Fundamentals of Isotope Geochemistry

2.1 Introduction
2.2 Fundamentals of Isotope Geochemistry
2.2.1 Definitions
2.2.2 Terminology
2.2.3 Standards
2.3 Stable Isotope Fractionation
2.3.1 Properties of isotopic molecules
2.3.2 Fractionation accompanying chemical reactions and phase changes
2.3.3 The Rayleigh equations
2.3.4 Isotopic fractionation in open and closed systems
2.3.5 Biological fractionations
2.4 Sample Collection, Analysis, and Quality Assurance
2.4.1 Sampling guidelines
2.4.2 Analytical methods and instrumentation
2.4.3 Quality assurance of contract laboratories
2.5 Applications of Isotope Tracers in Catchment Hydrology
2.5.1 Water isotope hydrology
2.5.2 Solute isotope biogeochemistry
2.5.3 Mixing
2.5.4 Isotopically labeled materials
2.5.5 Stable isotopes in geochemical modeling
2.5.6 Use of a multi-isotope approach for the determination of flowpaths
2.6 Summary
References

Of all the methods used to understand hydrologic processes in small catchments, applications of tracers--in particular isotope tracers--have been the most useful in terms of providing new insights into hydrologic processes. This is because they integrate small-scale variability to give an effective indication of catchment-scale processes (McDonnell and Kendall, 1992; Buttle, 1994). In contrast, internal watershed point measurements, such as those of water level or groundwater composition, cannot be used without extrapolation or additional assumptions about catchment behavior. Isotopes are also "applied" at the watershed scale (i.e., they are within all components of the hydrologic cycle). In particular, ¹⁸O, ²H, and ³H are integral parts of natural water molecules that fall as rain or snow (meteoric water) each year over a watershed and, consequently, are ideal tracers of water. This no cost, long-term, and wide-spread application of these natural tracers allows hydrologists to study runoff generation on scales ranging from macropores to hillslopes to first- and higher- order streams (Sklash, 1990).

Environmental isotopes are natural and anthropogenic isotopes whose wide distribution in the hydrosphere can assist in the solution of hydrogeochemical problems. Typical uses of environmental isotopes in hydrology include:

• identification of mechanisms responsible for streamflow generation
• testing flowpath and water-budget models developed using hydrometric data
• characterization of flowpaths that water follows from the time precipitation hits the ground until discharge at the stream
• determination of weathering reactions that mobilize solutes along those flowpaths
• determination of the role of atmospheric deposition in controlling water chemistry
• identification of the sources of solutes in contaminated systems and
• assessment of biologic cycling of nutrients within an ecosystem

Waters recharged at different times, in different locations, or that followed different flowpaths are often isotopically distinct; in other words, they have distinctive "fingerprints".

Unlike most chemical tracers, environmental isotopes are relatively conservative in reactions with catchment materials. This is especially true of oxygen and hydrogen isotopes in water; meteoric waters retain their distinctive fingerprints until they mix with waters of different compositions or, in the case of isotopes of dissolved species, there are reactions with minerals or other fluids.

Solutes in catchment waters that are derived from atmospheric sources are commonly isotopically distinct from solutes derived from geologic and biologic sources within the catchment.

Both biological cycling of solutes and water/rock reactions often change isotopic ratios of the solutes in predictable and recognizable directions; these interactions often can be reconstructed from the isotopic compositions.

If water from an isotopically distinctive source (e.g., rain with an unusual isotopic composition) is found along a flowpath, it provides proof for a hydrologic connection, despite any hydraulic measurements or models to the contrary.

Before embarking on the fundamentals of isotope geochemistry, we would like to close this introduction with a cautionary note first presented in 1983 at a short-course on Isotope Hydrology co-taught by Tyler B. Coplen and Carol Kendall to scientists at the U.S. Geological Survey. This very appropriate note has been repeated at each subsequent USGS course we have taught:

Fretwell's Law: "Warning! Stable isotope data may cause severe and contagious stomach upset if taken alone. To prevent upsetting reviewers' stomachs and your own, take stable isotope data with a healthy dose of other hydrologic, geologic, and geochemical information. Then, you will find stable isotope data very beneficial." (Marvin O. Fretwell, pers. comm. 1983).

2.2 Fundamentals of Isotope Geochemistry

The following section presents a very brief discussion of the fundamentals of stable and radio-isotope geochemistry, intended to provide readers with the necessary background information to understand the succeeding chapters. Many of the topics below are discussed in more detail in individual chapters. For more information on isotope systematics, the readers are encouraged to examine the following isotope reference books: Clark and Fritz (1997), Dickin (1995), Faure (1986), and Gat and Gonfiantini (1981). The first of these is a textbook intended for upper-division and graduate hydrogeology students, and contains problem sets.

