Geol. 656 Isotope Geochemistry

Transcription

1 STABLE ISOTOPE THEORY: EQUILIBRIUM FRACTIONATION INTRODUCTION Stable sotope geochemstry s concerned wth varatons of the sotopc compostons of lght elements arsng from chemcal fractonatons rather than nuclear processes. The elements most commonly studed are H, L, B, C, N, O, S, S and Cl. Of these, O, H, C, and S are by far the most mportant. These elements have several common characterstcs: They have low atomc mass. The relatve mass dfference between the sotopes s large. They form bonds wth a hgh degree of covalent character. The elements exst n more than one oxdaton state (C, N, and S), form a wde varety of compounds (O), or are mportant consttuents of naturally-occurrng solds and fluds. The abundance of the rare sotope s suffcently hgh (generally at least tenths of a percent) to facltate analyss. It was once thought that elements not meetng these crtera would not show measurable varaton n sotopc composton. However, as new technques offerng greater senstvty and hgher precson have become avalable (partcularly use of the MC-ICP-MS), geochemsts have begun to explore sotopc varatons of metals such as Mg, Ca, T, Cr, Fe, Zn, Cu, Ge, Mo, T, and Tl. The sotopc varatons observed n these metals have generally been qute small, except n materals affected or produced by bologcally processes, where fractonatons are a lttle larger, but stll smaller than the former group of elements. Nevertheless, some geologcally useful nformaton has been obtaned from sotopc study of these metals and exploraton of ther sotope geochemstry contnues. Stable sotopes can be appled to a varety of problems. One of the most common s geothermometry. Ths use derves from the extent of sotopc fractonaton varyng nversely wth temperature: fractonatons are large at low temperature and small at hgh temperature. Another applcaton s process dentfcaton. For nstance, plants that produce C 4 hydrocarbon chans (that s, hydrocarbon chans 4 carbons long) as ther prmary photosynthetc products fractonate carbon dfferently than to plants that produce C 3 chans. Ths fractonaton s retaned up the food chan. Ths allows us to draw some nferences about the det of fossl mammals from the stable sotope ratos n ther bones. Sometmes stable sotopes are used as 'tracers' much as radogenc sotopes are. So, for example, we can use oxygen sotope ratos n gneous rocks to determne whether they have assmlated crustal materal. The δ Notaton NOTATION AND DEFINITIONS Varatons n stable sotope ratos are typcally n the parts per thousand range and hence are generally reported as perml varatons, δ, from some standard. Oxygen sotope fractonatons are generally reported n perml devatons from SMOW (standard mean ocean water): δ 18 O = (18 O/ 16 O) sam ( 18 O/ 16 O) SMOW ( 18 O/ 16 O) SMOW The same formula s used to report other stable sotope ratos. Hydrogen sotope ratos, δd, are reported relatve to SMOW, carbon sotope ratos relatve to Pee Dee Belemte carbonate (PDB), ntrogen sotope ratos relatve to atmospherc ntrogen, and sulfur sotope ratos relatve to trolte n the Canyon Dablo ron meteorte. Cl sotopes are also reported relatve to seawater; L and B are reported relatve to NBS (whch has now become NIST: Natonal Insttute of Standards and Technology) standards /7/09

