Introduction to Aquatic and Marine Geochemistry
Introduction to Aquatic and Marine Geochemistry EPS 103
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3 ABUNDANCE AND FRACTIONATION OF STABLE ISOTOPES In classical chemistry isotopes of an element are regarded as having equal chemical properties In reality variations in isotopic abundances occur far exceeding measuring precision This phenomenon is the subject of this section 31 ISOTOPE RATIOS AND CONCENTRATIONS Before we can give a more quantitative description of isotope effects we must de ne isotope abundances more carefully Isotope abundance ratios are de ned by the expression R 7 abundance of rare isotope 7 31 abundance of abundant 1sotope The ratio carries a superscript before the ratio symbol R which refers to the isotope under consideration For instance 13 18 16 ERGO 1 002 18R CO C 0 0 lt 2 12002 lt 2 CMOZ 32 2 1 18 ZRHOJHHO ISRHOJHZO 2 le0 2 H2160 We should clearly distinguish between an isotope ratio and an isotope concentration For C02 for instance the latter is defined by 13 13 13 CO CO R 13 122 2 13 33933 C02ll C02 C02 1 R Especially if the rare isotope concentration is very large as in the case of labeled compounds the rare isotope concentration is often given in atom This is then related to the isotope ratio by R atom 100 1 7 atom 100 33b 32 ISOTOPE FRACTIONATION According to classical chemistry the chemical characteristics of isotopes or rather of molecules that contain different isotopes of the same element such as 13CO and 12C02 are 31 Chapter 3 equal To a large extent this is true However if a measurement is sufficiently accurate and this is the case with the modern mass spectrometers Chapter ll we observe tiny differences in chemical as well as physical behaviour of socalled isotopic molecules or isotopic compounds The phenomenon that these isotopic differences exist is called isotope fractionation Some authors refer to this phenomenon as isotope discrimination however we see no reason to deviate from the original expression This can occur as a change in isotopic composition by the transition of a compound from one state to another liquid water to water vapour or into another compound carbon dioxide into plant organic carbon or it can manifest itself as a difference in isotopic composition between two compounds in chemical equilibrium dissolved bicarbonate and carbon dioxide or in physical equilibrium liquid water and water vapour Throughout this volume examples of all these phenomena will be discussed The differences in physical and chemical properties of isotopic compounds ie chemical r J of 39 39 39 39 different isotopes of the same element are brought about by mass differences of the atomic nuclei The consequences of these mass differences are twofold 1 The heavier isotopic molecules have a lower mobility The kinetic energy of a molecule is solely determined by temperature kT 12mv2 k Boltzmann constant T absolute temperature m molecular mass v average molecular velocity Therefore molecules have the same 12mv2 regardless of their isotope content This means that the molecules with larger m necessarily have a smaller v Some practical consequences are a heavier molecules have a lower diffusion velocity b the collision frequency with other molecules the primary condition for chemical reaction is smaller for heavier molecules this is one of the reasons why as a rule lighter molecules react faster 2 The heavier molecules generally have higher binding energies The chemical bond between two molecules in a liquid or a crystal for instance or between two atoms in a molecule can be represented by the following simple model Two particles exhibit competing forces on each other The one force is repulsive and rapidly increases with decreasing distance N lr13 The other is attractive and increases less rapidly with decreasing distance in ionic crystals N lrz between uncharged particles N lr7 As a result of these forces the two particles will be located at a certain distance from each other In Fig3l the potential energies corresponding to each force and to the resulting net force are drawn schematically If one particle is located at the origin of the co ordinate system the other will be in the energy well Escape from the well is possible only if it obtains sufficient kinetic energy to overcome the net attractive force This energy is called the binding energy of the particle A simple example is the heat of evaporation 32 Stable Isotope Fractionation potential energy repmsmn n7 I ratgtra