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. Most of 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 est n more than one odaton 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 eplore 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, ecept n materals affected or produced by bologcally processes, where fractonatons are generally larger. Nevertheless, some geologcally useful nformaton has been obtaned from sotopc study of these metals and eploraton of ther sotope geochemstry contnues. Stable sotopes can be appled to a varety of problems. One of the most common s geothermometry. Another 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 eample, we can use oygen sotope ratos n gneous rocks to determne whether they have assmlated crustal materal. The etent of sotopc fractonaton vares nversely wth temperature: fractonatons are large at low temperature and small at hgh temperature. The d 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, d, from some standard. Oygen sotope fractonatons are generally reported n perml devatons from SMOW (standard mean ocean water): d 18 O = È Î (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, dd, 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. Unfortunately, a dual standard has developed for reportng O sotopes. Isotope 190 4/2/03

2 Table Isotope Ratos of Stable Isotopes Element Notaton Rato Standard Absolute Rato Hydrogen dd D/H ( 2 H/ 1 H) SMOW Lthum d 6 L 6 l/ 7 L NBS L-SVEC Boron d 11 B 11 B/ 10 B NBS Carbon d 13 C 13 C/ 12 C PDB Ntrogen d 15 N 15 N/ 14 N atmosphere Oygen d 18 O 18 O/ 16 O SMOW, PDB d 17 O 17 O/ 16 O SMOW Chlorne d 37 Cl 37 Cl/ 35 Cl seawater ~ Sulfur d 34 S 34 S/ 32 S CDT ratos of carbonates are reported relatve to the PDB carbonate standard. Ths value s related to SMOW by: d 18 O PDB = d 18 O SMOW Table 26.1 lsts the values for standards used n stable sotope analyss. The Fractonaton Factor An mportant parameter n stable sotope geochemstry s the fractonaton factor, a. It s defned as: a A- B R A R B 26.3 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 = d A d B. The relatonshp between and a s: (a - 1)10 3 or 10 3 ln a 26.4 We derve t as follows. Rearrangng equ. 26.1, we have: R A = (d A )R STD / where R denotes an sotope rato. Thus a may be epressed as: a = (d A +103 )R STD /10 3 (d B )R STD /10 = (d A +103 ) 3 (d B ) Subtractng 1 from each sde and rearrangng, we obtan: a -1= (d A - d ) B (d B (d A - d B ) = D snce d s generally << The second equaton n 26.4 results from the appromaton that for 1, ln 1. As we wll see, a s related to the equlbrum constant of thermodynamcs by a A-B = (K/K ) 1/n 26.8 where n s the number of atoms echanged, K s the equlbrum constant at nfnte temperature, and K s the equlbrum constant s wrtten n the usual way (ecept 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 epected (snce for eample, we can readly understand that a lghter sotope wll dffuse faster than a heaver one), but the latter may be somewhat surprsng. After all, we /2/03

3 have been taught that oygen s oygen, 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 n vbratonal motons. These effects are, as one mght epect, generally small. For eample, the equlbrum constant for the reacton 1 2 C16 O 2 + H 18 2 O = 1 2 C18 O 2 + H 16 2 O 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). s only about 1.04 at 25 C. Fgure 26.1 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 mamum when potental energy s at a mnmum). The datomc oscllator, for eample consstng of an Na and a Cl on, works n an analogous way. At small n /2/03

4 teratomc 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 n 0. The system wll have an energy of 1 / 2 hn 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 the net lecture. We wll no 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 echange 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 molecules havng nternal energy E to those havng the zero pont energy E 0 s: n = g e -E / kt 26.9 n 0 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: Â Â n E n Âg E e -E / kt = Âg 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 /2/03

5 q = g e -E Â / kt Substtutng nto 26.10, we can rewrte n terms of the partal dervatves of q: lnq E = kt 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 Rlnq T 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: DU - TDS = -Rln q N where 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 echange reacton), so that DG = -Rln q N The Gbbs Free Energy change s related to the equlbrum constant, K, by: DG = -RT lnk so the equlbrum constant for an sotope echange reacton s related to the partton functon as: K = q N For eample, n the reacton nvolvng echange of 18 O between H 2 O and CO 2, the equlbrum constant s smply: K = q C 16 O q H 2 18 O q C 18 O q H 2 16 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 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 /2/03

