither the product of nor undergo radioactive decay 16O, 17O and 18O are the stable isotopes of oxygen 12O (t1/2 ~ 10-21 s) is a radioactive (non-stable) O isotope Isotopologues: ‘Versions’ of a single molecule having different proportions and/or positions of isotopes The stable isotopologues of molecular oxygen include: 16O16O, 17O16O, 18O16O, 18O17O, 17O17O, 18O18O “normal” “singly substituted” “doubly substituted” Isotopomers: ‘Versions’ of an isotopic stoichiometry having non-equivalent locations of isotopes within the molecule (think of ‘isomers’) 14N15N16O and 15N14N16O are isotopomers of each other 18O16O and 16O18O are not isotopomers, as they are equivalent (related by simple symmetry operations) Isotopocule: A neologism that is synonymous with my definition of isotopologue; its users consider isotopologue to refer only to all isotopocules that share an isotopic stoichiometry

  3. Isotope abundances vs. isotope ratios [isotope i] = Ri [reference isotope] Where [i] denotes the concentration (mole fraction) of isotope i in a population of atoms • Usual convention is that the reference isotope is the most abundant stable isotope of that element, so that Ri values approximately equal mixing proportions of i • In general, the most abundant stable isotope of elements of interest is also the lightest, so Ri values always ratio a heavy isotope to a light one • This rule is violated for Li, B, some transition metals, U and others. Nomenclature is mixed in these cases. The least confusing thing to do for such elements is toplace heavier isotopes in the numerator and the lightest one in the denominator • The identity of the element whose isotopes are being ‘ratioed’, and the isotopic mass of the denominator species aregenerally not indicated. One must figure out what element is meant from memory or context. 7Li 6Li 17O 16O 11B 10B 13C 12C 15N 14N 34S 32S D H R7= R13= R34 = R17= RD = R11= R15= 18O 16O 6Li 7Li R18= …or R6=

  4. Abundances and ratios of isotopologues Consider a pool of H2 molecules where 50 % of the atoms are H and 50 % of the atoms are D ? ? D ? H ? First randomly ‘fill’ left atomic site: 50 % 50 % Then randomly ‘fill’ right atomic site: H H H D D H D D 25 % 25 % 25 % 25 % These two are symmetrically equivalent [H2]: 0.25 [HD]: 0.50 [D2]: 0.25 2x abundance of others because there are two ways in which the symmetrically equivalent formula can be made.

  5. Abundances and ratios of isotopologues General rule Fraction of all isotopes of the element of interest Fraction of all molecules [Isotopologue] = { ∏[constituent isotopes]} x {# of equivalent arrangements} [H2] = [H]2 [HD] = [H]x[D]x2 [D2] = [D]2 NH NH+ND ND NH+ND NHD NH2+NHD+ND2 Where: [H]= [D]= [HD]=

  6. Abundances and ratios of isotopologues Another example [12CH3D] = [12C]x[H]3x[D]x4 NH NH+ND ND NH+ND N12C N12C+N13C [12CH2D2] = [12C]x[H]2x[D]2x6 ~ [D]2x6 (6 equivalent arrangements of 2H’s and 2D’s) Where: [H]= [D]= [12C]= N12CH3D N12CH4+N12CH3D+N12CH2D2+N12CHD3+N12CD4+N13CH4+N13CH3D+N13CH2D2+N13CHD3+N13CD4 [12CH3D]=

  7. Abundances and ratios of isotopologues Abundance ratios of isotopologues generally use the unsubstituted form as the denominator RD2 of molecular hydrogen = [D2]/[H2] Sometimes concentrations of all isotoplogues sharing a cardinal mass will be combined and reported as a summed concentration: [50O3]: The fraction of all ozone molecules that have a mass of 50 AMU; equals the sum: [16O218O] + [17O216O] And these sums of isotopologues at each mass can be used to form ratios: R46: The ratio of the concentration of CO2 molecules having mass 46 to the concentration of ‘normal’ CO2 molecules having mass 44: R46 = {[12C16O18O] + [13C17O16O] + [12C17O2]}/[12C16O2]