tion refquot gt distance between particles Fig31 Schematic representation of the potential energy distribution caused by the repulsive and attractive forces between two particles in this case between oppositely charged ions The resulting potential is wellshaped solid line The one particle is situated at the origin r 0 The second particle is situated in the well The small horizontal lines in the well are the energy levels of the system the thin and heavy lines referring to the light and the heavy isotopic particle respectively The arrows indicate the respective binding energies at the zero temperature T0 and a higher temperature T respectively At higher temperatures the difference between the binding energies for the isotopic particles is smaller resulting in a smaller isotope effect Although the particle is in the energy well it is not at the bottom of the well even at the zero point of the absolute temperature scale 273150C All particles have three modes of motion translation ie displacement of the molecule as a whole vibrations of the atoms in the molecule with respect to each other and rotations of the molecule around certain molecular axes In the energy well representation the particle is not at the bottom of the well but at a certain energy level above the zero energy In Fig 31 an energy level is indicated by a horizontal line The higher the temperature of the substance the higher energy level will be occupied by the particle The energy required to leave the well ie to become separated from the other particle is indicated as the binding energy EB whereas E3 is the binding energy for the rare heavy particle The energy of a certain particle at a certain temperature depends on its mass The 33 Chapter 3 heavier isotopic particle is situated deeper in the energy well than the lighter heavy line in Fig3l and therefore escapes less easily the heavier isotopic particle atom or molecule generally has a higher binding energy 133 gt EB Fig32 normal case Examples of this phenomenon are 1 lelsO and 1HZHMO have lower vapour pressures than 1H2160 they also evaporate less easily and 2 V in most chemical reactions the light isotopic species reacts faster than the heavy For example Ca1 2CO3 dissolves faster in an acid solution than does Cal3CO3 In an isotope equilibrium between two chemical compounds the heavy isotope is generally concentrated in the compound which has the largest molecular weight The depth of the energy well may also depend on the particle masses in a more complicated manner Under certain conditions with polyatomic molecules the potential energy well is deeper for the light than for the heavy isotopic molecule Due principally to this phenomenon the binding energy of the heavy molecule can be smaller Because the isotope effects discussed above generally result in lower vapour pressure for the isotopically heavy species normal isotope effect this effect is referred to as the inverse isotope effect Fig32 Practical examples of the inverse isotope effect are the higher vapour pressure of 13C02 in the liquid phase as well as the lower solubility of 13C02 in water than of 12C02 both at room temperature Vogel et al 1970 At high temperatures the differences between binding energies of isotopic molecules becomes smaller resulting in smaller an ultimately disappearing isotope effects Fig3 1 33 KINETIC AND EQUILIBRIUM ISOTOPE FRACTIONATION The process of isotope fractionation is mathematically described by comparing the isotope ratios of the two compounds in chemical equilibrium A ltgt B or of the compounds before and after a physical or chemical transition process A gt B The isotope fractionation factor is then defined as the ratio of the two isotope ratios RB R713 RA 7 RA 3394 1MB 0 BA which expresses the isotope ratio in the phase or compound B relative to that in A Ifwe are dealing with changes in isotopic composition for instance C is oxidised to C02 the carbon isotope fractionation refers the quotnewquot 13RCOz value to the quotoldquot 13RC in other words not 13RCOz13RC 34 Stable Isotope Fractionation 13339 EB 13339 EB normal effect inverse effect Fig32 Schematic representation of the normal left and the inverse right isotope effect For some interactions the potential energy well for the heavy particle heavy line is less deep than for the light particle thin line Depending on the speci c interaction the binding energy for the isotopically heavy particle can be larger lefthand side normal effect or smaller righthand side inverse effect than for the