An Introduction to Isotopic Calculations

Transcription

1 An Introduction to Isotopic Calculations John M. Hayes Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA, 30 Septeber 2004 Abstract. These notes provide an introduction to: Methods for the expression of isotopic abundances, Isotopic ass balances, and Isotope effects and their consequences in open and closed systes. Notation. Absolute abundances of isotopes are coonly reported in ters of ato percent. For exaple, ato percent 13 C = [ 13 C/( 12 C + 13 C)]100 (1) A closely related ter is the fractional abundance fractional abundance of 13 C 13 F 13 F = 13 C/( 12 C + 13 C) (2) These variables deserve attention because they provide the only basis for perfectly accurate ass balances. Isotope ratios are also easures of the absolute abundance of isotopes; they are usually arranged so that the ore abundant isotope appears in the denoinator carbon isotope ratio = 13 C/ 12 C 13 (3) For eleents with only two stable nuclides (H, C, and N, for exaple), the relationship between fractional abundances and isotope ratios is straightforward 13 = 13 F/(1-13 F) (4) 13 F = 13 /( ) (5) Equations 4 and 5 also introduce a style of notation. In atheatical expressions dealing with isotopes, it is convenient to follow the cheical convention and to use left superscripts to designate the isotope of interest, thus avoiding confusion with exponents and retaining the option of defining subscripts. Here, for exaple, we have written 13 F rather than F 13 or F 13. Parallel version of equations 4 and 5 pertain to ultiisotopic eleents. In the case of oxygen, for exaple, F = (4a) F F 18 F = (5a) Natural variations of isotopic abundances. The isotopes of any eleent participate in the sae cheical reactions. ates of reaction and transport, however, depend on nuclidic ass, and isotopic substitutions subtly affect the partitioning of energy within olecules. These deviations fro perfect cheical equivalence are tered isotope effects. As a result of such effects, the natural abundances of the stable isotopes of practically all eleents involved in low-teperature geocheical (< 200 C) and biological processes are not precisely constant. Taking carbon as an exaple, the range of interest is roughly F Within that range, differences as sall as 0001 can provide inforation about the source of the carbon and about processes in which the carbon has participated. The delta notation. Because the interesting isotopic differences between natural saples usually occur at and beyond the third significant figure of the isotope ratio, it has becoe conventional to express isotopic abundances using a differential notation. To provide a concrete exaple, it is far easier to say and to reeber that the isotope ratios of saples A and B differ by one part per thousand than to say that saple A has % 15 N and saple B has % 15 N. The notation that provides this advantage is indicated in general for below [this eans of describing isotopic abundances was first used by Urey (1948) in an address to the Aerican Association for the Advanceent of Science, and first forally defined by McKinney et al. (1950)] A δ A Saple X STD = 1 (6) A STD Where δ expresses the abundance of isotope A of eleent X in a saple relative to the abundance of that sae isotope in an arbitrarily designated reference aterial, or isotopic standard. For hydrogen and oxygen, that reference aterial was initially Standard Mean Ocean Water (SMOW). For carbon, it was initially a particular calcareous fossil, the PeeDee Belenite (PDB; the sae standard served for oxygen isotopes in carbonate inerals). For nitrogen it is air (AI). Supplies of PDB and of the water that defined SMOW have been exhausted. Practical scales of isotopic abundance are now defined in ters of surrogate standards distributed by the International Atoic Energy Authority s laboratories in Vienna. Accordingly, odern reports often present values of δ VSMOW and δ VPDB. These are equal to values of δ SMOW and δ PDB. The original definition of δ (McKinney et al., 1950) ultiplied the right-hand side of equation 6 by Isotopic variations were thus expressed in parts per thousand and assigned the sybol (peril, fro the Latin per ille by analogy with per centu, percent). Equa- 1