2.2.1 Definitions

Isotopes are atoms of the same element that have different numbers of neutrons. Differences in the number of neutrons among the various isotopes of an element mean that the various isotopes have different masses. The superscript number to the left of the element designation is called the mass number and is the sum of the number of protons and neutrons in the isotope (Figure 2.1). For example, among the hydrogen isotopes, deuterium (denoted as D or ²H) has one neutron and one proton. This mass number of 2 is approximately equal to twice the mass of protium (¹H), whereas tritium (³H) has two neutrons and its mass is approximately three times the mass of protium. Isotopes of the same element have the same number of protons. For example, all isotopes of oxygen have 8 protons; however, an oxygen atom with a mass of 18 (denoted ¹⁸O) has 2 more neutrons than oxygen with a mass of 16 (¹⁶O). Isotope names are usually pronounced with the element name first, as in "oxygen-18" instead of "18-oxygen." In many texts, especially older ones typeset without superscripts, the mass number is shown to the right of the element abbreviation, as in C-13 or C¹³ for carbon-13.

The original isotopic compositions of planetary systems are a function of nuclear processes in stars. Over time, isotopic compositions in terrestrial environments change by the processes of radioactive decay, cosmic ray interactions, mass-dependent fractionations that accompany inorganic and biological reactions, and anthropogenic activities such as the processing of nuclear fuels, reactor accidents, and nuclear-weapons testing. Radioactive (unstable) isotopes are nuclides (isotope-specific atoms) that spontaneously disintegrate over time to form other isotopes. During the disintegration, radioactive isotopes emit alpha or beta particles and sometimes also gamma rays. Stable isotopes are nuclides that do not appear to decay to other isotopes on geologic time scales, but may themselves be produced by the decay of radioactive isotopes.

Naturally occurring nuclides define a path in the chart of nuclides, corresponding to the greatest stability of the neutron/proton (N/Z) ratio. For nuclides of low atomic mass, the greatest stability is achieved when the number of neutrons and protons are approximately equal (N = Z); these are the so-called stable isotopes (denoted as shaded nuclides in Figure 2.1). However, as the atomic mass increases, the stable neutron/proton ratio increases until N/Z = 1.5. Radioactive decay occurs when changes in N and Z of an unstable nuclide cause the trans-formation of an atom of one nuclide into that of another, more stable nuclide; these radioactive nuclides are called unstable nuclides (denoted as the non-shaded nuclides in Figure 2.1).

Atoms produced by the radioactive decay of other nuclides are termed radiogenic. A few nuclides are produced by cosmic ray bombardment of stable nuclides in the atmosphere and are termed cosmogenic. Other nuclides may be created by the addition of neutrons produced by the alpha decay of other nuclides (neutron activation). Alternatively, the neutron addition can displace a proton in the nucleus, creating a nuclide of the same atomic mass but lower atomic number. Nuclides produced by these two processes are termed lithogenic. If the daughter product is radioactive, it will decay to form an isotope of yet another element. This process will continue until a stable nuclide is produced. For example, uranium and thorium decay to form other radionuclides that are themselves radioactive and decay to other radionuclides, and so on until stable lead isotopes are formed (see Chapter 20 for uranium decay chains). Although the terms parent and daughter nuclides are commonly used, these terms can be misleading. Only one atom is involved during radioactive decay; that is, the daughter nuclide is the same nuclide as the parent atom. However, after radioactive decay it has a different number of neutrons in its nucleus.

The change in the number of neutrons can occur in a variety of ways (Figure 2.1). However, the four mechanisms described below are the most common and produce the radiogenic nuclides most relevant to hydrologic and geologic studies:

Beta decay occurs when nuclides deficient in protons transform a neutron into a proton and an electron, and expel the electron from the nucleus as a negative b particle (b^-), thereby increasing the atomic number by one while the number of neutrons is reduced by one.

Positron decay occurs when nuclides deficient in neutrons transform a proton into a neutron, an electron (b⁺), and a neutrino, thereby decreasing the atomic number by one and increasing the neutron number by one. The daughters are isobars (nuclides of equal mass) of their parent and are isotopes of different elements.

Beta capture (or electron capture) occurs when nuclides deficient in neutrons transform a proton into a neutron plus neutrino by the capture of an electron by a proton, thereby decreasing the number of protons in the nucleus by one. Both this and positron decay yield a radiogenic nuclide that is an isobar of the parent nuclide.

Alpha decay occurs when heavy atoms above Z = 83 in the nuclide chart emit an alpha particle, which consists of a helium nuclei with two neutrons, two protons, and a 2⁺ charge. This radiogenic daughter product in not an isobar of its parent nuclide because its mass is reduced by four (see Figure 2.1).