light particle In general isotope effects are small X m 1 Therefore the deviation of X from 1 is widely used rather than the fractionation factor This quantity to which we refer as the fractionation is defined by R sBA uBAi1iBi1 X1030 35 RA 8 represents the enrichment s gt 0 or the depletion s lt 0 of the rare isotope in B with respect to A The symbols OLBA and sBA are equivalent to OLAB and SAB In the oneway process A gt B s is the change in isotopic composition in other words the new isotopic composition compared to the old Because 8 is a small number it is generally given in 0 per mill equivalent to 1073 Note that we do not de ne g in o as many authors claim they do In fact they do not as they always add the o symbol An 8 value of for instance 50 is equal to 0005 The consequence is that in mathematical equations it is incorrect to use 6103 instead of merely g The student is to be reminded that s is a small number in equations one may numerically write for instance quot 7250 quot instead ofquot 725103 quot 35 Chapter 3 Again the fractionation of B with respect to A is denoted by sBA or SAB From the definition of s we simply derive is SBA 7MB E SAB 36 1sAB the last step because in natural processes the s values are small It is important to distinguish between two kinds of isotope fractionation kinetic fractionation and equilibrium fractionation Kinetic fractionation results from irreversible ie one way physical or chemical processes Examples include the evaporation of water with immediate withdrawal of the vapour from further contact with the water the absorption and diffusion of gases and such irreversible chemical reactions as the bacterial decay of plants or rapid calcite precipitation These fractionation effects are primarily determined by the binding energies of the original compounds Sect32 in that during physical processes isotopically lighter molecules have higher velocities and smaller binding energies in chemical processes light molecules react more rapidly than the heavy In some cases however the opposite is true This inverse kinetic isotope effect occurs most commonly in reactions involving hydrogen atoms Bigeleisen and Wolfsberg 1958 The second type of fractionation is equilibrium or thermodynamic fractionation This is essentially the isotope effect involved in a thermodynamic equilibrium reaction As a formal example we choose the isotope exchange reaction ABltgtAB 37 where the asterisk points to the presence of the rare isotope The fractionation factor for this equilibrium between phases or compounds A and B is the equilibrium constant for the exchange reaction of Eq37 K A BBB R713 A B AlA RA 0 BA 38 If sufficient information about the binding energies of atoms and molecules is available the fractionation effect can be calculated the kinetic effect Bigeleisen 1952 as well as the equilibrium effect Urey 1947 In practice however these data are often not known in sufficient detail With kinetic isotope effects we are confronted with an additional difficulty which arises from the fact that natural processes are often not purely kinetic or irreversible Moreover kinetic fractionation is difficult to measure in the laboratory because i complete irreversibility can not be guaranteed part of the water vapour will return to the liquid nor can the degree of irreversibility be quantified ii the vanishing phase or compound will have a nonhomogeneous and often immeasurable isotopic composition because the isotope effect occurs at the surface of the compound For example the surface layer of an evaporating water 36 Stable Isotope Fractionation mass may become enriched in 18O and 2H if the mixing within the water mass is not rapid enough to keep its content homogeneous throughout Isotope fractionation processes in nature which are not purely kinetic ie oneway processes will be referred to as non equilibrium fractionations An example is the evaporation of ocean or fresh surface water bodies the evaporation is not a oneway process certainly water vapour condenses neither an equilibrium process as there is a net evaporation Equilibrium fractionation on the other hand can be determined by laboratory experiments and in several cases reasonable agreement has been shown between experimental data and thermodynamic calculations The general condition for the establishment of isotopic equilibrium between two compounds is the existence of an isotope exchange mechanism This can be a reversible chemical equilibrium such as H2160 0180160 lt2 H2C1801602 lt2 H2180 