2 tions based on that definition are often littered with factors of 1000 and 01. To avoid this, it is ore convenient to define δ as in equation 6. According to this view (represented, for exaple, by Farquhar et al., 1989 and Mook, 2000), the sybol iplies the factor of 1000 and we can equivalently write either δ = -25 or δ = -25. To avoid clutter in atheatical expressions, sybols for δ should always be siplified. For exaple, δ b - 3 can represent δ 13 C PDB ( HCO ). If ultiple eleents and isotopes are being discussed, variables like 13 δ and 15 δ are explicit and allow use of right subscripts for designating cheical species. Mass-Balance Calculations. Exaples include (i) the calculation of isotopic abundances in pools derived by the cobination of isotopically differing aterials, (ii) isotope-dilution analyses, and (iii) the correction of experiental results for the effects of blanks. A single, aster equation is relevant in all of these cases. Its concept is rudientary: Heavy isotopes in product = Su of heavy isotopes in precursors. In atheatical ters: Σ F Σ = 1 F F (7) where the ters represent olar quantities of the eleent of interest and the F ters represent fractional isotopic abundances. The subscript Σ refers to total saple derived by cobination of subsaples 1, 2,... etc. The sae equation can be written in approxiate for by replacing the Fs with δs. Then, for any cobination of isotopically distinct aterials: = Σ i δ i /Σ i (8) Equation 8, though not exact, will serve in alost all calculations dealing with natural isotopic abundances. The errors can be deterined by coparing the results obtained using equations 7 and 8. Taking a twocoponent ixture as the siplest test case, errors are found to be largest when differs axially fro both δ 1 and δ 2, i. e., when 1 = 2. For this worst case, the error is given by - * = ( STD )[(δ 1 - δ 2 )/2] 2 (9) Where is the result obtained fro eq. 8; * is the exact result, obtained by converting δ 1 and δ 2 to fractional abundances, applying eq. 7, and converting the result to a δ value; and STD is the isotopic ratio in the standard that establishes the zero point for the δ scale used in the calculation. The errors are always positive. They are only weakly dependent on the absolute value of δ 1 or δ 2 and becoe saller than the result of eq. 9 as δ >> 0. If STD is low, as for 2 H ( 2 VSMOW = ), the errors are less than 4 for any δ 1 - δ For carbon ( 13 PDB = 11), errors are less than 3 for any δ 1 - δ The excellent accuracy of eq. 8 does not ean that all siple calculations based on δ will be siilarly accurate. As noted in a following section, particular care is required in the calculation and expression of isotopic fractionations. And, when isotopically labeled aterials are present, calculations based on δ should be avoided in favor of eq. 7. Isotope dilution. In isotope-dilution analyses, an isotopic spike is added to a saple and the ixture is then analyzed. The original saple ight be a aterial fro which a representative subsaple could be obtained but which could not be quantitatively isolated (say, total body water). For the ixture (Σ) of the saple (x) and the spike (k), we can write: ( x + k )F Σ = x F x + k F k (10) earrangeent yields an expression for x (e. g., oles of total body water) in ters of k, the size of the spike, and easurable isotopic abundances: x = k (F k - F Σ )/(F Σ - F x ) (11) Since we are dealing here with a ass balance, δ can be substituted for F unless heavily labeled spikes are involved. Blank corrections. When a saple has been containated during its preparation by contributions fro an analytical blank, the isotopic abundance actually deterined during the ass spectroetric easureent is that of the saple plus the blank. Using Σ to represent the saple prepared for ass spectroscopic analysis and x and b to represent the saple and blank, we can write Σ = x δ x + b δ b (12) Substituting x = Σ - b and rearranging yields = δ x - b (δ x - δ b )/n Σ (13) an equation of the for y = a + bx. If ultiple analyses are obtained, plotting vs. 1/ Σ will yield the accurate (i. e., blank-corrected) value of δ x as the intercept. Calculations related to isotope effects An isotope effect is a physical phenoenon. It cannot be easured directly, but its consequences a partial separation of the isotopes, a fractionation can soeties be observed. This relationship is indicated scheatically in Figure 1. Fractionation. Two conditions ust be et before an isotope effect can result in fractionation. First, the syste in which the isotope effect is occurring ust be arranged so that an isotopic separation can occur. If a reactant is transfored copletely to yield soe product, an isotopic separation which ight have been 2

