Isotope Effects on Vapor Pressure Curves

1. GE 140a 2019 Lecture 10 VAPOR PRESSURE ISOTOPE EFFECTS

2. Vapor pressure curves Note those for water and ‘heavy’ (deuterated) water are different

3. Clausius-Clapeyron Relation — the first-order description of vapor pressure curves Difference in vapor pressure between two temperatures: The heat of vaporization can be thought of as the energy cost of breaking an intermolecular bond in the condensed phase The essential question is whether the heat of vaporization is a function of isotopic composition — i.e., do bond strengths differ among intermolecular bonds involving the various isotopologues

4. Early (pre- and early-quantum mechanics) expectations about isotope effects on vapor pressure curves Chemical bonds as classical springs Boltzmann-Maxwell description of a gas: mean velocity depends on T and mass; mean energy depends only on temperature. 1/2kx2 + 1/2mv2 = E (energy is exchanged between potential in spring and kinetic energy of masses) w = (k/m)1/2 (frequency of vibration is function of spring constant and mass) Etot= 1/2kxmax2 (but the total energy of the system depends on spring constant and maximum ‘stretch’ of the spring, but not on mass) Velocity distribution for mono-isotopic ideal gas at various T’s K = 3/2kT where k = R/NA K = 1/2mv2 No isotope effect here either No isotope effects on mean energy of a bulk system Thus, in a classical world, energy of evaporation, sublimation and condensation should be independent of isotopic mass

5. 20Ne and 22Ne ices have different vapor pressures T (K) 20 25 50 acond.-vapor 20Ne 22Ne T-1 Similarly, liquid water boiling points: H2O = 373.15 K; D2O = 374.55 K H2 boiling points: H2 = 20.65 K; D2 = 23.45 K So isotopes must differ in heat of vaporization; greater for heavy isotopes (at least so far…)

6. Easily explained through an isotope effect on vibrational frequencies, and therefore energies of interatomic bonds in the condensed phase • K.E. equal or 20Ne and 22Ne • No interatomic bonds • Energies of interatomic bonds scale as hn • When reduced mas rises, n and E drop System energy 20Ne—20Ne 20Ne—22Ne 22Ne—22Ne

7. The Lindemann model A first-approximation based on the Debye-Einstein treatment of condensed phase ‘phonons’ 2 3 40 q T m’-m m’ ln(a) = 1/2 K m 1 2p hnm k vm = q = Where K is the spring constant of the condensed phase bonds and m and m’ are isotopologue masses Rcondensed Rvapor a = R = n’/n

8. Some vapor pressure isotope effects well described by Lindemann theory T (K) 20 100 25 50 36Ar 40Ar liquid ice 20Ne 22Ne liquid acond.-vapor ice T-1 Others examples where it does well 15N2 14N2 13C16O 12C16O 15N16O 14N16O 12C18O 12C16O 15N14N 14N2 14N18O 14N16O 15N18O 14N16O

9. Oops Some simple molecules have ‘reversed’ vapor pressure isotope effects 10BF3 11BF3 acond.-vapor 36Ar 40Ar liquid ice T-1 Other examples showing reversed fractionation 13CH4 12CH4 16O13C16O 16O12C16O 13CCl4 12CCl4 In general, this is important for isotopic substitution of a central atom in a symmetric molecule, and for hydrocarbons

10. The madness of hydrocarbon VPIE’s Many hydrogen-bearing molecules have ‘cross-overs’ (reversed fractionations at some temperatures but not others) and generally complex temperature dependence. Recall that when we encountered this in heterogeneous equilibria the explanation was that different fundamental modes contributed different temperature dependencies to each b value

11. Adsorption of CO2 at Mars-surface conditions is an example of a ‘reversed’ VPIE relevant to natural problems Temperature (K) 200 175 150 125 Ice, O Sorbate, O Liquid, O Ice, C Liquid, C Sorbate, C (hydrocarbon lakes on Titan is another obvious case where this matters) Eiler et al., 2000; Rahn and Eiler, 2001