01602 39a or such a reversible physical process as evaporationcondensation H2160vapour H21801iquid lt9 H2180vapour H2160liquid 39b The reaction rates of exchange processes and consequently the periods of time required to reach isotopic equilibrium vary greatly For instance the exchange of H20 ltgt C02 proceeds on a scale of minutes to hours at room temperature while that of H20 ltgt 80239 requires millennia The fractionation resulting from kinetic isotope effects generally exceeds that from equilibrium processes Moreover in a kinetic process the compound formed may be depleted in the rare isotope while it is enriched in the equivalent equilibrium process This can be understood by comparing the fractionation factor in a reversible equilibrium with the kinetic fractionation factors involved in the two opposed single reactions As an example we take the carbonic acid equilibrium of Eq39a CO2 H20 Hl HCO3T For the single reactions 12 12 7 CO2 HZO gtH H CO3 and 13CO2 H20 gt H H13CO3 37 Chapter 3 the reaction rates are respectively r 121412002 and 13r 131413002 310 where 12k and 13k are the reaction rate constants The isotope ratio of the bicarbonate formed AHCO339 is 13f 13k13CO2 7 13 13 R AHCO 7 or R CO 311 3 lzr 12k12CO2 k 2 where xk is the kinetic fractionation factor for this reaction Conversely for H H1200 gt 12CO2 H20 and H H13CO3 gt13CO2 H20 the reaction rates are 12r39 121lt39H CO3 and 13r 131lt39H13CO3 312 The carbon dioxide formed ACOZ has an isotope ratio 13 13 v 13 7 v k H CO 13RACOZHQI 13RHCO3 313 r39 k H co3 At isotopic equilibrium the isotope effects balance in other words 13RAHCO3 13RACOZ 314 so that combining Eqs3 11 and 313 results in uk39 1311002 13COZH CO on 13RHCO IZCOZHH13003 a 315 which shows that the isotopic equilibrium fractionation Xe is equivalent to the equilibrium constant of the isotope exchange reaction H1300 12CO2 lt2 H12C03 13002 316 38 Stable Isotope Fractionation Most xk and uk values are less than 1 by more than one per cent Differences between their ratios are less in the order of several per mill Later we will see that rapid evaporation of water might cause the water vapour to be about twice as depleted in 18O as the vapour in equilibrium with the water The reason for this is that H2160 is favoured also in the condensation process From Eq315 it is obvious that while xk for a certain phase transition is smaller than 1 0Le might be larger than 1 An example is to be found in the system dissolvedgaseous C02 the 13C content of C02 rapidly withdrawn from an aqueous C02 solution is smaller than that of the dissolved C02 ie 12C moves faster uklt1 while under equilibrium conditions the gaseous C02 contains relatively more 13C oregt1 inverse isotope effect Sect32 34 THEORETICAL BACKGROUND OF EQUILIBRIUM FRACTIONATION In this section we will present a discussion of the origin of isotope effects in particular the massdependent isotope fractionation based on some principles from thermodynamics and statistical mechanics The treatment here will not be exhaustive and so for full details the reader is referred to textbooks on these subjects A basic discussion has been given by Broecker and Oversby 1971 and Richet et al 1977 A basic principle of quantum physics is that the energy of the particle can take only certain discrete values This is true for any kind of motion These discrete energy values were already mentioned in Sect32 where we discussed the position of a particle at a certain time at a certain level in the energy well Figs31 and 32 A basic principle of statistical mechanics states that the chance that a particle is at a certain energy level Sr is eierkT Pr 7 q where k is the Boltzmann constant and T the absolute temperature of the compound The value of q the partition function is determined by the requirement that the sum of all chances equals unity Zpr r0 so that q Ee ErkT 317 r0 39 Chapter 3 The partition function is thus the summing up of the population of all energy levels of a certain system and so determines the energy state of the system It is an important quantity because thermodynamic considerations show that equilibrium conditions can be expressed as a ratio of partition functions Consider again the isotope exchange reaction of Eq37 A B Q A B where A and B denote two different compounds for instance C02 and HCOg or two phases of the same compound for instance liquid H20 and water vapour and the asterisk refers to the presence of a rare isotope in the molecule 13C 180 2H etc In the definition of the fractionation