For example, the radioisotope (radioactive isotope) ¹⁴C is produced in the atmosphere by cosmic ray neutron interaction with ¹⁴N. ¹⁴C has a half-life of about 5730 years, and decays back to stable ¹⁴N by emission of a beta particle. The decay equation below expresses the change in the concentration (activity) of the nuclide over time:

A_t = A_o · e^-lt (Eq. 2.1)

2.2.2 Terminology

Stable isotope compositions of low-mass (light) elements such as oxygen, hydrogen, carbon, nitrogen, and sulfur are normally reported as d values. The term "d" is spelled and pronounced delta not del. The word del describes either of two mathematical terms: an operator ( ) or a partial derivative (d). d values are reported in units of parts per thousand (denoted as ‰ or permil, or per mil, or per mille -- or even recently, per mill) relative to a standard of known composition. d values are calculated by:

d (in ‰) = (R_x / R_s - 1) ·1000 (Eq. 2.2)

There are several commonly used ways for making comparisons between the d values of two materials. The first two are preferred because of their clarity, and the fourth should be avoided:

(1) high vs. low values

(2) more/less positive vs. more/less negative (e.g., -10‰ is more positive than -20‰)

(3) heavier vs. lighter (the "heavy" material is the one with the higher d value)

(4) enriched vs. depleted (always remember to state what isotope is in short supply, e.g., a material is enriched in ¹⁸O or ¹⁶O relative to some other material, and that the enrichment or depletion is a result of some reaction or process). For example, to say that "one sample is enriched in ³⁴S relative to another because of sulfate reduction" is proper usage. Phrases such as "a sample has an enriched d¹⁵N value" are misuses of terminology.

The isotopic compositions of materials analyzed on mass spectrometers are usually reported relative to some international reference standard. Samples are either analyzed at the same time as this reference standard or with some internal laboratory standard that has been calibrated relative to the international standard. Alternatively, the absolute ratios of isotopes can be reported. Small quantities of these reference standards are available for calibration purposes from either the National Institute of Standards and Technology (NIST) in the USA (Web site: http://www.nist.gov/), or the International Atomic Energy Agency (IAEA) in Vienna (Web site: http://www.iaea.or.at/).

Various isotope standards are used for reporting light stable-isotopic compositions (Table 2.1). The d values of each of the standards have been defined as 0‰. dD and d¹⁸O values are normally reported relative to the SMOW standard (Standard Mean Ocean Water; Craig, 1961) or the equivalent VSMOW (Vienna-SMOW) standard. d¹³C values are reported relative to either the PDB (Pee Dee Belemnite) or the equivalent VPDB (Vienna-PDB) standard. d¹⁸O values of low-temperature carbonates are also commonly reported relative to PDB or VPDB.

Table 2.1. Abundance ratios and reference standards for some environmental isotopes.

Isotope	Ratio measured	Reference Standard	Abundance ratio  of standard
2H	2H/1H	VSMOW	1.5575 · 10-4
3He	3He/4He	atmospheric He	1.3 · 10-6
6Li	6Li/7Li	L-SVEC	8.32 · 10-2
11B	11B/10B	NBS 951	4.04362
13C	13C/12C	VPDB	1.1237 · 10-2
15N	15N/14N	atmospheric N2	3.677 · 10-3
18O	18O/16O	VSMOW, or	2.0052 · 10-3
		VPDB	2.0672 · 10-3
34S	34S/32S	CDT	4.5005 · 10-2
37Cl	37Cl/35Cl	SMOC	0.324
87Sr	87Sr/86Sr	Absolute ratio, or
		various materials

VSMOW and VPDB are virtually identical to the SMOW and PDB standards. Use of VSMOW and VPDB is supposed to imply that the measurements were calibrated according to IAEA guidelines for expression of d values relative to available reference materials on normalized permil scales (Coplen, 1994; 1995; 1996). Laboratories accustomed to analyzing synthetic compounds that are highly enriched in the heavy (or, less commonly, the light) isotope may report absolute isotope abundances in atomic-weight percent or ppm, instead of relative ratios in permil. In general, radioisotopes are reported as absolute concentrations or ratios. Tritium (³H) values are typically reported as absolute concentrations, called Tritium Units (TU) where one TU corresponds to 1 tritium atom per 10¹⁸ hydrogen atoms. Tritium values may also be expressed in terms of activity (pico-Curies/liter, pCi/L) or decay (disintegrations per minute/liter, dpm/L), where 1 TU = 3.2 pCi/L = 7.2 dpm/L. ¹⁴C contents are referenced to an international standard known as "modern carbon" and are typically expressed as a percent of modern carbon (pmc).

2.3 Stable Isotope Fractionation

2.3.1 Properties of isotopic molecules

The various isotopes of an element have slightly different chemical and physical properties because of their mass differences. Under the proper circumstances, such differences can manifest themselves as a mass-dependent isotope fractionation effect. Nuclear interactions, on the other hand, lead to a non-mass-dependent effect in the sense that they depend on the nuclear structure, rather than on the weight difference per se. In the first case, for example, the properties of molecules with ¹⁷O will be intermediate between those of molecules with ¹⁶O and ¹⁸O; this is not necessarily the case for the non-mass-dependent effects. For elements of low atomic numbers, these mass differences are large enough for many physical, chemical, and biological processes or reactions to fractionate or change the relative proportions of different isotopes of the same element in various compounds. As a result of fractionation processes, waters and solutes often develop unique isotopic compositions (ratios of heavy to light isotopes) that may be indicative of their source or the processes that formed them.