2 Table Isotope Ratos of Stable Isotopes Element Notaton Rato Standard Absolute Rato Hydrogen δd D/H ( 2 H/ 1 H) SMOW Lthum δ 6 L 6 l/ 7 L NBS L-SVEC Boron δ 11 B 11 B/ 10 B NBS Carbon δ 13 C 13 C/ 12 C PDB Ntrogen δ 15 N 15 N/ 14 N atmosphere Oxygen δ 18 O 18 O/ 16 O SMOW, PDB δ 17 O 17 O/ 16 O SMOW Chlorne δ 37 Cl 37 Cl/ 35 Cl seawater ~ Sulfur δ 34 S 34 S/ 32 S CDT Unfortunately, a dual standard has developed for reportng O sotopes. Isotope ratos of carbonates are reported relatve to the PDB carbonate standard. Ths value s related to SMOW by: δ 18 O PDB = δ 18 O SMOW Table 19.1 lsts the values for standards used n stable sotope analyss. The Fractonaton Factor An mportant parameter n stable sotope geochemstry s the fractonaton factor, α. It s defned as: α A B R A R B 19.3 where R A and R B are the sotope ratos of two phases, A and B. The fractonaton of sotopes between two phases s often also reported as A-B = δ A δ B. The relatonshp between and α s: (α - 1)10 3 or 10 3 ln α 19.4 We derve t as follows. Rearrangng equ. 19.1, we have: R A = (δ A )R STD / where R denotes an sotope rato. Thus α may be expressed as: α = (δ Α +103 )R STD /10 3 (δ Β )R STD /10 = (δ Α +103 ) 3 (δ Β ) Subtractng 1 from each sde and rearrangng, and snce δ s generally << 10 3, we obtan: α 1= (δ Α δ ) Β (δ Β ) (δ Α δ Β ) = Δ The second equaton n 19.4 results from the approxmaton that for x 1, ln x 1 x. As we wll see, α s related to the equlbrum constant of thermodynamcs by α A-B = (K/K ) 1/n 19.8 where n s the number of atoms exchanged, K s the equlbrum constant at nfnte temperature, and K s the equlbrum constant s wrtten n the usual way (except that concentratons are used rather than actvtes because the ratos of the actvty coeffcents are equal to 1,.e., there are no sotopc effects on the actvty coeffcent). THEORY OF ISOTOPIC FRACTIONATIONS Isotope fractonaton can orgnate from both knetc effects and equlbrum effects. The former mght be ntutvely expected (snce for example, we can readly understand that a lghter sotope wll dffuse 220 4/7/09

3 faster than a heaver one), but the latter may be somewhat surprsng. After all, we have been taught that oxygen s oxygen, and ts propertes are dctated by ts electronc structure. In the followng sectons, we wll see that quantum mechancs predcts that mass affects the strength of chemcal bonds and the vbratonal, rotatonal, and translatonal motons of atoms. These quantum mechancal effects predct the small dfferences n the chemcal propertes of sotopes qute accurately. We shall now consder the manner n whch sotopc fractonatons arse. The electronc structures of all sotopes of an element are dentcal and snce the electronc structure governs chemcal propertes, these propertes are generally dentcal as well. Nevertheless, small dfferences n chemcal behavor arse when ths behavor depends on the frequences of atomc and molecular vbratons. The energy of a molecule can be descrbed n terms of several components: electronc, nuclear spn, translatonal, rotatonal and vbratonal. The frst two terms are neglgble and play no role n sotopc fractonatons. The last three terms are the modes of moton avalable to a molecule and are the cause of dfferences n chemcal behavor among sotopes of the same element. Of the three, vbraton moton plays the most mportant role n sotopc fractonatons. Translatonal and rotatonal moton can be descrbed by classcal mechancs, but an adequate descrpton of vbratonal motons of atoms n a lattce or molecule requres the applcaton of quantum theory. As we shall see, temperature-dependent equlbrum sotope fractonatons arse from quantum mechancal effects on vbratonal motons. These effects are, as one mght expect, generally small. For example, the equlbrum constant for the reacton: Fgure Energy-level dagram for the hydrogen atom. Fundamental vbraton frequences are 4405 cm -1 for H 2, 3817 cm -1 for HD, and 3119 cm -1 for D 2. The zero-pont energy of H 2 s greater than that for HD whch s greater than that for D 2. From O'Nel (1986). 1 2 C16 O 2 + H 18 2 O = 1 2 C18 O 2 + H 16 2 O s only about 1.04 at 25 C. Fgure 19.1 s a plot of the potental energy of a datomc molecule as a functon of dstance between the two atoms. Ths plot looks broadly smlar to one we mght construct for two masses connected by a sprng. When the dstance between masses s small, the sprng s compressed, and the potental energy of the system correspondngly hgh. At great dstances between the masses, the sprng s stretched and the energy of the system also hgh. At some ntermedate dstance, there s no stress on the sprng, and the potental energy of the system s at a mnmum (energy would be nevertheless be conserved because knetc energy s at a maxmum when potental energy s at a mn /7/09