factor ie the equilibrium constant the concentration or rather the activities are now represented by the partition functions gtxlt gtxlt 0LAB K qA qB qB qB 318 qA qB qAqA We must now evaluate the various partition functions In Sect32 the three modes of motion were mentioned translation rotation and vibration The total partition function of a system of one compound or one phase is related to the partition functions of the different motions by q qtransqrotqvibr Each partition function results from intramolecular internal motions such as vibrations of atoms with respect to each other and rotations of the molecule around molecular axes and intermolecular external motions such as vibrations and rotations of molecules with respect to each other in a crystal lattice for instance Most systems are too complicated to allow an exact calculation of the partition functions The only simple systems are ideal monoatomic or diatomic gases The translations are not hindered by neighbouring atoms or molecules and intermolecular vibrations and rotations do not exist so that only the internal components of the partition functions have to be calculated In order to point out the mass dependence of fractionation factors and to illustrate the in uence of temperature on fractionation we will brie y mention the various partition functions Translation of a molecule as a whole has the partition function 32 271ka qms hz V 320 40 Stable Isotope Fractionation V volume in which the molecules are free to move m is the mass of the molecule h Planck39s constant In a gas the translational contribution to the partition function ratio then is M molar weight injwz jwz 321 q trans m M which is temperature independent The partition function for internal rotational motion of a diatomic molecule is 8112 upsz qrot T 322 where r0 is the equilibrium distance between the two atoms and u is the reduced mass of the molecule mlmzml m2 the energy of rotation is equal to that of a system where u rotates around the centre of mass instead of m1 and m around each other We should further remember that pro2 is the moment of inertia of a rotating mass and that s 1 unless the molecule consists of two equal atoms in which case s 2 In a gas the rotational contribution to the partition function ratio thus is 1 Lufml i 323 q rot H m1MS where ml is the rare isotope of atom m1 This ratio also is independent of temperature The vibrational partition function is eihvZkT qvibr 1 7 6mm 324 where v is the frequency of vibration of the two atoms with respect to each other The frequency is generally known from experimental spectroscopic data The temperature dependence of the fractionation effect can be shown to be caused primarily by the temperature dependence of the vibration The mass dependence of the vibration frequency of the harmonic oscillator is given by 12 1 k 7 t 325 where again u is the reduced mass of the molecule and k is the force constant In a rst order approximation k is not altered by an isotope substitution in the molecule and so 41 Chapter 3 12 1 L 326 At normal temperatures T lt hVk the exponential function in the denominator in Eq324 can be neglected The partition function ratio then is e11v7quotv2kT q vibr or inserting Eq326 q 7 LV 1 qivibreXkaTll ml 328 where u mlmzml ml The partition function ratio of a diatomic gas at normal temperatures is thus obtained by combining Eqs3l9 321 323 and 328 12gtxlt A L4 1116 hv 1 L 329 q M m s 2kT p where m and m refer to the exchanging isotopes The equilibrium fractionation factor between two diatomic gases A and B is given by Eq318 It should be noted that if A or B consists of two equal atoms such as in 02 or C02 the true relation between X and K contains a factor 2 such that the factor 2 arising from ss is cancelled in calculating CL A simple numerical example is given in Sect37 A general approximate expression for the fractionation factor as a function of the temperature is obtained from Eqs318 and 329 a AeBT 330 where the coefficients A and B do not depend on temperature but contain all temperature independent quantities mass Vibration frequency The natural logarithm of the fractionation factor is approximated by the power series 1nd C1ii 331a with the often used approximation for the fractionation s 1n1s C1 CzT 331b 42 Stable Isotope Fractionation It can further be shown that at very high temperatures the vibrational contribution to the fractionation balances the product of the translational and rotational factors so that nally at very high temperatures 