  8. Why do we talk about ratios rather than concentrations? • Technologies for measuring ratios (e.g., multi-collector mass spectrometry) are more precise than technologies for measuring concentrations (e.g., gravimetry) • Generally, when we discuss ‘accuracy’ in measurements of isotope ratios, we really mean relative to a poorly known or assumed value of a standard: RASample=[RASample/RAStandard] x RAStandard This reduces uncertainty in the ratio to the error in measuring differences in ratios (ca. 10-5 if done with care). But, how do I know RA for my reference standard? Answer: You don’t ! (at least not well). Field is based on highly precise knowledge of relative differences, but poorly known absolute abundances. Measured quantity Assumed or loosely known

  9. Things to remember about [i] and Ri values: (1) Concentrations of isotopes sometimes approach, but do not equal Ri values R13 1+R13 [13C]= ~ 0.99xR13 for natural materials These differences between [Ci] and Ri become extreme when the ratio is close to or more than 1, leading to much arithmetic mayhem: R238 = 137.9, whereas [238U]= 0.9928 (2) Concentrations of isotopologuesgenerally do not equal abundances of isotopes [D] ≠ [HDO] i.e., the mole fraction of D in the population of all hydrogen atoms does not equal the mole fraction of HDO in the population of all water molecules. This is because D atoms are (nearly) randomly mixed among water molecules, so relatively little occurs as D2O

  10. Things to remember about [i] and Ri values: The discrepancy between concentration and ratio blows up when heavy isotopes are more abundant (so Ri values are greater than 1) Concentration Ratio 10-4 10-3 10-2 10-1 100 101 102 103 104 This is a serious issue for Li, B, S, Cl, many metal isotope systems

  11. The d value Risample -1} x 1000 distandard= { Ristandard Usually stated using units of ‘per mil’ or ‰ Where i is an isotope of interest (e.g., 18O or D) • distandard values have meaning only in the context of their known reference frame (Ristandard) • Always, always, always write distandardvalues with the subscript denoting its standard • A distandard value without a known reference frame is worse than meaningless: It is often also misleading • Never, ever, ever, for any reason, even after suffering a head blow, treat distandard values as if they were Rior [i] values. Some commonly used Ristandard values: RDVSMOW = 0.00015575 R7L-SVEC = 12.019 R11NBS 951 = 4.04362 R13PDB = 0.0112372 R15Air = 0.0036747 R17VSMOW = 0.0003799 R18VSMOW = 0.0020052 R34CDT = 0.0450045

  12. ’Epsilon’ and ‘per meg’ versions of the ‘delta’ style notation eistandard= Risample -1} x 10,000 { Ristandard Developed for reporting variations in radiogenic isotope ingrowth of Nd and Hf isotopes, but starting to be used for stable isotope effects for iron and other metals Risample distandard= ‘per meg’ -1} x 1,000,000 { Ristandard (yes, both the x1000 and x1,000,000 scales are called ‘delta’ values) All the ideas we’ll raise regarding ’normal’ per-mil di notation also apply to the ei and ‘per meg’ scales.

  13. Reporting isotopic differences: aj-k, ∆j-kand e values Riof j Riof k aij-k= ∆ij-k = distandard of j–distandardof k e.g., ∆18Oj-k = d18OVSMOW of j - d18OVSMOW of k ∆ij-k ~ 1000x(aij-k-1) ∆ij-k ~ 1000x(ln[aij-k]) The error in these approximations is small (~0.02 ‰) if dij and dikare both within 10 ‰ of 0, and within 10 ‰ of each other, but large in other cases 1000 + diof j 1000 + dikof k aij-k= This, however, is true: eij-k: = (aj-k-1)x1000 Note eij-kapproximately (but not exactly) equals ∆j-k (yes, this is another, and different, use of the ei notation)

  14. Another way to report isotopic differences K: Equilibrium constant for an isotope exchange reaction [HDO][H2] [H2O][HD] HD + H2O = H2 + HDO K = Note K is related to aH2O-H2(dependence varies with # of atoms exchanged) We’ll work through the details of this in a couple of weeks. Note also that this definition of K is not thermodynamically rigorous, as it assumes that thermodynamic activity (ai = gim[i]n) is equal to concentration (or that the activity coefficients, gi, and powers, m,n, of the isotopologues are equal to each other). this is an approximation, but one that is universally applied.