4 mum). The datomc oscllator, for example consstng of a Na and a Cl on, works n an analogous way. At small nteratomc dstances, the electron clouds repel each other (the atoms are compressed); at large dstances, the atoms are attracted to each other by the net charge on atoms. At ntermedate dstances, the potental energy s at a mnmum. The energy and the dstance over whch the atoms vbrate are related to temperature. In quantum theory, a datomc oscllator cannot assume just any energy: only dscrete energy levels may be occuped. The permssble energy levels, as we shall see, depend on mass. Quantum theory also tells us that even at absolute 0 the atoms wll vbrate at a ground frequency ν 0. The system wll have energy of 1 / 2 hν 0, where h s Planck's constant. Ths energy level s called the Zero Pont Energy (ZPE). Its energy depends the electronc arrangements, the nuclear charges, and the postons of the atoms n the molecule or lattce, all of whch wll be dentcal for sotopes of the same element. However, the energy also depends on the masses of the atoms nvolved, and thus wll be dfferent for dfferent for sotopes. The vbratonal energy level for a gven quantum number wll be lower for a bond nvolvng a heaver sotope of an element, as suggested n Fgure Thus bonds nvolvng heaver sotopes wll be stronger. If a system conssts of two possble atomc stes wth dfferent bond energes and two sotopes of an element avalable to fll those stes, the energy of the system s mnmzed when the heavy sotope occupes the ste wth the stronger bond. Thus at equlbrum, the heavy sotope wll tend to occupy the ste wth the stronger bond. Ths, n bref, s why equlbrum fractonatons arse. Because bonds nvolvng lghter sotopes are weaker and more readly broken, the lghter sotopes of an element partcpate more readly n a gven chemcal reacton. If the reacton fals to go to completon, whch s often the case, ths tendency gves rse to knetc fractonatons of sotopes. There are other causes of knetc fractonatons as well, and wll consder them n due course. We wll now consder n greater detal the bass for equlbrum fractonaton, and see that they can be predcted from statstcal mechancs. Equlbrum Fractonatons Urey (1947) and Bgelesen and Mayer (1947) ponted out the possblty of calculatng the equlbrum constant for sotopc exchange reactons from the partton functon, q, of statstcal mechancs. In the followng dscusson, bear n mnd that quantum theory states that only dscrete energes are avalable to an atom or molecule. At equlbrum, the rato of the number of molecules havng nternal energy E to the number havng the zero pont energy E 0 s: n n 0 = g e E / kt where n 0 s the number of molecules wth ground-state or zero pont energy, n s the number of molecules wth energy E and k s Boltzmann's constant, T s the thermodynamc, or absolute, temperature, and g s a statstcal weght factor used to account for possble degenerate energy levels* (g s equal to the number of states havng energy E ). The average energy (per molecule) n a system s gven by the Boltzmann dstrbuton functon, whch s just the sum of the energy of all possble states tmes the number of partcles n that state dvded by the number of partcles n those states: E = n E n g E e E /kt = g e E /kt The partton functon, q, s the denomnator of ths equaton: * The energy level s sad to be 'degenerate' f two or more states have the same energy level E /7/09

5 q = g e E / kt Substtutng nto 19.10, we can rewrte n terms of the partal dervatves of q: lnq E = kt 2 T We wll return to these equatons shortly, but frst let s see how all ths relates to some parameters that are more famlar from thermodynamcs and physcal chemstry. It can also be shown (but we won't) from statstcal mechancs that entropy s related to energy and q by S = U T Rlnq Where R s the deal gas constant and U s the nternal energy of a system. We can rearrange ths as: U TS = Rlnq And for the entropy and energy changes of a reacton, we have: ξ ΔU T ΔS = Rln q n where ξ n ths case s the stochometrc coeffcent. In ths notaton, the stochometrc coeffcent s taken to have a negatve sgn for reactants (left sde of reacton) and a postve sgn for products (rght sde of reacton). The left hand sde of ths equaton s smply the Gbbs Free Energy change of reacton under condtons of constant volume (as would be the case for an sotopc exchange reacton), so that ξ ΔG = Rln q n The Gbbs Free Energy change s related to the equlbrum constant, K, by: ΔG = RT lnk so the equlbrum constant for an sotope exchange reacton s related to the partton functon as: ξ K = q n For example, n the reacton nvolvng exchange of 18 O between H 2 O and CO 2, the equlbrum constant s smply: 1/2 K = q C 16 O 2 1/2 q q H2 18 O q C H2 O O The pont of all ths s smply that: the usefulness of the partton functon s that t can be calculated from quantum mechancs, and from t we can calculate equlbrum fractonatons of sotopes. The partton functon can be wrtten as: q total = q tr q vb q rot e., the product of the translatonal, rotatonal and vbratonal partton functons. It s convenent to treat these three modes of moton separately. Let's now do so. Entropy s defned n the second law of thermodynamcs, whch states: ds = dq rev T where Q rev s heat ganed by a system n a reversble process. Entropy can be thought of as a measure of the randomness of a system /7/09