12. Why does this weird thing keep happening? Condensed phases often have slower intramolecular frequencies than gases

13. This results in a difference between vapor and condensate in contributions to Q’/Q arising from intramolecular motions — in the opposite direction of the ‘normal’ VPIE µ µ’ The frequency drop of a given fundamental mode is proportional to the frequency of that mode 1/2 n-n’ = n.(1- ) The energy drop for a given fundamental mode just scales as the frequency drop, and thus is also proportional to n Ei - Ei’ = [(ni+1/2)h].(n-n’) The reduced masses of each intramolecular fundamental mode are the same in the gas and condensed phase (to first order…) Thus, heavy isotopes will be preferentially concentrated in the phase with the higher frequencies of the intramolecular fundamental modes. Often this is the gas.

14. This is a nuanced point, so let’s say it again but with cartoons Intramolecular bond in vapor – high n; big ∆Ei; good place to put heavy isotope 11B Intermolecular motion term Intramolecular motion term System energy 11B 10BF3 11BF3 Intramolecular bond in condensate; low n; low ∆Ei; bad place to put heavy isotope System energy 10BF3 11BF3

15. Two more terms that tend to push heavy isotopes toward the vapor, counter-acting ‘normal’ VPIE’s Slows the ‘rovibrational’ coupled motions •Hindered rotations in liquids • Differences in London’s forces in the condensed phase due to polarizability Molecule is ‘stickier’ when in its asymmetric stretched position. Isotopologues with a heavy central molecule spend less of their time in this state, and so have higher vapor pressures

16. A ‘corrected’ form of the Lindemannequation to account for these factors Q is the partition function of the intramolecular motions of the constituent molecules ] [ L-L’ is the change in the binding energy arising from tricky intermolecular force constant terms, like rotational hindering and polarizability. Think of it as a fudge factor hnm k Terms associated with intermolecular bonds having spring constant K q = 1/2 K m 1 2p vm = 2 3 40 q T m’-m m’ L-L’ RT QgQ’l Q’gQl ln(a) = — + ln

17. IR spectrum of water vapor The ‘phonon spectra’ of condensed phases containe many modes that blend together into a continuum. Most models of fundamental lattice vibrations deal with this complexity through rough approximations Raman spectrum of water ice This is an inexact game and can lead to errors larger than measurements of stable isotope compositions of volatiles. Thus, once more theory gives us physical insight but lets us down when it comes time for quantification

18. For this reason, we again turn to experimental or empirical calibrations of fractionations Manometry Isotope exchange HV D2O H2O Thermostat

19. Earth’s atmospheric water vapor distribution (a microwave spectroscopy remote sensing product) Lots Not lots

20. Average vertical gradient in Earth’s atmospheric water vapor

21. The most important vapor-pressure isotope effects: D and 18O for liquid water 1000ln(aA-B) = C1 + 1000(C2/T) + 1,000,000 (C3/T2) D/H, liquid—vapor: C1=52.612; C2=-76.248; C3=24.844 18O/16O, liquid-vapor: C1=-2.0667, C2=-0.4156; C3=1.137 See Table 3.2, page 103 of Criss for others

22. Beware known unknowns of the vapor pressure isotope effect of water 450 Earth’s stratosphere, Mars, most of the early solar nebula, surfaces of icy moons and comets, etc… 350 D/H VPIE for water ice Ellehøj, ‘11 e(‰) =1000(a-1) 250 150 Matsuo, ‘64, Merlivat, ’67, Johansson, ‘69 180 200 220 240 260 Temperature (K) Adsorbate-vapor VPIE for water is also very important for some environments (because water is so ‘sticky’ on mineral surfaces). It is often guestimated but, as far as I can find, has never been experimentally observed with useful precision.