6 .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. Translatonal Partton Functon Wrtng a verson of equaton for translatonal energy, q trans s epressed as: q trans = g tr, e -E tr, Â / kt Now all that remans s to fnd and epresson 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 bo 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 Ú 2 kt dn = 2pmkT d h gves an epresson for q trans for each dmenson. The total three-dmensonal translatonal partton functon s then: q trans = ( 2pmkT )3/2 V h 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 26.19, all terms n ecept mass wll eventually cancel.) If translaton moton were the only component of energy, the equlbrum constant for echange 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: K tr = q tr K tr reduces to the product of the molecular masses rased to the stochometrc coeffcent tmes threehalves: 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: 2 j( j +1)h E trans = 8p 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: 195 4/2/03

7 m = m 1 m 2 m 1 + m A datomc molecule wll have two rotatonal aes, one along the bond as, 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 ecept 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 /8p 2 I kt  Agan the spacng between energy levels s relatvely small (ecept for hydrogen) and may be evaluated as an ntegral. For a datomc molecule, the partton functon for rotaton s gven by: q rot = 8p 2 IkT sh where s 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 = 8p 2 (8p 2 ABC) 1/ 2 (kt) 3 / sh 3 where A, B, and C are the prncpal moments of nerta of the molecule and s 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 ecept 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: Ê I K tr = ˆ Á Ë s 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 appromaton n most cases). In ths case the quantum energy levels are gven by: Ê E vb = n + 1 ˆ Á hn Ë 2 where n s the vbratonal quantum number and n 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. For datomc molecules the summaton s smply equal to: /2kT e-hn q vb = e -hn /2kT For a non-lnear polyatomc molecule consstng of atoms and the product s performed over all vbratonal modes, l (there are only 3-5 modes of moton for a lnear polyatomc molecule, hence the product s carred out only to 3-5): 196 4/2/03

8 3-6 e -hn l /2kT q vb = e -hn l /2kT l At room temperature, the eponental term n the denomnator appromates to 0, and the denomnator therefore appromates to 1, so the relaton smplfes to: q e -hn /2kT Thus at low temperature, the vbratonal contrbuton to the equlbrum constant appromates to: K vb = e - l hn l /2kT l whch has an eponental temperature dependence. The full epresson for the equlbrum constant calculated from partton functons for datomc molecules s then: Ê K = q tr q rot vb ( q ) È 3/2 I = M Í e-hn /2kT ˆ Á Î s 1- e -hn Ë /2kT By use of the Teller-Redlch spectroscopc theorem*, ths equaton smplfes to: -U /2 Ê 1 3/2 e ˆ K = Á m Ë s 1- e -U where m s the mass of the sotope echanged and U s defned as: U = hn kt = hcw kt and w s the vbratonal wave number. Eample 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 echange 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 comple model, such as CO 2 would complcate the calculaton because there are more vbratonal modes possble. Valley (2001) provdes an eample of the calculaton for more comple molecules. The equlbrum constant s: K = [16 O 2 ][C 18 O] [ 18 O 16 O][C 16 O] 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, a, s defned as: * 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: 3/2 3/2 Ê m 2 ˆ I1 Ê M 1 ˆ Á Á = U 1 Ë m 1 I 2 Ë M 2 U 2 where m s the sotope mass and M s the molecular mass. We need not concern ourselves wth ts detals /2/03