6 Translatonal Partton Functon Wrtng a verson of equaton for translatonal energy, q trans s expressed as: q trans = g tr, e E tr, / kt Now all that remans s to fnd and expresson for translatonal energy and a way to do the summaton. At temperatures above about 2 K, translatonal energy levels are so closely spaced that they essentally form a contnuum, so we can use a classcal mechancal approach to calculatng the energy. The quantum translatonal energy of a partcle n a cubcal box s gven by: E trans = n 2 h 2 8md where n s the quantum energy level, h s Planck s constant, d s the length of the sde of the cube, and m s mass of the partcle. Substtutng nto and ntegratng: ( ) 1/ 2 q trans = e n 2 h 2 8md 2 kt dn = 2πmkT 0 h d gves an expresson for q trans for each dmenson. The total three-dmensonal translatonal partton functon s then: ( q trans = 2πmkT ) 3/2 h V where V s volume and s equal to d 3. (It may seem odd that the volume should enter nto the calculaton, but snce t s the rato of partton functons that are mportant n equatons such as 19.19, all terms n except mass wll eventually cancel.) If translaton moton were the only component of energy, the equlbrum constant for exchange of sotopes would be smply the rato of the molecular weghts rased to the 3 / 2 power. If we defne the translatonal contrbuton to the equlbrum constant as K tr as: ξ K tr = q tr K tr reduces to the product of the molecular masses rased to the stochometrc coeffcent tmes threehalves: ξ K tr = M 3/ where we have replace m wth M, the molecular mass. Thus the translatonal contrbuton to the partton functon and fractonaton factor s ndependent of temperature. Rotatonal Partton Functon The allowed quantum rotatonal energy states are: E rot = j( j + 1)h2 8π 2 I where j s the rotatonal quantum number and I s the moment of nerta. For a datomc molecule, I= µd 2, where d s the bond length, m s the atomc mass of atom, and µ s reduced mass: µ = m 1 m 2 m 1 + m /7/09

7 A datomc molecule wll have two rotatonal axes, one along the bond axs, the other perpendcular to t. Hence n a datomc molecule, j quanta of energy may be dstrbuted 2j+1 ways because there are two possbltes for every value of j except j = 0, for whch there s only one possble way. The statstcal weght factor s therefore 2j + 1. Hence: q rot = (2 j +1)e j( j+1)h 2 /8π 2 I kt Agan the spacng between energy levels s relatvely small (except for hydrogen) and may be evaluated as an ntegral. For a datomc molecule, the partton functon for rotaton s gven by: q rot = 8π 2 IkT σh where σ s a symmetry factor. It s 1 for a heteronuclear datomc molecule (such as CO or 18 O 16 O), and 2 for a homonuclear datomc molecule such as 16 O 2. Equ also holds for lnear polyatomc molecules wth the symmetry factor equal to 2 f the molecule has a plane of symmetry (e.g., CO 2 ) and 1 f t does not. For non-lnear polyatomc molecules, the partton functon s gven by: q rot = 8π 2 (8π 2 ABC) 1/ 2 (kt) 3 / 2 σh where A, B, and C are the prncpal moments of nerta of the molecule and σ s equal to the number of equvalent ways of orentng the molecule n space (e.g., 2 for H 2 O, 12 for CH 4 ). In calculatng the rotatonal contrbuton to the partton functon and equlbrum constant, all terms cancel except for moment of nerta and the symmetry factor, and the contrbuton of rotatonal moton to sotope fractonaton s also ndependent of temperature. For datomc molecules we may wrte: I K rot = σ In general, bond lengths are also ndependent of the sotope nvolved, so the moment of nerta term may be replaced by the reduced masses. Vbratonal Partton Functon We wll smplfy the calculaton of the vbratonal partton functon by treatng the datomc molecule as a harmonc oscllator (as Fg suggests, ths s a good approxmaton n most cases). In ths case the quantum energy levels are gven by: ξ E vb = n + 1 hν where n s the vbratonal quantum number and ν s vbratonal frequency. Unlke rotatonal and vbratonal energes, the spacng between vbratonal energy levels s large at geologc temperatures, so the partton functon cannot be ntegrated. Instead, t must be summed over all avalable energy levels. Fortunately, the sum has a smple form: for datomc molecules the summaton s smply equal to: q vb = /2kT e hν hν /2kT 1 e For molecules consstng of more than two atoms, there are many vbratonal motons possble. In ths case, the vbratonal partton functon s the product of the partton functons for each mode of moton, wth the ndvdual partton functons gven by For a non-lnear polyatomc molecule consstng of atoms and the product s performed over all vbratonal modes,, the partton functon s gven by: 225 4/7/09