3 Figure 1. Scheatic representation of the relationship between an isotope effect (a physical phenoenon) and the occurrence of isotopic fractionation (an observable quantity). visible at soe interediate point will not be observable because the isotopic coposition of the product ust eventually duplicate that of the reactant. Second, since isotope effects are sall enough that they don t upset blanket stateents about cheical properties, the techniques of easureent ust be precise enough to detect very sall isotopic differences. Observed fractionations are proportional to the agnitudes of the associated isotope effects. Exaples of equilibriu and kinetic isotope effects are shown in Figure 2. As indicated, the agnitude of an equilibriu isotope effect can be represented by an equilibriu constant. In this exaple, the reactants and products are cheically identical (carbon dioxide and bicarbonate in each case). However, due to the isotope effect, the equilibriu constant is not exactly 1. There is, however, a proble with the use of equilibriu constants to quantify isotope effects. An equilibriu constant always pertains to a specific cheical reaction, and, for any particular isotopic exchange, the reaction can be forulated in various ways, for exaple: H 18 2 O + 1/3 CaC 16 O 3 H 16 2 O + 1/3 CaC 18 O 3 (14) vs. H 18 2 O + CaC 16 O 3 H 16 2 O + CaC 18 O 16 O 2 (15) To avoid such abiguities, EIEs are ore coonly described in ters of fractionation factors. A fractionation factor is always a ratio of isotope ratios. For the exaple in Figure 2, the fractionation factor would be: HCO 3 /CO 2 13 C = 12 C HCO 3 13 C 12 C CO α (16) A ore copact and typical representation would be 2 Figure 2. Exaples of equilibriu and kinetic isotope effects. 13 α b/g = b / g (17) Here, subscripts have been used to designate the cheical species (b for bicarbonate, g for gas-phase CO 2 ) and the superscript 13 indicates that the exchange of 13 C is being considered. The resulting equation is ore copact. It does not cobine cheical and atheatical systes of notation and, as a result, is less cluttered and ore readable. For either reaction 14 or 15, the fractionation factor would be 18 α c/w = c / w (18) where c and w designate calcite and water, respectively. For a kinetic isotope effect, the fractionation factor is equal to the ratio of the isotope-specific rate constants. It is soeties denoted by β rather than α. In physical organic cheistry and enzyology, however, β is generally reserved for the ratio of isotope-specific equilibriu constants while α stands as the observed isotopic fractionation factor. There are no fir conventions about what goes in the nuerator and the denoinator of a fractionation factor. For equilibriu isotope effects it s best to add subscripts to α in order to indicate clearly which substance is in the nuerator and, for kinetic isotope effects, readers ust exaine the equations to deterine whether an author has placed the heavy or the light isotope in the nuerator. eversibility and systes. Given a fractionation factor, isotopic copositions of reactants and products can usually be calculated once two questions have been answered: 1. Is the reaction reversible? 2. Is the syste open or closed? 3

4 An open syste is one in which both atter and energy are exchanged with the surroundings. In contrast, only energy crosses the boundaries of a closed syste. eversibility and openness vs. closure are key concepts. Each is forally crisp, but authors often sew confusion. If a syste is described as partially closed, readers should be on guard. If the description eans that only soe eleents can cross the boundaries of the syste, the concept ight be helpful. If partial closure instead refers to ipeded transport of aterials into and out of the syste, the concept is invalid and likely to cause probles. Openness covers a range of possibilities. Soe open systes are at steady state, with inputs and outputs balanced so that the inventory of aterial reains constant. In others, there are no inputs but products are lost as soon as they are created. These cases can be treated siply, others require specific odels. eversible reaction, closed syste. In such a syste, (1) the isotopic difference between products and reactants will be controlled by the fractionation factor and (2) ass balance will prevail. To develop a quantitative treatent, we will consider an equilibriu between substances A and B, A B (19) The isotopic relationship between these aterials is defined in ters of a fractionation factor: A δ A + 1 αa/b = = (20) B δ B + 1 where the s are isotope ratios and the δs are the corresponding δ values on any scale of abundances (the sae scale ust be