Two main types of phenomena produce isotopic fractionations: isotope exchange reactions and kinetic processes. Isotope exchange reactions can be viewed as a subset of kinetic isotope reactions where the reactants and products remain in contact in a closed, well-mixed system such that back reactions can occur and chemical equilibrium can be established. Under such circumstances, isotopic equilibrium can be also established. Detailed discussions of isotope fractionations are found in O'Neil (1986), Gat and Gonfiantini (1981), Gat (1980), and other texts.

2.3.2 Fractionation accompanying chemical reactions and phase changes

The strength of chemical bonds involving different isotopic species will usually be different. Molecules containing heavy isotopes are more stable (i.e., have a higher dissociation energy) than molecules with lighter isotopes. Hence, isotopic fractionations between molecules can be explained by differences in their zero point energies (ZPE). For example, there is about a 2 kcal/mole difference in ZPE associated with the breaking of the H-H bond compared to the D-D bond (Figure 2.2). Hence, H-H bonds are broken more easily and D-D bonds are more stable. Chemical reaction rates where such a bond is broken will also show an isotope effect. These are quantum effects that become appreciable at low temperatures and disappear at higher temperatures. The energy differences associated with isotope effects are about 1000 times smaller than the DG for chemical reactions, and hence cannot be the driving force for chemical equilibrium.

Equilibrium fractionations

Equilibrium isotope-exchange reactions involve the redistribution of isotopes of an element among various species or compounds (in a strict sense, this only occurs in a closed, well-mixed system at chemical equilibrium). At isotopic equilibrium, the forward and backward reaction rates of any particular isotope are identical. This does not mean that the isotopic compositions of two compounds at equilibrium are identical, but only that the ratios of the different isotopes in each compound are constant for a particular temperature.

During equilibrium reactions, the heavier isotope generally preferentially accumulates in the species or compound with the higher oxidation state. For example, sulfate becomes enriched in ³⁴S relative to sulfide (i.e., has a more positive various materials d³⁴S value); consequently, the residual sulfide becomes depleted in various materials d³⁴S. As a "rule of thumb," among different phases of the same compound or different species of the same element, the more dense the material, the more it tends to be enriched in the heavier isotope. For example, for the various phases of water, at equilibrium, d¹⁸O_S > d¹⁸O_L > d¹⁸O_V. Also, the d¹³C and d¹⁸O values of CO₂ < HCO₃- < CaCO₃.

During phase changes, the ratio of heavy to light isotopes in the molecules in the two phases changes. For example, as water vapor condenses in rain clouds (a process typically viewed as an equilibrium process), the heavier water isotopes (¹⁸O and ²H) become enriched in the liquid phase while the lighter isotopes (¹⁶O and ¹H) remain in the vapor phase. In general, the higher the temperature, the less the difference between the equilibrium isotopic compositions of any two species (because the differences in ZPE between the species become smaller). The fractionation associated with the equilibrium exchange reaction between two substances A and B (i.e., the fractionation of A relative to B) can be expressed by use of the isotope fractionation factor a (alpha):

a_A-B = R_A / R_B (Eq. 2.3)

The value of such an equilibrium fractionation factor can be calculated on the basis of spectral data of the isotopic molecular species, at least for simple molecules. The a values generally differ by just a few percent from the equal-energy value of 1.00, except for exchange reactions involving hydrogen isotopes where a values may be as large as 4 at room temperature (see Friedman and O'Neil, 1977). The sign and magnitude of a are dependent on many factors, of which temperature is generally the most important. Other factors include chemical composition, crystal structure, and pressure.

The equilibrium fractionation factors (a_l-v) for the water liquid-vapor phase transition are 1.0098 and 1.084 at 20ºC and 1.0117 and 1.111 at 0ºC for ¹⁸O and ²H, respectively (Majoube, 1971). In both cases, a_l-v > 1, which means that the first phase (the liquid water) is "heavier" than the second phase (e.g., for a_l-v = 1.0098, the d¹⁸O of water is +9.8‰ higher than the d¹⁸O value of vapor at equilibrium). For the ice-water transition (0ºC), the values are 1.0035 and 1.0208, respectively (Arnason, 1969).

A useful equation that relates d values and fractionation factors is:

a_A-B = (1000 + d_A) / (1000 + d_B) . (Eq. 2.4)

a* = 1/a = a_B-A = R_B / R_A (Eq. 2.5)

e_A-B = ( a_A-B - 1) · 1000 . (Eq. 2.6)

e_A-B ~ d_A - d_B ~ 1000 ln a_A-B . (Eq. 2.7)

Kinetic fractionations
Chemical, physical, and biological processes can be viewed as either reversible equilibrium reactions or irreversible unidirectional kinetic reactions. In systems out of chemical and isotopic equilibrium, forward and backward reaction rates are not identical, and isotope reactions may, in fact, be unidirectional if reaction products become physically isolated from the reactants. Such reaction rates are dependent on the ratios of the masses of the isotopes and their vibrational energies, and hence are called kinetic isotope fractionations.