9 We must also consder the echange reacton: 18 O 18 O + 16 O 16 O = 2 16 O 18 O a = (18 O / 16 O) CO ( 18 O / 16 O) O for whch we can wrte a second equlbrum constant, K 2. It turns out that when both reactons are consdered, a 2K. The reason for ths s as follows. The sotope rato n molecular oygen s related to the concentraton of the 2 molecular speces as: Ê 18 Oˆ [ 18 O 16 O] Á Ë 16 O [ 18 O 16 O]+ 2[ 16 O 2 ] O 2 = ( 16 O 2 has 2 16 O atoms, so t must be counted twce) whereas the rato n CO s smply: Ê 18 Oˆ Á = [C18 O] Ë 16 O [C 16 O] CO Lettng the sotope rato equal R, we can solve for [ 18 O 16 O]: and substtute t nto 26.42: K = (1- R O 2 )[C 18 O] 2R O2 [C 16 O] 198 4/2/ [ 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 R CO 2R O2 = a 2 We can calculate K from the partton functons as: = 2 (1- R O 2 )R CO 2R O K = q q 16 O 2 C 18 O q q O 16 O 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 Ê = M M 16 O 2 C ˆ 18 O Á M M Ë 18 O 16 O C 16 O 3/2 Ê 32 30ˆ = Á Ë /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 epresson for the rato of rotatonal partton functons, all terms cancel ecept the moment of nerta and the symmetry factors. The symmetry factor s 1 for all the molecules nvolved ecept 16 O 2. In ths case, the terms for bond length also cancel, so the epresson nvolves only the reduced masses. So the epresson for the rotatonal equlbrum constant becomes: K tr = q 16 O 2 q C 18 O q 18 O 16 O q C 16 O Ê I 16 I O = 2 C ˆ 18 O Á 2I I Ë 18 O 16 O C 16 O 3/2 Ê ˆ = 1 Á Á Á Ë /2 =

10 (gnore the 1/2, t wll cancel out later). If rotatonal were the only mode of moton, 18 O would be 8 per ml more abundant n O 2. The vbratonal equlbrum constant may be epressed as: -h(n 16 O2 +n C 18 O -n C 16 O -n 18 O 16 O ) K vb = q q 16 O 2 C 18 O 2KT q q = e K vb = e - l hn l 18 /2kT O 16 O C 16 O l Snce we epect the dfference n vbratonal frequences to be qute small, we may make the appromaton e = + 1. Hence: K h 2KT [{ n -n C 16 O C 18 O}-{ n 16 -n 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: n = 1 2p k m where k s the forcng constant, and depends on bond strength, and wll be dentcal for all sotopes of an element. In ths case, we may wrte: n C 18 O = n C 16 O m C 18 O m C 16 O = n C 16 O = 0.976n C 16 O A smlar epresson may be wrtten relatng the vbratonal frequences of the oygen molecule: n16 O 18 O = n16 O2 Substtutng these epressons n the equlbrum constant epresson, we have: K vb =1+ h ( 2kT n [ ]-n [ ] 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 = T At 300 K (room temperature), ths evaluates to We may now evaluate the total echange equlbrum constant at 300 K as: K = K tr K rot K vb = = 2 2 Snce a = 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 appromatons hold only at low temperature). Ths temperature dependence s llustrated n Fgure Thus CO would be 24 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 /2/03

11 1000 lna Qtz H2O T ( C) b quartz a quartz /T2 K-2 Fgure Calculated temperature dependences of the fractonaton of oygen between water and quartz. From Kawabe (1978). 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 coestng 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, whch for large N appromates 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 eample, 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 epermental determnaton of fractonaton therefore dffcult. Fgure 26.3 shows the calculated fractonaton factor between quartz and water as a functon of temperature. REFERENCES AND SUGGESTIONS FOR FURTHER READING Chacko, T., D. Cole and J. Horta, Equlbrum oygen, hydrogen and carbon sotope fractonaton factors applcable to geologc systems, n J. W. Valley (ed.), Stable Isotope Geochemstry, pp. 1-61, Bgelesen, J. and M. G. Mayer, Calculaton of equlbrum constants for sotopc echange reactons, J. Chem. Phys., 15: , a T, K Fgure Fractonaton factor, a= ( 18 O/ 16 O) CO / ( 18 O/ 16 O) O2, calculated from partton functons as a functon of temperature /2/03

12 Broecker, W. and V. Oversby, Chemcal Equlbra n the Earth, McGraw-Hll, New York, 318 pp., Kawabe, I., Calculaton of oygen sotopc fractonaton n quartz-water system wth specal reference to the low temperature fractonaton, Geochm. Cosmochm. Acta, 42: , O'Nel, J. R., Theoretcal and epermental 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: , /2/03