Gravitational and Centrifugal Separations of Isotopes

1. GE 14a 2019 Lecture 6 GRAVITATIONAL AND CENTRIFUGAL SEPARATIONS OF ISOTOPES

2. Gravitational separation of molecules in gases is ubiquitous near the upper and lower boundary layers of planetary atmospheres, and the similar phenomenon of centrifugal separation is the breakout enrichment technology that has enabled a global nuclear arms race Upper atmospheres Firn Sand dunes Gas centrifuges

3. Derivation of the gravitational isotope effect Consider a gas-filled cylinder that is at internal equilibrium and has no significant gradient in gravitational potential energy (e.g., a thin cylinder lying flat): End 1 End 2 Isothermal: T1 = T2 Chemical equilibrium: µi1 = µi2, µj1= µj2, etc. dGi dni Gibbs free energy (joules) µi = Chemical potential Partial pressure of i (bars) Mols Chemical potential for a gas: µi = µi0 + RTln(pi) Standard state: pure gas, 1 bar, 298.15 K At equilibrium, pi1 = pi2, pj1 = pj2, etc. Thus ( ) ( ) pi pj pi pj = 2 1

4. Derivation of the gravitational isotope effect Now stand the gas-filled cylinder on its head so that its long axis lies along a gravitational potential energy gradient (i.e., make it vertical), and wait until it re-establishes equilibrium: End 1 Isothermal: T1 = T2 Chemical equilibrium in a gravitational gradient: dA + dF = 0 Assume g constant between ends 1 and 2 (not necessary but simple) Helmholtz free energy, A, = U – TS (U is internal energy; S is entropy) This is a measure of work potential of a chemical system Gravitational potential energy, F= mgh dA + dF = 0 dA = µi2dni - µi1dni (change in A moving n moles of i from 1 to 2) dF = Mig(h2-h1)dni (change in F moving n moles of i from 1 to 2) Thus µi2 + Migh2 = µi1 + Migh1 End 2

5. Derivation of the gravitational isotope effect That’s all there is to it, but let’s make substitutions until we see something that reminds us of a fractionation factor: End 1 µi2 + Migh2 = µi1 + Migh1 Recall µi= µi0 + RTln(pi) µi0 + RTln(pi)2+ Migh2 = µi0 + RTln(pi)1+ Migh1 (recalling T1 = T2) ( ) pi1 pi2 = Mig(h2– h1) RTln Simplify and re-arrange: ( ) pi1 pi2 Mig RT = (h2-h1) ln [ ] Mig(h2-h1) RT ( ) pi1 pi2 e = End 2

6. Derivation of the gravitational isotope effect In mixtures of high and low mass gases, the pressure gradient with height depends on mass, and thus the stable gas column differs in mixing ratios of the components from top to bottom Mig(h2-h1) RT ( ) pi1 pi2 e = Easily applied to problems where your atmosphere is ‘pinned’ at the top or bottom Reference partial pressure Reference partial pressure Reference height Low Mj h h High Mi High Mi Low Mj Reference height pi pi

7. Derivation of the gravitational isotope effect Turning this into familiar units Mig(h2-h1) RT ( ) ( ) pi1 pj1 pi1 pi2 e ai/j1-2 = = = ( ) ( ) pi2 pj2 Mjg(h2-h1) RT pj1 pj2 e (Mi-Mj)g(h2-h1) RT Note Ri = [i]/[j] e ai1-2 = (Mi-Mj)g(h2-h1) RT = ~ ln(ai1-2) Simplest units M: kg/mol g: m/s2 H: m R: J/molK T: K [ ] (Mi-Mj)g(h2-h1) RT ∆i1-2 1000x