used for both the product and the reactant). The requireent that the inventory of aterials reain constant can be expressed inexactly, but with good accuracy, in ters of a ass-balance equation: = f B δ B + (1 f B )δ A (21) where refers to the weighted-average isotopic coposition of all of the aterial involved in the equilibriu (for equation 14 or 15, it would refer to all of the oxygen in the syste) and f B refers to the fraction of that aterial which is in the for of B. By difference, the fraction of aterial in the for of A is 1 f B. earrangeent of equation 20 yields: δ A = α A/B δ B + (α A/B 1) (22) The second ter on the right-hand side of this equation is coonly denoted by a special sybol: α - 1 ε (23) Equation 22 then becoes: δ A = α A/B δ B + ε A/B (24) (Like δ, ε is often expressed in parts per thousand. For exaple, if α = 077, then ε = 077 or 7.7. Diensional analysis reliably indicates the for required in any calculation. In equations 24-29, all ters could be diensionless or could be expressed in peril units. The diensionless for is required in equation 20, where δ is added to a diensionless constant, in equations 39 and 40, where use of peril units would unbalance the equations, and in equations 41 and 42, where δ and/or ε appear as exponents or in the arguents of transcendental functions.) Substituting for δ A in equation 21, we obtain = f B δ B + (1 f B )(α A/B δ B + ε A/B ) (25) Solving for δ B, we obtain an expression which allows calculation of δ B as a function of f B, given α A/B and : δ ( 1 f B ) ε δ (26) ( 1 f ) + B = Σ α A/B B A/B f B If equation 24 is rearranged to express δ B in ters of δ A, α A/B, and ε A/B, and the result is substituted for δ B in equation 21, we obtain a copleentary expression for δ A : α δ + f ε δ A = (27) α A/B Σ B A/B A/B( 1 f B ) + f B Equations 26 and 27 are generally applicable but rarely eployed because adoption of the approxiation α A/B 1 yields far sipler results, naely: δ A = + f B ε A/B (28) and δ B = - (1 f B )ε A/B (29) Graphs depicting these relationships can be constructed very easily. An exaple is shown in Figure 3. For f B = 1 we ust have δ B = and for f B = 0 we ust have δ A =. As shown in Figure 3, the intercepts for the other ends of the lines representing δ A and δ B will be ± ε A/B. If α A/B < 1, ε A/B will be negative and A will be isotopically depleted relative to B. The lines representing δ A and δ B will slope upward as f B 0. If α A/B > 1 then A would be enriched relative to B and the lines would slope downward. There are two circustances in which the approxiation leading to equations 28 and 29 (i. e., α A/B 1) ust be exained. The first involves highly precise studies and the second alost any hydrogen-isotopic fractionations. As an exaple of the first case, Figure 4 copares the results of equations 26 and 27 to those of 28 and 29 for a syste with α A/B = 412. This is, in fact, the fractionation factor for the exchange of oxygen 4

5 f B δ B δ B + - ε A/B 40 δ, ε A/B δ A δ A 20 δ A δ, ε A/B ff B Figure 3. General for of a diagra depicting isotopic copositions of species related by a reversible cheical reaction acting in a closed syste. between water and gas-phase carbon dioxide at 25 C (where A is CO 2 and B is H 2 O). The solid and dotted lines in the larger graph represent the ore accurate and the approxiate results, respectively. The lines in the saller graph in Figure 4 represent the errors, which significantly exceed the precision of easureent (which is typically better than 0.1 ). For any hydrogen-isotopic fractionations, α differs fro by ore than 10%. In such cases, the approxiation fails and a agnifying lens is not needed to see the difference between the accurate and the approxiate results. An exaple is shown in Figure 5. Irreversible reaction, closed syste. The goal is to deterine the isotopic relationship between the reactants and products over the course of the reaction, starting with only reactants and ending with a 100% yield of products. We ll consider the general reaction P (eactants Products). If the rate is sensitive to isotopic substitution, we ll have α P/ 1, where α P/ P,i / (30) In this equation, α P/ is the fractionation factor, P,i is the isotope ratio (e. g., 13 C/ 12 C) of an increent of product and is the isotope ratio of the reactant at the sae tie. Following the approach of Mariotti et al. (1981) and expressing P,i in ters of differential quantities, we can write Error, α A/B = 412 = 0 Figure 4. Graphs showing results of both accurate and approxiate calculations of isotopic copositions of reactants and products of a reversible reaction in a closed syste. The dotted lines in the upper graph are drawn as indicated in Figure 3 and by equations 28 and 29. The solid lines are based on equations 26 and 27. The differences between these treatents are shown in the saller graph. P/ = δ A f B dhp / dlp h / l α (31) where the s represent olar quantities and the subscripts designate the heavy and light isotopic species of the Product and eactant. In absence of side reactions, d hp = -d h and d lp = -d l. Eploying these substitutions, the equation can be expressed entirely in ters of quantities of reactant: δ B δ B 5