0L l and thus isotope effects disappear at suf ciently high temperatures From the foregoing we can draw the following conclusions 1 In a kinetic oneway or irreversible process the phase or compound formed is depleted in the heavy isotope with respect to the original phase or compound OLk lt l theoretical predictions about the degree of fractionation can only be qualitative fast evaporating water is depleted in 180 relative to the water itself 2 V In an isotopic equilibrium reversible process it can not with certainty be predicted whether the one phase or compound is enriched or depleted in the heavy isotope However the dense phase liquid rather than vapour or the compound having the largest molecular mass CaC03 versus C02 usually contains the highest abundance of the heavy isotope 3 V Under equilibrium conditions and provided suf cient spectroscopic data on the binding energies are available Xe may be calculated measurements on Xe of various isotope equilibria will be reported in chapter 6 4 As a rule fractionation decreases with increasing temperature in Chapter 6 this is shown V for instance for the exchange equilibria of C02 ltgt H20 and H201iquid ltgt H20vap0ur At very high temperatures the isotopic differences between the compounds disappear 35 FRACTIONATION BY DIFFUSION As was mentioned in Sect32 isotope fractionation might occur because of the different mobilities of isotopic molecules An example in nature is the diffusion of C02 or H20 through air According to Fick39s law the net ux of gas F through a unit surface area is dC F 7D7 332 dx where dCdx is the concentration gradient in the direction of diffusion and D is the diffusion constant The latter is proportional to the temperature and to lVm where m is the molecular mass This proportionality results from the fact that all molecules in a gas mixture have equal temperature and thus equal average 12mv2 The average velocity of the molecules and thus their mobilities are inversely proportional to Vm Ifthe diffusion process of interest involves the movement of gas A through gas B however m has to be replaced by the reduced mass 43 Chapter 3 H w 333 mA mB see textbooks on the kinetic theory of gases The above equations hold for the abundant as well as for the rare isotope The resulting fractionation is then given by the ratio of the diffusion coef cient for the two isotopic species Furthermore the molecular masses can be replaced by the molar weights M in numerator and denominator D M M M M 0L 7 A B A B 334 D M A MB M A MB In the example of water vapour diffusing through air the resulting fractionation factor for oxygen is MB of air is taken to be 29 MA 18 MA 20 180k 2029 18x29 20x29 1829 12 0969 335 By diffusion through air water vapour thus will become depleted in 18O in agreement with the rules given at the end of Sect34 by 31 o 188 731 o The 13C fractionation factor for diffusion of C02 through air is 13 4529 44x29 0L 12 7 09956 336 45 X 29 44 29 in other words 138 744 o a depletion of 44 o 36 RELATION BETWEEN ATOMIC AND MOLECULAR ISOTOPE RATIOS We want to have a closer look at the meaning of the isotope ratio in the context of the abundance of rare isotopes of an element in polyatomic molecules containing at least two atoms of this element The isotope ratio is the chance that the abundant isotope has been replaced by the rare isotope To be more specific we de ne the atomic isotope ratio Ramm as the abundance of all rare isotopic atoms in the compound divided by the abundance of all abundant isotopic atoms For the sake of clarity we select a simple example the isotopes of hydrogen in water 2H1601H2H1701H2H1801H22H1602H 2 R 337 m 21H1601H21H1701H21H1801H1H1601H 44 Stable Isotope Fractionation The quotsecondorderquot abundances such as for instance 2H2180 in the nominator and 1H2H180 in the denominator have been neglected In this context the molecular isotope ratio Rmol is de ned as the abundance of the rare isotope of a speci c element in a speci c type of molecule divided by the abundance of the abundant isotope In our example of deuterium in water 2H1601H 2 Rmol 7 21H1601H 338 The realistic value of this de nition refers for instance to the measurement of isotope ratios by laser spectrometry see Sect102l2 where the light absorption is compared of two isotopic molecules only differing in one isotope contrary to mass spectrometry Kerstel et al 1999 priVcomm Eq337 can be rewritten as 2H17O1H 2H1801H 22H1502H 21611216121612161 HOH HOHHOH HOH zRatom 1 16 1 1 17 1 1 18 1 1 16 2 339 2HOH HOHHOH