8. The mass law of gravitational fractionation Scales with absolute, not relative mass difference! In this respect, differs radically from most other mass-dependent physical and chemical effects; e.g., the 238U/235U fractionation is about 3x the strength of the D/H fractionation (all other things being equal) [ ] (Mi-Mj)g(h2-h1) RT ∆i1-2 1000x The mass exponent, li/j, = (Mi–Mref)/(Mj–Mref) E.g., for the 16O-17O-18O system,l17/18 = ((16.9991-15.9959)/(17.9992-15.9959) = 0.50077 This is generally ’flatter’ than any other mass-dependent fractionation, and so can lead to relatively large ‘mass anomalous’ effects: Reference lA/B ~ + ∆A Gravitational lA/B RA ~ proportional todAi-j - ∆A RB ~ proportional todBi-j

9. Gravitational site preference? A rational assumption is that there is no such thing as gravitational site preferences because all isotopologues having the same molecular mass presumably have identical gravitational (or centripetal) potential energy. A fun (though presumably imaginary ?) case to consider is the vicinity of a black hole or other extreme gravitational anomaly. In this case, significant differences in gravitational potential energy arise over length scales similar to individual molecules, and the lowest free energy state may involve preferred orientations of molecules relative to their centers of mass. This really can’t be treated with Newton’s laws and probably violates other premises of our equation for the gravitational isotope effect, but if we just plug and chug the smallest radius black holes, with a ‘g’ of ~1014 m/s2, at thermal equilibrium with their surroundings (2.73 K) act on 13C/12C substitution (∆M = 1.0033 x10-3 Kg/mol) in long molecules (~10-9 m), leadinng to a from ’top’ to ‘bottom’ of ~80 (!)

10. Clumped isotope effects of gravitational separation Gravitational separations don’t change clumped isotope anomalies in any system that’s been considered (but there might be exceptions for systems with unusual proportions and mass differences of isotopes) 4 X10-8 random 3 [D2] ([H2]+[HD]+[D2]) settled 2 ∆Z assumed 10,000 m T assumed 273 K g assumed 9.81 m/s2 ∆M based on masses of H2, HD, D2 T-dependent vibrational term ignored (to emphasize gravitation) 1 1.0 1.2 1.4 1.6 1.8 2.0 2.2 [D] ([H]+[D]) X10-4 120 100 80 60 ∆D2 (‰) 40 20 settled random 0 -300 -200 -100 0 100 200 300 dDVSMOW (‰)

11. A first scientifically impactful application: Glacial firn

12. The discovery observation Craig et al., 1988, Science

13. ‘Bubble close off’ appears to modify molecular abundances in trapped gases due to differences in molecular volume, but (hopefully!) doesn’t have an isotope effect Craig et al., 1988, Science

14. A more in-depth follow up of several ice cores Sowers et al., 1989, JGR

15. An analysis of a ‘typical’ firn column R15atmosphere — effectively fixed Snow Firn Z Firn Ice R15of trapped N2 1000.(∆Mi-j)g(∆h2-1) RT ∆1-2 ~ For ∆h = 100 m, T = 260 K, ∆1-2 ~ 0.45 ‰ per 1 g/mol ∆Mi-j (or ~0.9 ‰ for R18 of O2)

16. Something peculiar: ‘spikes’ in 15N at sharp ‘warming’ events. Why? d15N of air trapped in Greenland ice Severinghaus et al., 1998, Nature

17. Recall the thermal, or ‘Soret’ fractionation Thermal separator a T2 T1 R1 R2 = Where a is an empirical constant that depends on the gas mixture (this a is not equal to the fractionation factor, ai-j used elsewhere in this class!!!) Grew, 1952, Chemical Diffusion in Gases

18. What happens when the top and bottom of the gas column have different temperatures? Strictly speaking, our analysis of the equilibrium energetics breaks down. But simply adding the gravitational and Soret effects results in a reasonable (?) approximation of what happens when the gas sorts itself to the quasi-equilibrium state of a vertical column with a T gradient a T2 T1 R1 R2 = a is experimentally measured to be 0.0065 for 15N14N/14N2 in air If the surface warms to 270 when the interior stays at 260, the surface-interior fractionation will be: 0.0065 260 270 = -0.24 a1-2 = So, something like ½ the amplitude of the gravitational effect in typical firn