6 δ, α A/B = 00 = +100 Accurate δ B Approxiate δ A Accurate δ A d = h d α l P/ (32) h l Separating variables and integrating fro initial conditions to any arbitrary point in the reaction, we obtain l = α h P/ ln ln (33) l,0 h,0 where the subscript zeroes indicate quantities of l and h at tie = 0. It is convenient to define f as the fraction of reactant which reains unutilized (i. e., 1 - f is the fractional yield). Specifically: h + l f = = (34),0 h,0 + l,0 Avoiding approxiations introduced by Mariotti et al. (1981), we rearrange this equation to yield exact substitutions for the arguents of the logariths in equation 33. For exaple: Equation 41 is often presented as The ayleigh Equation. It was developed by Lord ayleigh (John Willia Strutt, ) in his treatent of the distillation of liquid air and was exploited by Sir Willia asey ( ) in his isolation first of argon and then of neon, krypton, and xenon. The ratios in these cases were not isotopic. The concept of isotopes was introduced by Frederick Soddy ( ) in 1913, long after Strutt and asey had been awarded the Nobel Prizes in physics and cheistry, respectively, in Instead, the ratios in ayleigh s conception pertained to the abunf B Approxiate δ B Figure 5. Graph depicting isotopic copositions of reactants and products of a reversible reaction in a closed syste with an α value typical of hydrogen isotope effects. h l + 1 l l ( + 1) = = f ( + 1) h,0 l,0,0 + 1 l,0 l,0 Which yields (35) l 1+,0 = f (36) l,0 1+ Siilar rearrangeents yield h h,0 1+ = f (37) 1+,0,0 So that equation 33 becoes 1 +,0 1 + =,0 α P/ ln f ln f (38) ,0 Which can be rearranged to yield 1+,0 = ε P/ln f ln (39) 1+,0 an expression which duplicates exactly equation V.17 in the rigorous treatent by Bigeleisen and Wolfsburg (1958). Without approxiation, this equation relates the isotopic coposition of the residual reactant ( ) to that of the initial reactant (,0 ), to the isotope effect (α P/ ), and to f, the extent of reaction. If the abundance of the rare isotope is low, the coefficient for f, (1 +,0 )/(1 + ), is alost exactly 1. In that case, equation 39 becoes ε P/ ln f = ln (40),0 Because quantities with equal logariths ust theselves be equal, we can write ε f P/ =,0 (41) 6

7 2 1 ε P/ = -30 (eq. 43) = -30 (eq. 42) δ,0 = -8 (eq. 43) = -08 (eq. 42) STD = Error in δ, 0 Equation 42-1 Equation 43-2 f Figure 7. Errors in values of δ calculated using equations 42 and 43. Figure 6. Scheatic representation of isotopic copositions of reactants and products for an irreversible reaction in a closed syste. dances of Ar, Ne, Kr, and Xe relative to N 2 and, as the distillation proceeded, relative to each other. Figure 6 indicates values of that are in accordance with equation 41 (viz., the line arked ; on the horizontal axis, yield of P = 1 f). If the objective is to use observed values of δ P and/or δ in order to evaluate α P/, a linear for is often desirable. One which is linear and exact is provided by equation 39. Values of δ ust be converted to isotope ratios and the initial coposition of the reactant ust be known. egression of ln( /,0 ) on ln[f(1 +,0 )/(1 + )] will then yield ε P/ as the slope. By use of approxiations, Mariotti et al. (1981) developed less cubersoe fors. The first is derived by rewriting equation 40 with the arguent of the logarith in the delta notation. earrangeent then yields ln ( δ 1) = ln( δ + 1) + ε ln f + (42),0 P/ This is an expression of the for y = a + bx. egression of ln(δ + 1) on lnf yields a straight line with slope ε P/. A second for is very widely applied. Noting that ln[(1 + u)/(1 + v)] u v when u and v are sall relative to 1, Mariotti et al. (1981) siplified eq. 42 to obtain δ = δ,0 + ε P/ lnf (43) This is also an expression of the for y = a + bx. Moreover, it is an equation in which all ters can be expressed in peril units. egression of δ on lnf will yield ε P/ as the slope and δ,0, which need not be known