The magnitude of a kinetic isotope fractionation depends on the reaction pathway, the reaction rate, and the relative bond energies of the bonds being severed or formed by the reaction. Kinetic fractionations, especially unidirectional ones, are usually larger than the equilibrium fractionation factor for the same reaction in most low-temperature environments. As a rule, bonds between the lighter isotopes are broken more easily than equivalent bonds of heavier isotopes. Hence, the light isotopes react faster and become concentrated in the products, causing the residual reactants to become enriched in the heavy isotopes. In contrast, reversible equilibrium reactions can produce products heavier or lighter than the original reactants.

Many reactions can take place either under purely equilibrium conditions or be affected by an additional kinetic isotope fractionation. For example, although isotopic exchange between water and vapor can take place under more-or-less equilibrium conditions (i.e., at 100% humidity when the air is still and the system is almost chemically closed), more typically the system is out of chemical equilibrium (i.e., < 100% humidity) or the products become partially isolated from the reactants (e.g., the resultant vapor is blown downwind). Under these conditions, the isotopic compositions of the water and vapor are affected by an additional kinetic isotope fractionation of variable magnitude (see below).

Isotope fractionation factors can be defined as:

a = R_p / R_s (Eq. 2.8)

e_p-s = (a - 1) · 1000 . (Eq. 2.9)

e_p-s ~ D = d_p - d_s (Eq. 2.10)

The same formulations apply not only when part of the system is removed by a chemical or biological reaction, but also when material escapes by diffusion or outflow (e.g., by effusion through an aperture). In the latter cases the term transport fractionation factor may be preferred. The transport fractionation, like the equilibrium factors, is temperature dependent. However, unlike true kinetic fractionation factors, which can be quite appreciable, transport fractionations have only slight (positive) temperature coefficients

2.3.3 The Rayleigh equation

The isotopic literature abounds with different approximations of the Rayleigh equations, including the three equations below. These equations are so-named because the original equation was derived by Lord Rayleigh (pronounced "raylee") for the case of fractional distillation of mixed liquids. This is an exponential relation that describes the partitioning of isotopes between two reservoirs as one reservoir decreases in size. The equations can be used to describe an isotope fractionation process if: (1) material is continuously removed from a mixed system containing molecules of two or more isotopic species (e.g., water with ¹⁸O and ¹⁶O, or sulfate with ³⁴S and ³²S), (2) the fractionation accompanying the removal process at any instance is described by the fractionation factor a, and (3) a does not change during the process. Under these conditions, the evolution of the isotopic composition in the residual (reactant) material is described by:

(R / R_º) = (X₁ / X_1º) ^a-1 (Eq. 2.11)

R = R_º ƒ^(a-1) . (Eq. 2.12)

d ~ d_º ƒ^(a-1) (Eq. 2.13)

In a strict sense, the term "Rayleigh fractionation" should only be used for chemically open systems where the isotopic species removed at every instant were in thermodynamic and isotopic equilibrium with those remaining in the system at the moment of removal. Furthermore, such an "ideal" Rayleigh distillation is one where the reactant reservoir is finite and well mixed, and does not re-react with the product (Clark and Fritz, 1997). However, the term "Rayleigh fractionation" is commonly applied to equilibrium closed systems and kinetic fractionations as well (as described below) because the situations may be computationally identical.

2.3.4 Isotopic fractionation in open and closed systems

The Rayleigh equation applies to an open system from which material is removed continuously under condition of a constant fractionation factor. However, such processes can proceed under different boundary conditions, even when the fractionation factors are the same. One such system is the so-called "closed" system (or 2-phase equilibrium model), where the material removed from one reservoir accumulates in a second reservoir in such a manner that isotopic equilibrium is maintained throughout the process (Gat and Gonfiantini, 1981). An example is the condensation of vapor to droplets in a cloud where there is continuous exchange between the isotopes in the vapor and water droplets.

The isotope enrichment achieved can be very different in closed vs. open systems. For example, Figure 2.3 shows the changes in the d¹⁸O of water and vapor during evaporation (an open-system process) where the vapor is continuously removed (i.e., isolated from the water) with a constant fractionation factor a_l-v = 1.010 (i.e., the newly formed vapor is always 10‰ lighter than the residual water). As evaporation progresses (i.e., ƒ-->O), the d¹⁸O of the remaining water (solid line A), becomes heavier and heavier. The d¹⁸O of the instantaneously formed vapor (solid line B) describes a curve parallel to that of the remaining water, but lower than it (for all values of ƒ) by the precise amount dictated by the fractionation factor for ambient temperature, in this case by 10‰. For higher temperatures, the a value would be smaller and the curves closer together. The integrated curve, giving the isotopic composition of the accumulated vapor thus removed, is shown as solid line C. Mass balance considerations require that the isotope content of the total accumulated vapor approaches the initial water d¹⁸O value as ƒ-->O; hence, any process should be carried out to completion (with 100% yield) to avoid isotopic fractionation.

