Equilibrium stable isotope fractionation

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$Equilibrium stable isotope fractionation Edwin A.$

1 Equilibrium stable isotope fractionation Edwin A. Schauble UCLA 37 Cl 54 Fe Notation for fractionation factors: α XA-XB XA-XB = R XA XA /R XB δ XA δ XB 1000 (α XA-XB 1) 1000 ln( ln(α XA-XB ) For equilibrium isotopic fractionation, α is related to the equilibrium constant of a one-atom exchange reaction: light XA + heavy XB heavy XA + light XB K eq = [ heavy XA][ light XB] [ light XA][ heavy XB] XB] = R XA 1 R XB = α XA-XB This relationship is slightly more complicated if A or B has more than one atom of X! 1

$Theory of equilibrium isotopic fractionation Cl O Vibrations are quantized, E(vib vib) ) = hν(n+1/2) n=0,1,2,3 1/2 hν is called$ length Rotation and Translation: Both types of motion are also quantized, but there is no zero-point energy, and the quanta are

length Rotation and Translation: Both types of motion are also quantized, but there is no zero-point energy, and the quanta are

2 Theory of equilibrium isotopic fractionation Cl O Vibrations are quantized, E(vib vib) ) = hν(n+1/2) n=0,1,2,3 1/2 hν is called the Zero-Point Energy (ZPE) Energy n=2,, E(vib vib) ) = 5/2 hν n=1, E(vib vib) ) = 3/2 hν n=0, E(vib vib) ) = 1/2 hν R o Bond length Rotation and Translation: Both types of motion are also quantized, but there is no zero-point energy, and the quanta are much smaller than for vibrations. Rotational and translational quanta are much smaller than thermal energy at room temperature. 2

3 Cl O 298 K Rotational and translational quanta are so small relative to relevant thermal energies that classical mechanics gives a reasonably accurate description of non-vibrational motion. ClO ν = cm 1 ( Hz) 35 Cl ClO ν = cm 1 ( Hz) 37 Cl E(vib vib) ) = 1/2 hν E(vib vib) ) = 1/2 hν 5106 J/mol 5063 J/mol At Equilibrium, for Cl-isotope exchange between monoatomic Cl and ClO: 37 Cl + 35 ClO 35 Cl + 37 ClO ΔG 0 ΔE (vib vib) = E (vib vib) products E (vib vib) reactants E (vib E (vib vib) products = 5063 J/mol ( 37 ClO) vib) reactants = 5106 J/mol ( 35 ClO) ΔE( E(vib) ) = 43 J/mol, driving the reaction to the right and concentrating 37 Cl in ClO! 3

4 Cl O To get from ΔG 0 to a fractionation factor, we can use the standard thermodynamic formula: ΔG 0 = kt ln(k eq ) = kt ln(α ClO-Cl ) α ClO-Cl = exp( ΔG 0 /kt) So, considering only vibrations, and if all molecules are in the ground vibrational state, α ClO-Cl = exp( ΔG 0 /kt) = exp( ΔE vib /kt) exp( {1/2 hν 37ClO 1/2 hν 35ClO }/kt) h = exp( {ν 35ClO ν 37ClO }) = at 298 K 2kT Cl O Σ n=0 y n = 1/(1-y) In reality, some molecules will be vibrationally excited: α ClO-Cl = exp( ΔG 0 /kt) = exp( ΔE vib /kt) E vib = ktln(q vib ) Q vib = Σ n=0 exp( E n /kt) Q vib = Σ n=0 exp( hν(n+1/2)/kt) = Σ n=0 exp( hv/2kt) exp( hνn/kt) = exp( hv/2kt) Σ n=0 exp( hν/kt) n = exp( hv/2kt) hv/2kt) 1/{1 exp( exp( hν ZPE Excited states 4

5 So by including excited vibrational states, exp( hv ClO-Cl = exp( hv α ClO-Cl hv 37ClO hv 35ClO exp( hν 37ClO exp( hν 35ClO We re almost out of the woods! The final step is to include a simplified accounting for rotational and translational energies, α ClO-Cl = v 37ClO v 35ClO Rotation and translation exp( hv exp( hv hv 37ClO hv 35ClO exp( hν 37ClO exp( hν 35ClO The effect of isotopic substitution on rotational and translational energies can be expressed in terms of vibrational frequencies! Cl O α ClO-Cl = at 298 K (the difference is due mainly to excited rotation & translation) 5

6 It is necessary to know vibrational frequencies of all relevant isotopic forms of a molecule or mineral (i.e. Na 35 Cl and Na 37 Cl). Generally, vibrational frequencies have not been measured or are incomplete for rare isotopic forms. Predicting equilibrium isotope fractionations requires us to predict unknown vibrational frequencies THz w/ 35 Cl, no shift when 37 Cl is substituted THz w/ 35 Cl, 0.07 THz shift when 37 Cl is substituted Qualitative rules governing equilibrium isotope fractionation: 1. Fractionations are largest at low T-- scaling roughly as 1/T 2 Theory can be particularly useful for extending the temperature-range of experiments 2. Fractionations are largest for low mass elements -- scaling roughly as (m( heavy m light )/m 2 6

7 Qualitative rules governing equilibrium isotope fractionation: 3. Heavy isotopes prefer stiff chemical bonds. Typically this means short, strong bonds, correlating with a. Oxidation state: high charge stiff bonds C 4+ C C 4, S 6+ S 2, Fe 3+ Fe 2+ b. Bond partner oxidation state: high charge stiff bonds SiO 2 Al 2 O 3 CaO c. Bond partners: hard, low atomic # stiff bonds SiO 2 TiO UO 2 2 d. Bond type: covalent stiff bonds e. Coordination number: fewer bonds, smaller site stiff bonds 1-fold < 2-fold < 3-fold > 4-fold > 6-fold > 8-fold coordination 7

Applying stable isotope fractionation theory to heavy elements

$Applying stable isotope fractionation theory to heavy elements$ Applying stable isotope fractionation theory to heavy elements Edwin A. Schauble UCLA 37 Cl 54 Fe With help from: Elizabeth Johnson Hugh Taylor, George Rossman, John Eiler, Clark Johnson, Joe Kirschvink,