HOH 1H1601H 1H1601H 21H1601H The rst factor on the right in Eq339 is the molecular isotope ratio To a very good approximation the molecular isotope ratios in the second factor may be replaced by the atomic isotope ratios 117R18R2R 117R18R2R 2 2 2 Ratom E Rmol Rmol This result means that to a rstorder approximation the atomic isotope ratio is equal to a molecular isotope ratio Although it was shown for one speci c molecule the reasoning is equally valid for the other molecules in Eq339 Also for more complicated molecules the demonstration of the proof is analogous The approximations made are indicated as quot rstorderquot However if we use 5 values the approximation is even better because almost equal approximations occur in the R values of the nominator as well as the denominator of Eq340 cf Sect43 l 37 RELATION BE I WEEN FRACTIONATIONS FOR THREE ISOTOPIC MOLECULES For some isotope studies it is of interest to know the relation between the fractionations for more than two isotopic molecules For instance water contains three different isotopic molecules as far as oxygen is concerned H2160 H2170 and H2180 carbon dioxide consists of 12C02 13C02 as well 14C02 The question now is whether there is a theoretical relation 45 Chapter 3 between the 170 and 180 fractionation with respect to 160 for instance during evaporation and between the 13C and 14C fractionation with respect to 12C for instance during the uptake of CO2 by plants Generally the fractionation for the heaviest isotope is taken twice as large as that for the less heavy isotope 14a lt13OL2 and 18a lt17OL2 341 while 14a 1 148 113s2 1213s13s2 12 13s so that for the C02 uptake by plants as well as the isotope exchange between C02 and water respectively a m 2132 and 188 m 2 17s 342 The approximation that these fractionations differ by a factor of 2 originates from the relation between the molecular masses as presented in Eqs 321 and 323 For the heaviest isotope quotNor is a function of M2 M while quotNor is a similar function of M 1M Having discussed the theoretical background of isotope effects in the preceding sections we are able to make an approximation of the exponent 6 in the general relation 8 m2RA MIRA 1n m20LAB 6 343 WRB m1RB or where m 12 for carbon or 16 for oxygen The fractionations by diffusion are easily calculated For the dz usion of water vapour through air Eqs335 and 343 result in ln180L1n170L 193 whereas according to Eqs336 and 343 for the dz usz39on ofCO2 through air In 140L1n130L 196 In terms of fractionation this comes to 188 19317s and 14s 19613s 46 Stable Isotope Fractionation Calculating the exponent 0 for fractionation effects in general is quite elaborate The relatively simple case for the CO ltgt Oz isotopic equilibrium may serve as an example cf Broecker and Oversby 1971 The isotope exchange reactions are 18016O12C16O 160212C18O and 17016O12C16O 160212C17O Inserting the proper mass numbers and the vibration frequencies v12C160 650 x 1013s and v160160 474 x 1013s in Eqs318 and 329 the 01 values at 20 C are 18 16 O 0 1n CO 18 1023363 2 and l7 l6 WacoOz W ln 1012245 1n 2 so that In18 OLCOOz ln17 OLCOOz 190 h 6626 103934 Js k 138 103923 JK Applying the equations to polyatomic molecules can only result in approximate 0 values However calculations on the exchange equilibria such as CO ltgt C02 and C02 ltgt CH4 result in In quotMot1n W101 z 19 344 rather than in 0 2 Skaron and Wolfsberg 1980 In one case an experimental determination of the relation between 170 and 18O has been possible A series of natural water samples was electrolytically converted into oxygen which was then analysed mass spectrometrically The resulting relation between the 175 and 185 values was reported as Meijer and Li 1998 1185 1175189350005 345 Despite this in many routine cases where the obtainable precision is limited or irrelevant we may still use the relation 47 Chapter 3 lnm2 OLln 1 0L 2 and quotHZOL ltm10L2 346 or m28 2 mlg 347 at least with a precision of 10 In this respect we will continue to apply the factor of 2 in the relation between the 13C1 2C and 14C1 2C isotope ratios for natural processes 14 13 82 8 as originally used by Craig 1954 The precision of a 14C analysis is not suf cient to allow an experimental veri cation of the relation of Eq342 between the fractionations for 13C and 14C containing molecules On the other hand Eq344 is to be applied if the fractionations involved are large and the measurements sufficiently accurate 48
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