  15. distandard and ∆ij-kvaluesin different standard reference frames Water 1: RD = 0.00017133 Water 2: RD = 0.00014018 aD1-2= 1.222 1000x(aD1-2-1) = 222 ‰ 1000x(lnaD1-2) = 200.65 SLAP reference frame (RDSLAP = 0.000089089) VSMOW reference frame (RDVSMOW = 0.00015575) dDSLAPof water 1 = +923 ‰ dDSLAPof water 2 = +573 ‰ ∆D1-2= 350 ‰ dDVSMOWof water 1 = +100 ‰ dDVSMOWof water 2 = -100 ‰ ∆D1-2= 200 ‰ Standard conversion identity (Craig, ‘57): d2 ofsample= d1of sample+ d2of std 1+ (d1of sample)(d2of std 1) 1000 (see page 32 of Criss)

  16. Another meaning of ∆i: mass-independent fractionation First, the reality behind the mess I’m about to show you: • aij-kvalues for elements having more than one isotope ratio are related to one another, generally through a power-law relationship with exponent l (or sometimes q or b): e.g., for the system 16O, 17O, 18O: a17i-j = a18i-jl17/18 The l (or q or b) value can be referred to as the ‘mass law exponent’. • The mass law exponent gets a subscript that denotes which masses are involved; e.g., in the case above the l should be written: l17/18. This is often omitted. • For most common mechanisms of isotopic fractionation, li1/i2 values are simple functions of mass. We’ll define them later, but take it as given for now that I could specify a relationship between l17/18 and the mass differences among 16O, 17O and 18O. To first order, these tend to be loosely proportional to mass differences. i.e., l17/18 is about 0.5 because the (17-16) is half (18-16). Similarly, l33/34 for S isotopes is about 0.5 because (33-32) is half (34-32).

  17. Now for the mess that happens when we try to apply these concepts to the mathematics of d notation and plots of distd values: Core concept Line defined by compositions fractionated by a common mass law An anomaly relative to mass-dependent line Slope ~ lA/B + ∆A dAi-j - ∆A dBi-j ∆17O: Deviation in d17OVSMOW from the ‘terrestrial mass fractionation line’ (a trend defined by the approximate relationship d17OVSMOW = 0.52xd18OVSMOW) Mass-dependent: a fractionation that moves products and residues parallel the reference line Mass-independent: a fractionation that separatesisotopes in a way that is not a simple function of mass. Often results in non-0 ∆ values Mass-anomalous: a fractionation that actually has mass dependent chemical physics but results in a non-0 ∆ value due to peculiarities of mass law or process.

  18. Mass-independent: a fractionation that separatesisotopes in a way that is not a simple function of mass. Often results in non-0 ∆ values Mass-anomalous: a fractionation that actually has mass dependent chemical physics but results in a non-0 ∆ value due to peculiarities of mass law or process. Crazy, truly mass-independent process This anomaly is ‘mass independent’ This anomaly is ‘mass anomalous’ (and really just an artifact of the reference frame) Slightly different mass-dependent slope dAi-j Reference slope dBi-j

  19. But this is wrong in detail because fractionations are related by a power law, not a line Curvature a function of lA/B + ∆A RA ~ proportional todAi-j Mixing trend - ∆A RB ~ proportional todBi-j • In these spaces, mass-dependent fractionations actually follow curves • But some other common processes, like mixing, follow lines • Thus, anomalies can be generated just by mixing (we’ll return to this later…)

  20. To avoid this real effect, many studies of systems having 3 or more isotopes (O, S, etc.) work with a processed version of the d value, d’, where power law fractionations will plot as lines rather than curves. d’Ai-j = 1000 x (ln(dAi-j/1000+1) d’Bi-j = 1000 x (ln(dBi-j/1000+1) Slope =lA/B + ∆’A Mixing trend d’Ai-j - ∆’A d’Bi-j The disadvantage of this is that in this peculiar space simple processes like mixing follow curves. I.e., you’ve made yourself feel better, but actually just turned a complicated real thing into a complicated made up thing