23. Site preference of the VPIE Not very many systems of natural significance have been explored, but those that have show measurable site-specific isotope effects associated with the VPIE N2O at 184 K av-l = 0.9984 av-l = 0.9992 av-l = 0.9978 15N 18O 15N ‘Site preference’ of vapor favors central 15N position by 0.84 ‰ (beyond any site preference in liquid)

24. VPIE’s of clumped isotope species Extensively explored for simple compounds by chemists in the 50’s and 60’s, using manometry, distillation and theory. Often only modest effects are observed: NO at 115 K av-l = 0.94016 av-l = 0.91459 av-l = 0.86363 18O 15N 18O 15N ∆15N18O of vapor is = 4.4 ‰ higher than liquid

25. VPIE’s of clumped isotope species But the multiply deuterated species of volatile organics often exhibit peculiar behaviors Deuterated methane ice and liquid • Crossovers, with reversals at most temperatures • As T rises, approaches ‘rule of the geometric mean’ (clumped species are have a’s that scale with number of substitutions) • But clearly deviate from this rule at most temperatures (favoring ‘clumps’ in vapor) • Bizarre T dependencies, for reasons we’ve discussed

26. VPIE’s of clumped isotope species But the multiply deuterated species of volatile organics often exhibit peculiar behaviors Deuterated Ethanes • All ‘reversed’ • As T rises, approaches ‘rule of the geometric mean’ (clumped species are have a’s that scale with number of substitutions) • But clearly deviate from this rule at all temperatures, but up to 10’s of per mil (favoring ‘clumps’ in vapor) • Bizarre T dependencies, for reasons we’ve discussed C2D6 C2HD5 1,1,1,2 C2H2D4 1,1,2,2 C2H2D4 1,1,1 C2H3D3 1,1,2 C2H3D3 1,1 C2H4D2 1,2 C2H4D2 C2H5D

27. Mass laws of VPIE’s The mass law for l17/18 of liquid water at Earth-surface temperature is rather ordinary 11.4 ˚C Average l17/18 = 0.529 a17l-v 25.3 ˚C 41.5 ˚C a18l-v (compare with 0.5305 for high-T limit thermodynamic effect)

28. But some VPIE’s have peculiar mass laws SF6 ice, liquid and adsorbate 225 K 0 Ice; ~equilibrium Ice; partial equilibrium Adsorption Liquid 300 K M = 0.565 -0.4 d33Scondensate— d33Svapor -0.8 188, 180 K -1.2 155 K 150 K -3.0 -2.0 -1.0 0.0 d34Scondensate— d34Svapor (compare with 0.5152 for high-T limit thermodynamic effect)

29. But some VPIE’s have peculiar mass laws SF6 ice, liquid and adsorbate 0.58 Ice/vapor Sorbate/vapor 0.56 Slope 0.54 lna33 lna34 0.52 Canonical equilibrium Canonical kinetic 0.50 20 40 60 80 1000.ln(a) (‰)

30. Some possible drivers of the exotic mass law of SF6 VPIE’s Intra-molecular motions: • High frequency • Low reduced mass • High b • ‘Reversed’ fractionation (red shifted in ice) Inter-molecular motions: • Low frequency • High reduced mass • Low b • ‘Normal’ fractionation

31. Some possible drivers of the exotic mass law of SF6 VPIE’s Consider additive effects of two modes having different reduced masses, and thus l’s Low reduced mass mode (high l) l33/34 = 0.515 High reduced mass mode (low l) l33/34 = 0.505 d34S What if these modes have opposite signs of alpha? d33S l33/34 = 0.515 l33/34 = 0.505 d34S d33S

32. Another VPIE with a peculiar mass law The wildly aberrant mass law for the methane VPI 8 91 K Liquid Vapors 18-17-16 AMU canonical slope 95 K 4 T-D-H canonical slope 97.5 K 100 K CH3T/CH4 1000ln(p’/p) 0 105 K 110 K 115 K -4 120 K -4 -2 0 2 1000ln(p’/p) CH3D/CH4 Multiple sources; see Jansco, 1980 compilation