The dashed lines in Figure 2.3 show the d¹⁸O of vapor (E) and water (D) during equilibrium evaporation in a closed system (i.e., where the vapor and water are in contact for the entire phase change). Note that the d¹⁸O of vapor in the open system where the vapor is continuously removed (line B) is always heavier than the d¹⁸O of vapor in a closed system where the vapor (line E) and water (line D) remain in contact. In both cases, the evaporation takes place under equilibrium conditions with a = 1.010, but the cumulative vapor in the closed system remains in equilibrium with the water during the entire phase change. As a rule, fractionations in a true "open-system" Rayleigh process create a much larger range in the isotopic compositions of the products and reactants than in closed systems. This is because of the lack of back reactions in open systems. Natural processes will produce fractionations between these two "ideal" cases.

Other non-equilibrium fractionations may behave like Rayleigh fractionations in that there may be negligible back reaction between the reactant and product, regardless of whether the system is open or closed, because of kinetics. Such fractionations typically result in larger ranges of composition than for equivalent equilibrium reactions. An example of this process is biologically mediated denitrification (reduction) of nitrate to N₂ in groundwater; the N₂ is lost so it can't re-equilibrate with the nitrate, even if there was a back reaction by this organism, which there isn't. Figure 2.4 shows how Rayleigh-type fractionations affect the compositions of residual substrate, instantaneous product, and cumulative product (curved lines) during a closed-system kinetic reaction (e.g., denitrification, uptake of N by plants, or nitrification). Note that at all times, the d values of instantaneous product are " ‰" less than the corresponding d values of residual substrate. The parallel straight lines are the compositions for an open-system kinetic reaction where the supply of substrate is infinite and, hence, is not affected by the conversion of some substrate to product with a constant fractionation of e.
 
The curved lines on Figure 2.4 are very similar to the curved lines on Figure 2.3, which is reasonable since they are both solutions of the same Rayleigh equations. However, the ones in Figure 2.3 describe an open-system equilibrium process whereas the ones in Figure 2.4 describe a closed-system kinetic process. Furthermore, the straight lines in Figure 2.3 depict a closed system and the dissimilar but also straight lines in Figure 2.4 depict an open system. What is going on here? How can the same Rayleigh equations be used to produce fractionation curves described so differently?

The answer lies with where the boundary lines are drawn between the system being studied and the rest of the universe. In the case of the equilibrium fractionations illustrated in Figure 2.3, "open" means that the product, once formed at equilibrium, escapes to outside the system and does not interact again with the residual substrate (and, consequently, is no longer in equilibrium with the substrate). And "closed" means that the reactant and product remain in close contact, in their own closed (finite) system during the entire reaction, so that the two reservoirs are always in chemical and isotopic equilibrium. For the kinetic fractionations illustrated in Figure 2.4, "open" means that the supply of substrate is infinite (which it can't be in a closed system). The use of "closed" for kinetic reactions suggests that there is a limited supply of reactant, which is undergoing irreversible, quantitative, conversion to product in an isolated system.

Thusfar, a constant fractionation factor was assumed to apply throughout the process. However, this is not always the case. For example, rainout from an air mass is usually the result of a continuous cooling of the air parcel. The cooling increases the fractionation factor for the vapor-to-water (or vapor-to-ice) transition. Another conspicuous example of a changing "effective" fractionation factor is that of the evaporation of water from a surface water body to the atmosphere. As will be shown, the change in this situation is the result of the changing conditions (in this case, of the isotopic gradient) at the water-atmosphere boundary, rather than a change of the fractionation factors themselves.

Almost everyone finds the Rayleigh equations a bit confusing. Hence, we will now give some examples of how to calculate open- and closed-system fractionations, and how they affect the compositions of the residual substrate and the newly formed products of a reaction. Because much of the book focuses on water and its isotopes, we will demonstrate how to apply the Rayleigh equations by using the fractionations during water phase changes (i.e., during the condensation of vapor and the evaporation of water) as examples. For a more rigorous discussion of the topic, see Gat and Gonfiantini (1981) or Gat (1996); chapter 2 of Clark and Fritz (1997) provides a well-illustrated and exceptionally clear discussion of this fascinating topic. Many other reactions (e.g., sulfate reduction, methane oxidation, ammonia volatilization, and nitrification) can also be modeled with Rayleigh-type models; the same principles described here apply to these kinetic reactions.