8 3 6 e hν /2kT q vb = e hν /2kT (There are 3-5 modes of moton for lnear polyatomc molecules, hence the product s n carred out to 3-5 for such.) At room temperature, the exponental term n the denomnator approxmates to 0, and the denomnator therefore approxmates to 1, so the relaton smplfes to: q vb e hν /2kT Thus at low temperature, the vbratonal contrbuton to the equlbrum constant approxmates to: K vb = e ξ hν /2kT whch has an exponental temperature dependence. The full expresson for the equlbrum constant calculated from partton functons for datomc molecules s then: K = q tr q rot vb ( q ) ξ 3/2 I = M e ξhν /2kT ξ σ 1 e ξhν /2kT By use of the Teller-Redlch spectroscopc theorem*, ths equaton smplfes to: where m s the mass of the sotope exchanged and U s defned as: U /2 ξ 1 3/2 e K = m σ 1 e U U = hν kt = hcω kt and ω s the vbratonal wave number and c the speed of lght. Example of fractonaton factor calculated from partton functons To llustrate the use of partton functons n calculatng theoretcal fractonaton factors, we wll do the calculaton for a very smple reacton: the exchange of 18 O and 16 O between O 2 and CO: C 16 O + 18 O 16 O = C 18 O + 16 O The choce of datomc molecules greatly smplfes the equatons. Choosng even a slghtly more complex model, such as CO 2 would complcate the calculaton because there are more vbratonal modes possble. Chacko et al. (2001) provde an example of the calculaton for more complex molecules such as CO 2. The equlbrum constant for our reacton s: K = [16 O 2 ][C 18 O] [ 18 O 16 O][C 16 O] * The Teller-Redlch Theorem relates the products of the frequences for each symmetry type of the two sotopes to the ratos of ther masses and moments of nerta: m 2 m 1 3/2 I1 M 1 I 2 M 2 3/2 = U 1 U 2 where m s the sotope mass and M s the molecular mass. We need not concern ourselves wth ts detals /7/09

9 where we are usng the brackets n the unusual chemcal sense to denote concentraton. We can use concentratons rather than actvtes or fugactes because the actvty coeffcent of a phase s ndependent of ts sotopc compostons. The fractonaton factor, α, s defned as: α = (18 O / 16 O) CO ( 18 O / 16 O) O O 18 O + 16 O 16 O = 2 16 O 18 O We must also consder the exchange reacton: for whch we can wrte a second equlbrum constant, K 2. It turns out that when both reactons are consdered, α 2K. The reason for ths s as follows. The sotope rato n molecular oxygen s related to the concentraton of the 2 molecular speces as: 18 O 16 O = O 2 [ 18 O 16 O] [ 18 O 16 O]+ 2[ 16 O 2 ] ( 16 O 2 has 2 16 O atoms, so t must be counted twce) whereas the rato n CO s smply: 18 O 16 O CO = [C18 O] [C 16 O] Lettng the sotope rato equal R, we can solve for [ 18 O 16 O]: and substtute t nto 19.42: K = (1 R O 2 )[C 18 O] 2R O2 [C 16 O] [ 18 O / 16 O] = 2 [16 O 2 ]R O2 1 R O Snce the sotope rato s a small number, the term (1 R) 1, so that: = 2 (1 R O 2 )R CO 2R O K 2 R CO 2R O2 = α 2 We can calculate K from the partton functons as: K = q 16 O 2 q C 18 O q 18 O 16 O q C 16 O where each partton functon s the product of the translatonal, rotatonal, and vbratonal partton functons. However, we wll proceed by calculatng an equlbrum constant for each mode of moton. The total equlbrum constant wll then be the product of all three partal equlbrum constants. For translatonal moton, we noted the rato of partton functons reduces to the rato of molecular masses rased to the 3/2 power. Hence: K tr = q 16 O 2 q C 18 O q 18 O 16 O q C 16 O = M M 16 O 2 C 18 O M M 18 O 16 O C 16 O 3/ = /2 = We fnd that CO would be 12.6 rcher n 18 O f translatonal motons were the only modes of energy avalable. In the expresson for the rato of rotatonal partton functons, all terms cancel except the moment of nerta and the symmetry factors. The symmetry factor s 1 for all the molecules nvolved except 16 O 2. In ths case, the terms for bond length also cancel, so the expresson nvolves only the reduced masses. So the expresson for the rotatonal equlbrum constant becomes: 227 4/7/09