independently, as the intercept. The approxiations leading to equations 42 and 43 can lead to systeatic errors. These are exained in Figure 7, which is based on values of δ and ε that are typical for 13 C. Equation 42 is based on the approxiation that l / l,0 does not differ significantly fro /,0 (cf. eq. 34). This is evidently uch ore satisfactory than the approxiation leading to equation 43, which produces systeatic errors that are larger than analytical uncertainties for values of f <. Equation 43 should never be used in hydrogen-isotopic calculations, for which the approxiation u v << 1 is copletely invalid. Equations pertain to values of δ, the isotopic coposition of the reactant available within the closed syste. Figure 6 also shows lines arked P and P representing respectively the isotopic copositions of the pooled product and of the product foring at any point in tie. The relationship between and P always follows directly fro the fractionation factor: 7

8 ε P/ = -30 δ,0 = -8 Error in δ P, Equation 46 P δ P + 1 αp/ = = (44) δ + 1 As noted in Figure 6, this leads to a constant isotopic difference (designated B in the Figure) between and P. Initially, both the pooled product, P, and P are depleted relative to by the sae aount. As the reaction proceeds, the isotopic coposition of P steadily approaches that of the initial reactant. The functional relationship can be derived fro a ass balance:,0 δ,0 = δ + P δ P (45) substituting /,0 = f and P /,0 = 1 - f, cobination of equations 43 and 45 yields δ P = δ,0 [f/(1 f)]ε P/ lnf (46) an expression which shows that the regression of δ P on [f/(1 f)] lnf will yield ε P/ as the slope. Systeatic errors associated with equation 46, suarized graphically in Figure 8, are saller than those in equation 43 (note the differing vertical scales in Figs. 7 and 8). In practical studies of isotope effects, the ost vexing proble is usually not whether to accept approxiations but instead how to cobine observations fro ultiple experients. The proble has been explored systeatically and very helpfully by Scott et al. (2004), who have, in addition, concluded that calculations based on equation 42 are ost likely to provide the sallest uncertainties. Notably, they recoend regression of lnf on ln(δ + 1), thus necessitating transforation of the regression constants in order to obtain a value for ε P/. Although f Figure 8. Errors in values of δ P calculated using equation 46. Figure 9. Scheatic view of an open syste which is supplied with reactant and fro which products P and Q are withdrawn. both f and δ are subject to error, use of a Model I linear regression, and thus assuing that errors in δ are sall in coparison to those in f, is recoended. eversible reaction, open syste (product lost). Atospheric oisture provides the classical exaple of systes of this kind. Water vapor condenses to for a liquid or solid which precipitates fro the syste. Vapor pressure isotope effects lead to fractionation of the isotopes of both H and O. The process of condensation is reversible, but the evolution of isotopic copositions is described by the sae equations introduced above to describe fractionations caused by an irreversible reaction in a closed syste. The heavy isotopes accuulate in the condensed phases. Consequently, the residual reactant (the oisture vapor reaining in the atosphere) is described by increasingly negative values of δ. The curves describing the isotopic copositions have the sae shape as those in Figure 6, but bend downward rather than upward. eversible or irreversible reaction, open syste at steady state. A scheatic view of a syste of this kind is shown in Figure 9. A reactant,, flows steadily into a stirred reaction chaber. Products P and Q flow fro the chaber. The aount of aterial in the reaction chaber is constant and, therefore, J = J P + J Q (47) Where J is the flux of aterial, oles/unit tie. The relevant fractionation factors are δ + 1 α/p = = (48) P δ P + 1 and δ + 1 α/q = = (49) Q δq + 1 It doesn t atter whether these fractionations result fro reversible equilibria or fro kinetically liited processes that operate consistently because residence ties within the reactor are constant. As a result of these processes, the isotopic difference between P and Q can also be described by a fractionation factor 8