19. R15atmosphere Snow Firn Z Firn Ice TF-I = TS-F TF-I < TS-F (rapid warming of surface) TF-I > TS-F (rapid cooling of surface) R15

20. There is a time-dependence to this effect because a temperature change at the surface eventually propagates into the firn, and the ‘sorting’ of molecules depends on diffusion Combined thermo-gravitational effect of a 5˚C increase in T at the surface at t = 0 yrs Severinghaus et al., 1998, Nature

21. Maybe this is the accumulation rate effect instead? Note that times of inferred warming are also times of increased accumulation, which might correspond to times of thicker firn, and thus higher gravitational fractionation Severinghaus et al., 1998, Nature

22. 15N/14N effect is stronger than 40Ar/36Ar effect/4. Must be equal for gravitational effect acting alone. Suggests something that scales with m’/m(as Soret effects do) Correct critique: ‘Are you kidding me? Those two curves are almost identical!’

23. Gravitational separation of isotopes in the ‘heterosphere’ colder warmer Heavy molecule Light molecule z z Pz RH Base of the mixed layer (lower atmosphere)

24. Basic idea of fractionations during ‘non-thermal’ escape Species stripped from top by various reaction mechanisms, allowing gravitational effect to be expressed as net a Gravitational Effect (less w/ eddy diffusion) Ri(heavy/light)

25. Gravitational and ‘thermal’ escape fractionations generally reinforce each other Thermal escape effect Gravitational Effect (less w/ eddy diffusion) Ri(heavy/light)

26. Atmospheric escape on Mars D/H 15N/14N Roughly 5x Earth Roughly 1.7x Earth Roughly earth like

27. Nitrogen isotope data for Mars 15N/14N of N2, vs. terrestrial air Bieman et al., 1976 (GCMS): 740 ±1360 ‰ Nier et al., 1976 (neutral MS): 740 ±270 ‰ Viking final estimate (1992) 629±62 ‰ Curiosity rover 582±40 ‰ Meteorites Shergottite trapped gas: ≤300 ‰ Chassigny (interrior?): terrestrial ALH84001 (ancient atm.?) terrestrial

28. Simplest analysis of non-thermal escape of N2 ∆M = 0.001 Kg/mol g ~ 3.7 m/s2 ∆Z (turbopause to exobase): ~100,000 m R = 8.314 T of upper atmosphere ~ 200 k (hotter at top) Solve for a = 0.8 R = R0xF(a-1) (the Rayleigh law; we have seen this but will derive it later, when discussing fractionation networks)

29. Box-model view of isotopic budget of atmosphere during escape (a ‘teaser’ of the sorts of models we’ll start making in 2-3 weeks) Escape flux of isotopically light gas J2 aUA-esc Upper atmosphere (UA) MUA, RUA Flux of bulk atmosphere to upper atmosphere J1 Return flux of residual, heavy gas J3 Lower atmosphere (LA) MLA, RLA MLA >> MUA dMLA/dt= J3 – J1 RLAt1 = (MLAt0•RLAt0 - J1•RLAt0 + J3•RUAt0)/MLAt1 J1 = J2 + J3 Mass balance J1•RLA = J2•RUA•aUA-esc + J3•RUA Isotope balance Generally treated as a Rayleigh distillation that acts on MLA with aUA-esc

30. R/R0 F (fraction remaining) 75 % enrichment ~ 0.075 F Today N2 is 2.5 % of 6 mb, or 0.15 mb Thus original N2 must have been ~2 mbar

31. Recall ambiguities/complexities we noted re. atmospheric escape: • T in upper atmosphere and at exobase, which determines strength of gravitational isotope effect and which species can leave and how sharp is the mass cut off (and thus how strong the fractionation) • Relative importance of gravitational separation in upper atmosphere vs. thermal escape at exobase • Thickness of lower and upper atmospheres vary with total atmospheric P and radiative budget, both of which effect T within heterosphere and at exobase and thickness of the heterosphere Jeans Escape: a synthesis of these effects for idealized atmosphere. Has a complex derivation that we won’t go into. Just know it exists.