10 K rot = q 16 O 2 q C 18 O q 18 O 16 O q C 16 O I 16 I O = 2 C 18 O 2I I 18 O 16 O C 16 O = = (gnore the 1/2, t wll cancel out later). If rotaton were the only mode of moton, CO would be 8 poorer n 18 O. The vbratonal equlbrum constant may be expressed as: K vb = q q 16 O 2 C 18 O q q = e 18 O 16 O C 16 O h(ν 16 O2 +ν C 18 O ν C 16 O ν 18 O 16 O ) 2KT Snce we expect the dfference n vbratonal frequences to be qute small, we may make the approxmaton e x = x + 1. Hence: K vb 1+ h 2KT [{ ν ν C 16 O C 18 O} { ν 16 ν O 18 2 O 16 O} ] Let's make the smplfcaton that the vbraton frequences are related to reduced mass as n a smple Hooke's Law harmonc oscllator: ν = 1 2π k µ where k s the forcng constant, and depends on the nature of the bond, and wll be the same for all sotopes of an element. In ths case, we may wrte: ν C 18 O = ν C 16 O µ C 16 O µ C 18 O = ν C 16 O = 0.976ν C 16 O A smlar expresson may be wrtten relatng the vbratonal frequences of the oxygen molecule: ν16 O 18 O = ν16 O2 Substtutng these expressons n the equlbrum constant expresson, we have: ( ) K vb =1+ h 2kT ν [ ] ν [ ] C 16 O 16 O 2 The measured vbratonal frequences of CO and O 2 are sec -1 and sec -1. Substtutng these values and values for the Planck and Boltzmann constants, we obtan: K vb = T At 300 K (room temperature), ths evaluates to We may now wrte the total equlbrum constant expresson as: M 16 M O K = K tr K rot K vb 2 C 18 O M M 18 O 16 O C 16 O h 1+ 4πkT Evaluatng ths at 300 K we have: k µ C 16 O 3/2 k µ C 18 O µ 16 µ O 2 C 18 O 2µ µ 18 O 16 O C 16 O k k µ 16 O 2 µ 18 O 16 O /7/09

11 ! T, K Fgure Fractonaton factor, α= ( 18 O/ 16 O) CO / ( 18 O/ 16 O) O2, calculated from partton functons as a functon of temperature. K = = Snce α = 2K, the fractonaton factor s at 300 K and would decrease by about 6 per ml per 100 temperature ncrease (however, we must bear n mnd that our approxmatons hold only at low temperature). Ths temperature dependence s llustrated n Fgure Thus CO would be 23 perml rcher n the heavy sotope, 18 O, than O 2. Ths llustrates an mportant rule of stable sotope fractonatons: The heavy sotope goes preferentally n the chemcal compound n whch the element s most strongly bound. Translatonal and rotatonal energy modes are, of course, not avalable to solds. Thus sotopc fractonatons between solds are entrely controlled by the vbratonal partton functon. In prncple, fractonatons between coexstng solds could be calculated as we have done above. The task s consderably complcated by the varety of vbratonal modes avalable to a lattce. The lattce may be treated as a large polyatomc molecule havng 3N-6 vbratonal modes, where N s the number of atoms n the unt cell. For large N, ths approxmates to 3N. Vbratonal frequency and heat capacty are closely related because thermal energy n a crystal s stored as vbratonal energy of the atoms n the lattce. Ensten and Debye ndependently treated the problem by assumng the vbratons arse from ndependent harmonc oscllatons. Ther models can be used to predct heat capactes n solds. The vbratonal motons avalable to a lattce may be dvded nto 'nternal' or 'optcal' vbratons between ndvdual radcals or atomc groupngs wthn the lattce such as CO 3, and S O. The vbratonal frequences of these groups can be calculated from the Ensten functon and can be measured by optcal spectroscopy. In addton, there are vbratons of the lattce as a whole, called 'acoustcal' vbratons, whch can also be measured, but may be calculated from the Debye functon. From ether calculated or observed vbratonal frequences, partton functon ratos may be calculated, whch n turn are drectly related to the fractonaton factor. Generally, the optcal modes are the prmary contrbuton to the partton functon ratos. For example, for parttonng of 18 O between water and quartz, the contrbuton of the acoustcal modes s less than 10%. The ablty to calculate fractonaton factors s partcularly mportant at low temperatures where reacton rates are qute slow, and expermental determnaton of fractonaton therefore dffcult. Fgure 19.3 shows the cal lnα Qtz H2O T ( C) β quartz α quartz /T2 K-2 Fgure Calculated temperature dependences of the fractonaton of oxygen between water and quartz. From Kawabe (1978) /7/09