  21. Another common visualization of ‘mass independent/anomalous’ effects + ∆’A Any point on the reference lA/B slope will be on this horizontal line ∆’Ai-j Mixing trend - ∆’A d’Bi-j

  22. Yet another use of the ∆ notation: ‘clumped’ isotope anomalies First, some idiosyncratic nomenclature Stochastic distribution: a purely random distribution governed by sampling statistics Multiply substituted isotopologue: An isotopologue containing 2 or more rare isotopes ‘Clump’ or ‘clumped’ isotopic species: synonymous with multiply substituted isotopologue

  23. Rules for reporting ‘clumped’ isotopic compositions • Ni = number of atoms or mols of isotopologuei Ni Total of Ni values for all isotopologues of molecule j [i] = • N18O2 N16O2+N16O17O+N17O2+N18O16O+N18O2+N17O18O [18O2] = [isotopologuei] • = Ri [reference isotopologue] D2 H2 RD2= Note, for both [i] and Ri, ‘i’ can refer to the sum of all isotopologues having the a given cardinal mass [47CO2] = [13C16O18O]+[12C17O18O]+[13C17O2]

  24. The ‘stochastic reference frame’ A curve through composition space marking location of all samples having a stochastic distribution Stochastic (random) distribution 0.8 0.6 [D2] ([H2]+[HD]+[D2]) 0.4 0.2 0.6 0.2 0.4 0.8 [D] ([H]+[D])

  25. Same idea, more complicated system Abundances of 13C18O16O in CO2 having the stochastic distribution

  26. The ∆i value The difference between a sample’s true composition and the composition it would have if it possessed a stochastic distribution of isotopologues Rimeasured Ristochastic [ -1]x 1000 ∆i = The Ri value calculated based on a sample’s known bulk isotopic composition Stochastic (random) distribution 0.8 0.6 [D2] ([H2]+[HD]+[D2]) + ∆D2 0.4 – ∆D2 0.2 0.6 0.2 0.4 0.8 [D] ([H]+[D])

  27. ∆i values in a complex system +∆47 -∆47 ∆47 47

  28. The tangled nomenclature of ‘site specific’ isotopic fractionation The study of intramolecular differences in isotopic composition is a relatively small sub-discipline that is nowhere near having its act together. All of the following nomenclatures can be found in the literature (plus some others I couldn’t interpret): Consider the site-specific C isotope structure of acetic acid: • [13C] values of each site, given some idiosyncratic label like: [13C]1 or [13C]methyl • R13 values of each site, e.g.,: R131 or R13methyl • d13CVPDB values of each site, but replacing the standard subscript with a made-up subscript indicating the site, e.g.,: d13C1 or d13Cmethyl • An ‘alpha’ factor between the sites, such as a13Cmethyl-carboxyl • A difference between the sites, such as ∆13Cmethyl-carboxyl • An ‘epsilon’ between the sites, such as e13Cmethyl-carboxyl Any of the above could have peculiar or missing superscripts and/or subscripts, which might or might not be explained

  29. High dimensionality isotopic compositions There is well-established nomenclature and plotting conventions for systems with two independent isotopic properties, but what do we do elements having many isotopes (e.g., Xe, with 9), or molecules that have two or more measurable independent clumped isotope indices (e.g., O2 with 18O2 and 18O17O, or CH4 with 13CH3D and 12CH2D2). Or bigger molecules with large numbers of unique isotopologues (e.g., sucrose with ~1015) Two general categories of approaches have been suggested data data Anomaly (∆i or ai relative to a reference frame) 0 Anomaly (e.g., ∆i value) 2 Reference frame process Predicted trend for a process of interest Isotope (or isotopologue?) i Anomaly (e.g., ∆i value) 1 People are really just starting to experiment with these ideas; there is room for someone clever to come up with a better and more universal approach to complex composition spaces. Basically, all that’s been done is to make old-fashioned plots with arithmetic variables people made up 20-50 years ago

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