32. The complicated version in the literature is basically the same as our analysis Nier et al., 1976

33. Geomorphologic evidence clearly suggests the surface has contained much more than 1 mm of precipitable H2O in its past.

34. Geomorphological records of deep standing water The Eberswalde delta Malin and Edgett, 2003

35. Hydrogen isotope data for Mars dD of H2Ov or other H sources (vs. SMOW) IR Spectroscopy (1988; Keck) 5000±3000 ‰ IR Spectroscopy (1989; Kuiper) 4200±200 ‰ IR Spectroscopy (2003-2007; various) 1500-8000 ‰ (varying with season) Meteorites MS Shergottites: ≤4358 ‰ MS Chassigny (interrior?): ≤900 ‰ MS ALH84001 (ancient atm.?) ≤ 780 ‰ “IR” here refers to IR absorption spectra gathered by satellite or earth-based telescope; “MS” is mass spectrometry by any number of methods

36. Placing the Mars atmosphere in the broader solar system context

37. For Hydrogen, thermal escape occurs and multiple species are important (H and H2) so real solution is complex function of physics and chemistry • Sources of H and H2: H2O + hn = H2 + O2 H2 + hn = 2x H This photochemistry might have isotope effects of a type we’ll discuss later in class. Thought to be unimportant for this problem • H and H2 diffuse into upper atmosphere, become gravitationally fractionated, and undergo thermal escape • Gravitational component of net fractionation same as N2 (alpha of roughly 0.8, top/ bottom) • Thermal effect very big : average vHD/vH2 = ~0.82; average vD/vH ~ 0.71. Expression as fractionation depends strongly on T and proportions of H and H2, but clearly large • When all effects are combined, est. a is 0.32 (Yung et al.). This is mostly thermal effect on H

38. Evolution of Mars H pool assuming escape with an a of 0.32 R/R0 F (fraction remaining) 6x enrichment ~ 0.07 F. Today H2O in atmosphere is , or 100 µm, precipitable This could only be residue of loss from a negligibly small initial reservoir (~1 mm, precipitable layer) And we know modern escape rate is so high that this could be achieved in a few thousand years, which seems really, really recent

39. The answer must be a much larger reservoir of H2O, which both prevents D/H from skyrocketing in a few thousand years and explains geomorphologic evidence Revised box model view of the hydrogen problem Escape flux of isotopically light gas J2 aUA-esc MUA, RUA Flux of bulk atmosphere to upper atmosphere J1 Return flux of residual, heavy gas J3 MLA, RLA Fluxes between some big reservoir J4,5 Mmysteryreservoir, Rmysteryreservoir

41. Hydrodynamic atmospheric escape The escape processes discussed so far are quasi-steady-state phenomena that can occur throughout a planet’s history. Here is an alternative one that can only occur on terrestrial planets early in their history. Core idea: a primitive atmosphere of H2 flows into space in response to heating. Heavy constituents are caught up in this flow. Though the Peclet number is big, the fact that this is happening in a gravity well lets there be some gravitational mass fractionation of the minor, heavy constituents.

42. Hydrodynamic atmospheric escape • Continuity of flux, combined with pressure (i.e., density) gradient leads to accelerating velocity of bulk flow with increasing altitude • Drag force up balances gravity for a critical mass; lighter masses have positive velocity and leave with departing atmosphere at some rate that scales inversely with mass; all heavier ones settle (have a lower scale height and cannot escape). These are stratified in atmosphere during process, but don’t actually leave and end up non- fractionated in residual atmosphere Flow of H2 is presumed to be supported by reduction of water (photochemicallyor by heterogeneous reaction in crust)