12 culated fractonaton factor between quartz and water as a functon of temperature. FRACTIONATION OF SEVERAL ISOTOPES In the example n the prevous secton we consdered only the fractonaton between 18 O and 16 O, and ndeed almost all research on oxygen sotope fracton focuses on just these two sotopes. However, a thrd sotope of oxygen, 17 O, exsts, although t s an order of magntude less abundant that 18 O (whch s two orders of magntude less abundant than 16 O). The reason for ths focus s that, based on the theory we have just revewed, mass fractonaton should depend on mass dfference. The mass dfference between 17 O and 16 O s half the dfference between 18 O and 16 O, hence we expect the fractonaton between 17 O and 16 O to be half that between 18 O and 16 O. In the example of fractonaton between CO and O 2 n the prevous secton, t s easy to show from equaton that through the range of temperatures we expect near the surface of the Earth (or Mars) that the rato of fractonaton factors 17 O/ 18 O should be In the lmt of nfnte temperature, 17 O/ 18 O The emprcally observed rato for terrestral fractonaton (and also wthn classes of meteortes) s 17 O/ 18 O Because the fractonaton between 17 O and 16 O bears a smple relatonshp to that between 18 O and 16 O, the 17 O/ 16 O rato s rarely measured. However, as we saw n Lecture 12, not all O sotope varaton n solar system materals follows the expected mass-dependent fractonaton. Furthermore, we saw that there s laboratory evdence that mass-ndependent fractonaton can occur. Mass ndependent fractonaton has subsequently been demonstrated to occur n nature, and ndeed may provde mportant clues to Earth and Solar System processes and hstory, and we wll return to ths topc later. KINETIC FRACTIONATION Knetc effects are normally assocated wth fast, ncomplete, or undrectonal processes lke evaporaton, dffuson and dssocaton reactons. As an example, recall that temperature s related to the average knetc energy. In an deal gas, the average knetc energy of all molecules s the same. The knetc energy s gven by: E = 1 2 mv Consder two molecules of carbon doxde, 12 C 16 O 2 and 13 C 16 O 2, n such a gas. If ther energes are equal, the rato of ther veloctes s (45/44) 1/2, or Thus 12 C 16 O 2 can dffuse 1.1% further n a gven amount of tme at a gven temperature than 13 C 16 O 2. Ths result, however, s largely lmted to deal gases,.e., low pressures where collsons between molecules are nfrequent and ntermolecular forces neglgble. For the case of ar, where molecular collsons are mportant, the rato of the dffuson coeffcents of the two CO 2 speces s the rato of the square roots of the reduced masses of CO 2 and ar (mean molecular weght 28.8): D 12 CO 2 = µ 13 CO 2 = = D 13 CO 2 µ CO 2 Hence we would predct that gaseous dffuson wll lead to only a 4.4 fractonaton. In addton, molecules contanng the heavy sotope are more stable and have hgher dssocaton energes than those contanng the lght sotope. Ths can be readly seen n Fgure The energy requred to rase the D 2 molecule to the energy where the atoms dssocate s kj/mole, whereas the energy requred to dssocate the H 2 molecule s kj/mole. Therefore t s easer to break bonds such as C-H than C-D. Where reactons go to completon, ths dfference n bondng energy plays no role: sotopc fractonatons wll be governed by the consderatons of equlbrum dscussed n the prevous lecture. Where reactons do not acheve equlbrum the lghter sotope wll be preferentally concentrated n the reacton products, because of ths effect of the bonds nvolvng lght sotopes n the reactants beng 230 4/7/09