Condensation of water

The isotopic composition of moisture in the marine atmosphere is controlled by the air-sea interaction processes as described by Craig and Gordon (1965), Merlivat and Jouzel (1979), and others. As air masses move across continents and lose water by rainout, they become depleted in the heavy isotopic species (H₂¹⁸O and HDO) because the liquid phase is enriched in the heavy isotopic species relative to the vapor phase (see Chapter 3). The evolution of the isotopic composition is adequately described by a Rayleigh process (in this case it is condensation) in those cases where rainout is the only factor in the atmospheric-moisture budget (Dansgaard, 1964; Gat, 1980). A Rayleigh fractionation plot for condensation would be very similar to Figure 2.3, except that all the curves would bend down instead of up because the residual vapor and water condensed would become progressively lighter over time, not heavier (as they do for evaporation).

When the isotopic compositions of precipitation samples from all over the world are plotted relative to each other on d¹⁸O versus dD plots, the data form a linear band of data that can be described by the equation (Craig, 1961):

dD = 8 d¹⁸O + 10 (Eq. 2.14)

The y-intercept value of 10 in the GMWL equation is called the deuterium excess (or d-excess, or d parameter) value for this equation. The term only applies to the calculated y-intercept for sets of meteoric data "fitted" to a slope of 8; typical d-excess values range from 0 to 20 (see Chapter 3). The fact that the intercept of the GMWL is 10 instead of 0 means that the GMWL does not intersect d¹⁸O = dD = 0, which is the composition of average ocean water (VSMOW). The GMWL does not intersect the composition of the ocean, the source of most of the water vapor that produces rain, because of the 10‰ kinetic enrichment in D of vapor evaporating from the ocean at an average humidity of 85%.

The Rayleigh law is formulated in approximate differential form and using notation as:

dd ~ e^* · dlnƒ (Eq. 2.15)

e⁺ = (a⁺ - 1) · 10³ . (Eq. 2.16)

The equilibrium fractionation factor a between liquid and vapor can be defined in two ways, which are mathematical inverses: a = R_l /R_v or a = R_l /R_v, where R_l and R_v are the isotopic ratios of the liquid and vapor, respectively. However, Craig and Gordon (1965) defined equilibrium fractionation factors such that a⁺ = 1/a^*, so that a⁺ = R_l /R_v >1 and a^* = R_v /R_l <1 (and, consequently, e⁺>0 and e⁺ -e^*). This usage has become traditional when discussing atmospheric processes. In general, a⁺ (often abbreviated to simply a) is used for condensation problems, whereas a^* is commonly preferred for evaporation problems. Values for a⁺ can be calculated from Majoube (1971). Although the use of a⁺ vs. a^* may simplify calculations, many other people find it more convenient to use the definition of fractionation factor that produces a >1, despite tradition.

As rain condenses, the heavier isotopes of water (mainly HD¹⁶O and H₂¹⁸O) are preferentially removed from the air mass (and into the rain), and the air mass consequently becomes progressively lighter in isotopic composition (i.e., higher concentrations of H₂¹⁶O). Hence, the isotopic compositions of successive aliquots of rain become progressively lighter in the heavier isotope due to continuing rainout of the heavy isotopes. As will be described in Chapter 3, this is why the d values of rain become lighter as storms move inland from the ocean. At any point along the storm trajectory (i.e., for some specific fraction ƒ of the total original vapor mass), the d¹⁸O of the residual fraction of vapor in the air mass can be calculated by:

d ~ d_o + e ln (ƒ) (Eq. 2.17)

d¹⁸O_v ~ d_o¹⁸O_v + e_l-v · ln ƒ (Eq. 2.18)

d¹⁸O_l ~ d¹⁸O_v + e_l-v (Eq. 2.19)

Another commonly used formulation of the Rayleigh equation for systems with a constant fractionation factor is: d ~ d_o - e ln (ƒ). In this case, the enrichment factor in the Rayleigh equation has a negative sign, instead of the positive sign shown earlier (Equation 2.17), because of different definitions for a (and hence for values). The choice of either a^* or its reciprocal value a⁺ for the equilibrium fractionation factor is dictated only by convenience; there is no "right" way. If there is any confusion about how the fractionation terms are defined in some paper, just try a few test calculations to make sure the d values for a reaction change in the "right" direction (e.g., with biological reactions, residual reactants get heavier; during condensation of rain, residual vapor gets lighter; etc); if the d values don't change as expected, this probably means that the fractionation factor being used is the inverse of what should be used in the equation.

Evaporation of water

Evaporation from an open-water surface fractionates the isotopes of hydrogen and oxygen in a manner which depends on a number of environmental parameters, the most important of which is the ambient humidity. This is illustrated for various relative humidities in Figure 2.5. The higher the humidity, the smaller the change in d¹⁸O and dD during evaporation. For example, at 95% humidity, the d values are constant for evaporation of the last 85% of the water. Evaporation results in lines with slopes <8 on a d¹⁸O vs. dD plot (i.e., the data plot on lines below the MWL that intersect the MWL at the composition of the original water).