9 α α P/Q = = α P Q /Q /P (50) Given these conditions, isotopic fractionations for any syste of this kind are described by equations 26 and 27 (or by eq. 28 and 29 if α 1), with P and Q equivalent to A and B and equivalent to the aterial designated by Σ. The value of f B is given by J Q /J. Systes of this kind range widely in size. The treatent just described is conventional in odels of the global carbon cycle, in which the reaction chaber is the atosphere + hydrosphere + biosphere, is recycling carbon entering the syste in the for of CO 2, and P and Q are organic and carbonate carbon being buried in sedients. The value of α /Q, the fractionation between CO 2 and carbonate sedients is accordingly The value of α /P, the fractionation between CO2 and buried organic aterial is 15. Cobination of these factors as in equation 50 leads to the fractionation between organic and carbonate carbon, α P/Q The treatent is also appropriate for isotopic fractionations occurring at an enzyatic reaction site. In this case, the reaction chaber is the icroscopic pool of reactants available at the active site of the enzye, is the substrate, P is the product, and Q is unutilized substrate (thus α /Q = 00). Fractionations occuring in networks coprised of ultiple systes of this kind have been described by Hayes (2001). Irreversible reaction, open syste (product accuulated). Plants exeplify systes of this kind. Carbon and hydrogen are assiilated fro infinite supplies of CO 2 and H 2 O and accuulate in the bioass. The proble is trivial, but a particular detail requires ephasis. The fixation of C or H can be represented by P (51) The corresponding isotopic relationship can be described by a fractionation factor P δp + 1 α P/ = = (52) δ + 1 The relationship between δ and δ P is then described by δ P = αp/δ + εp/ (53) This equation is often siplified to this for: δ P = δ + ε P/ (54) For exaple, to estiate the δ value of the CO 2 that was available to the plant, an experienter will often refer to a one-to-one relationship and subtract ε P/ (typically -20 ) fro the δ value of the bioass. For carbon, no serious error will result. The relationship is nearly oneto-one (slope = α P/ 0.980). δ P, δ P, α P/ = C/ 12 C δ P = 0.980δ P D/H δ, -200 α P/ = δ P = 00δ Figure 10. Coparison of fractionations affecting carbon and hydrogen. In each case, the broken line represents a one-to-one approxiation and the solid line represents the accurate relationship described by the equation. The approxiation is essentially valid for carbon but seriously in error for hydrogen. For hydrogen, however, this approach leads to disaster. Any experienter dealing with ε P/ -200, a value typical of fractionations affecting deuteriu, should reeber that the slope of the corresponding relationship ust differ significantly fro 1 and that equation 53 cannot be replaced by equation 54. This coparison between fractionations affecting carbon and those affecting hydrogen is suarized graphically in Figure 10. 9

10 Acknowledgeents Figures 1, 2, 6, and 9 were prepared by Steve Studley. These notes derive fro teaching aterials prepared for classes at Indiana University, Blooington, and at Harvard University, Cabridge, Massachusetts. I a grateful to the students in those classes and for support provided by those institutions and by the National Science Foundation (OCE ). eferences Bigeleisen J. and Wolfsberg M. (1958) Theoretical and experiental aspects of isotope effects in cheical kinetics. Adv. Che. Phys. 1, Farquhar, G. D., Ehleringer, J.., Hubick, K. T. (1989) Carbon isotope discriination and photosynthesis. Annu. ev. Plant Physiol. Plant Mol. Biol. 40, Hayes, J. M. (2001) Fractionation of the isotopes of carbon and hydrogen in biosynthetic processes. pp in John W. Valley and David. Cole (eds.) Stable Isotope Geocheistry, eviews in Mineralogy and Geocheistry Vol. 43. Mineralogical Society of Aerica, Washington, D. C. pdf available at Mariotti A., Geron J. C., Hubert P., Kaiser P., Leto Ile., Tardieux A. and Tardieux P. (1981) Experiental deterination of nitrogen kinetic isotope fractionation: soe principles; illustration for the denitrification and nitrification processes. Pl. Soil 62, McKinney C.., McCrea J. M., Epstein S., Allen H. A. and Urey H. C. (1950) Iproveents in ass spectroeters for the easureent of sall differences in isotope abundance ratios. ev. Sci. Instru. 21, Mook, W. G. (2000) Environental Isotopes in the Hydrological Cycle, Principles and Applications. Available on line fro Scott, K. M., Lu, X., Cavanaugh, C. M., and Liu, J. S. (2004) Evaluation of the precision and accuracy of kinetic isotope effects estiated fro different fors of the ayleigh distillation equation. Geochi. Cosochi. Acta 68, Urey H. C. (1948) Oxygen isotopes in nature and in the laboratory. Science 108,