43. Hydrodynamic atmospheric escape

44. Hydrodynamic atmospheric escape

45. Hydrodynamic atmospheric escape This process can explain relative abundances of noble gases in martian atmosphere, so likely part of the story • Must have controlled isotopic fractionation of Martian Xe • Can’t have controlled Martian H or probably N isotopes(effects too subtle – tens of ‰ per amu) • Might have influenced Ar isotopes (along with non- thermal escape)

46. Gas centrifuges — the agents of modern nuclear energy and arms proliferation Enriched in 235U UF6 in Depleted in 235U Cascade of centrifuge tubes in a gas enrichment plant Photo from the Department of Energy Gas Centrifuge

47. ’Centrifugal’ isotopic fractionations The energy balance is similar to gravitational fractionations, with the complication that centripetal accelerations vary with the radius of rotation whereas g is usually close to constant across natural environments exhibiting gravitational fractionations: • A mass rotating in a circular orbit experiences an acceleration equal to rw2, where r is the radius of the circle and w is the angular velocity (in radians/s) • The work needed to move a mass along the radial direction equals mass x acceleration x distance. So, in place of the Migh term in the gravitational problem, we should see something like a Miw2r2 term in the centrifugal problem. Because the acceleration decreases as you approach the center of rotation, the change in centripetal potential energy for moving from the edge to the center (the integral of [(Force)dr] is (Miw2r2)/2 (whereas the gravitational potential energy change for moving up a column with constant g is just Mig∆h). Centrifugal force Radial motion Gas pressure effective provides balancing centripitalforce

48. ’Centrifugal’ isotopic fractionations As with the gravitational problem, at equilibrium, the gradient in centripetal potential energy is offset by a difference in RTln(pi). Skipping the intermediate steps from our preceding proof, this results in the following fractionation between the center and edge of a rotating gas filled cylinder: Simplest units M: kg/mol w: radians/s2 r: m R: J/molK T: K [ ] (Mi-Mj)w2r2 2RT ∆i1-2 1000x Note 1 RPM = (2p/60) radians/s ~ Modern gas centrifuge: 90,000 RPM, 0.073m radius. For the 0.003 kg/mol mass difference between 235U and 238U, a = 0.715 (285 ‰) at 300 K One of the most powerful things about this approach is that the strength of the fractionation depends on the absolute mass difference between the isotopes, not on the relative difference. So, the fact that UF6 weighs 352 amu doesn’t matter.

49. Gas centrifuges — the agents of modern nuclear energy and arms proliferation 1919: Idea first suggested and physical analysis presented. No practical experiment 1934: Jesse Beams at the University of Virginia separates 35Cl from 37Cl in Cl2 gas using the first functional gas centrifuge 1941: Manhattan project gives it a try to separate 235UF6 from 238UF6 1944: Attempt is abandoned because the ball bearings on their mechanical centrifuges keep failing due to overheating and deformation 1945-1956: Nazi captives and emigrees from eastern Europe, working with Soviet engineers, make the first ‘Zippe-type’ centrifuge. Magnetic bearings and magnetic pulse drive permit very high RPM with negligible solid-solid contacts and so low friction 1956: Zippe released by Soviets; comes to the US and builds magnetic centrifuges at the University of Virginia 1980’s: Various countries replicate Zippe’s design, most significantly Pakistan But it’s hard to do right! The Iraqi’s tried to do this in the late 80’s and early 90’s using graphite cylinders produced using machining hardware purchased from Switzerland. We now know that they wouldn’t go above 3000 RPM. A = 0.99968 (0.32 ‰). Totally useless.

50. The path to cheap modern nuclear proliferation 1989-2004: Abdul Kahn (Pakistan) sells plans to N. Korea, Iran and Libya. Whereas in the 1940’s only a massive industrial nation could marshal the power and hardware to make Kg quantities of 235U, now basically anyone can do it in a basement, provided they can get the magnetic bearings to work. 2006: Reports leak out that Iran’s centrifuges are exploding from ‘fingerprints on rotating parts’. This is now understood to have been cyber espionage that hacked their centrifuge control systems A rogue’s gallery of Iranian gas centrifuges