13 more easly broken. Large knetc effects are assocated wth bologcally medated reactons (e.g., bacteral reducton), because such reactons generally do not acheve equlbrum. Thus 12 C s enrched n the products of photosynthess n plants (hydrocarbons) relatve to atmospherc CO 2, and 32 S s enrched n H 2 S produced by bacteral reducton of sulfate. We can express ths n a more quanttatve sense. The rate at whch reactons occur s gven by: R = Ae E b /kt where A s a constant called the frequency factor and E b s the barrer energy. Referrng to Fgure 19.1, the barrer energy s the dfference between the dssocaton energy, ε, and the zero-pont energy. The constant A s ndependent of sotopc composton, thus the rato of reacton rates between the HD molecule and the H 2 molecule s: R D = e (ε 1 2hν D )/kt R H e (ε 1 2hν H )/kt or R D R H = e (ν H ν D )h /2kT Substtutng for the varous constants, and usng the wavenumbers gven n the capton to Fgure 19.1 (rememberng that ω = cν where c s the speed of lght) the rato s calculated as 0.24; n other words we expect the H 2 molecule to react four tmes faster than the HD molecule, a very large dfference. For heaver elements, the rate dfferences are smaller. For example, the same rato calculated for 16 O 2 and 18 O 16 O shows that the 16 O wll react about 15% faster than the 18 O 16 O molecule. The greater translatonal veloctes of lghter molecules also allows them to break through a lqud surface more readly and hence evaporate more quckly than a heavy molecule of the same composton. The transton from lqud to gas n the case of water also nvolves breakng hydrogen bonds that form between the hydrogen of one molecule and an oxygen of another. Ths bond s weaker f 16 O s nvolved rather than 18 O, and thus s broken more easly, meanng H 2 16 O s more readly avalable to transform nto the gas phase than H 2 18 O. Thus water vapor above the ocean typcally has δ 18 O around 13 per ml, whereas at equlbrum the vapor should only be about 9 per ml lghter than the lqud. Let's explore ths example a bt further. An nterestng example of a knetc effect s the fractonaton of O sotopes between water and water vapor. Ths s another example of Raylegh dstllaton (or condensaton), as s fractonal crystallzaton. Let A be the amount of the speces contanng the major sotope, H 2 16 O, and B be the amount of the speces contanng the mnor sotope, H 2 18 O. The rate at whch these speces evaporate s proportonal to the amount present: da=k A A 19.62a and db=k B B 19.62b Fgure Fractonaton of sotope ratos durng Raylegh and equlbrum condensaton. δ s the per ml dfference between the sotopc composton of orgnal vapor and the sotopc composton as a functon of ƒ, the fracton of vapor remanng. Snce the sotopc composton affects the reacton, or evaporaton, rate, k A k B. We'll call ths rato of the rate constants α. Then db da = α B A /7/09

14 Rearrangng and ntegratng, we have ln B B = α ln A A or B B = A A where A and B are the amount of A and B orgnally present. Dvdng both sdes by A/A B / A B / A = A A α 1 α Snce the amount of B makes up only a trace of the total amount of H 2 O present, A s essentally equal to the total water present, and A/A s essentally dentcal to ƒ, the fracton of the orgnal water remanng. Hence: B / A B / A = ƒα 1 Subtractng 1 from both sdes, we have B / A B / A B / A = ƒ α Comparng the left sde of the equaton to 26.1, we see the perml fractonaton s gven by: δ = 1000( f α 1 1) Of course, the same prncple apples when water condenses from vapor. Assumng a value of α of 1.01, δ wll vary wth ƒ, the fracton of vapor remanng, as shown n Fgure Even f the vapor and lqud reman n equlbrum throughout the condensaton process, the sotopc composton of the remanng vapor wll change contnuously. The relevant equaton s: δ = 1 1 (1 ƒ)/α + f The effect of equlbrum condensaton s also shown n Fgure REFERENCES AND SUGGESTIONS FOR FURTHER READING Chacko, T., D. Cole and J. Horta, Equlbrum oxygen, hydrogen and carbon sotope fractonaton factors applcable to geologc systems, n J. W. Valley and D. R. Cole (ed.), Stable Isotope Geochemstry, pp. 1-61, Bgelesen, J. and M. G. Mayer, Calculaton of equlbrum constants for sotopc exchange reactons, J. Chem. Phys., 15: , Broecker, W. and V. Oversby, Chemcal Equlbra n the Earth, McGraw-Hll, New York, 318 pp., Kawabe, I., Calculaton of oxygen sotopc fractonaton n quartz-water system wth specal reference to the low temperature fractonaton, Geochm. Cosmochm. Acta, 42: , O'Nel, J. R., Theoretcal and expermental aspects of sotopc fractonaton, n Stable Isotopes n Hgh Temperature Geologc Processes, Revews n Mneralogy Volume 16, J. W. Valley, et al., eds., Mneralogcal Socety of Amerca, Washngton, pp. 1-40, Urey, H., The thermodynamc propertes of sotopc substances, J. Chem. Soc. (London), 1947: , /7/09