Evaporation at 0% humidity describes open-system evaporation ("open" in terms of the definitions for Figure 2.3). Note that the two upper diagrams on Figure 2.5 are Rayleigh-type plots, similar to Figure 2.3 but with larger changes in d¹⁸O during open-system evaporation on Figure 2.5. The d values on the curved fractionation lines on the upper diagrams plot along nearly straight lines on the lower d¹⁸O vs. D plot. The "length" of the evaporation lines on the d¹⁸O vs. dD plot reflect the range of values of water produced during total evaporation under different humidities. For example, the short line for 95% humidity indicates that the water changes little during the entire evaporation process.
 
Evaporation under almost 100% humidity conditions is more-or-less equivalent to evaporation under closed-system conditions (i.e., isotopic equilibrium is possible), and data for waters plot along a slope of 8 (i.e., along the MWL). However, the shapes of the curves in the upper diagrams for 95% humidity are not the same as for closed-system equilibrium fractionation (Figure 2.3); instead, they are similar to the infinite-reservoir kinetic fractionation shown on Figure 2.4. This is because the curves in Figure 2.5 were calculated using both an equilibrium fractionation for the phase change and a kinetic fractionation (Equation 2.20) for the diffusion of water vapor across the water-atmosphere interface (Gat and Gonfiantini, 1981). This is also the explanation for the larger open-system fractionations on Figure 2.5 than on Figure 2.3.

The most useful model for the isotope fractionation during evaporation is that of Craig and Gordon (1965). It is schematically shown in Figure 2.6 (redrawn from Gat, 1996). This model assumes that equilibrium conditions apply at the air/water interface (where the humidity is 100%), that there is a constant vertical flux, and that there is no fractionation during fully turbulent transport. For a detailed derivation, see Craig and Gordon (1965) or Gat (1996).

At the water-air interface, there is a balance between two opposing water fluxes: one upward from the water surface and one downward consisting of atmospheric moisture. When the humidity is 100% (i.e., the air is saturated), the upward and downward physical fluxes can become equivalent and their isotopic compositions may then reach equilibrium (note that equilibrium does not mean that the d values of the two reservoirs are identical, only that they differ by the equilibrium enrichment factor); see Clark and Fritz (1997) pages 26-27.
 
The changes in humidity and corresponding changes in the isotopic composition of vapor across the transition between the water and the free atmosphere are given as dashed lines in Figure 2.6. Note that where h = 1 (in the so-called "equilibrium vapor" layer between the interface and the boundary layer where the humidity is 100%), the vapor is in equilibrium with the liquid (i.e., d_v = d - e^*). When the air is undersaturated (i.e., h < 1), a net evaporative flux is produced. The rate determining step for evaporation is the diffusion of water vapor across the air boundary layer, which occurs in response to the humidity gradient between the surface and the fully turbulent ambient air (Figure 2.6).

The isotopic composition of the evaporated moisture (for either oxygen or hydrogen isotopes) can be formulated as:

d_E = (a^*d_w - hd_a - e) / [(1- h) + D /1000] ~ (d_w - hd_a - e) / (1- h) (Eq. 2.20)

De = (1- h) · q · n · e_k (Eq. 2.21)

The values of d_E and d_w define a line in d¹⁸O vs. dD space called the evaporation line (Figures 2.5 and 2.7) whose slope is given by:

S = [h( d_a - d_w) + e]_2H / [h( d_a - d_w) + e]_18O . (Eq. 2.22)

2.3.5 Biological fractionations

Biological processes are generally unidirectional and are excellent examples of kinetic isotope reactions. Organisms preferentially use the lighter isotopic species because of the lower energy "costs" associated with breaking the bonds in these molecules, resulting in significant fractionations between the substrate (heavier) and the biologically mediated product (lighter). Kinetic isotopic fractionations of biologically-mediated processes vary in magnitude, depending on reaction rates, concentrations of products and reactants, environmental conditions, and -- in the case of metabolic transformations -- species of the organism. The variability of the fractionations makes interpretation of isotopic data difficult, particularly for nitrogen and sulfur. The fractionations are very different from, and typically larger than, the equivalent equilibrium reaction. The magnitude of the fractionation depends on the reaction pathway utilized (i.e., which is the rate-limiting step) and the relative energies of the bonds severed and formed by the reaction. In general, slower reaction steps show greater isotopic fractionation than faster steps because the organism has time to be more selective (i.e., the organism saves internal energy by preferentially breaking light-isotope bonds).

If the substrate concentration is large enough that the isotopic composition of the reservoir is insignificantly changed by the reaction (O'Leary, 1981) or if the isotopic ratio of the product is measured within an infinitely short time period (Mariotti et al., 1981), the fractionation factor can be defined as in Equation 2.8 (i.e., the straight lines on Figure 2.4 for the "open system model"). For unidirectional reactions (Figure 2.4, curved lines), the change in the isotope ratio of the substrate relative to the fraction of the unreacted substrate can be described by the Rayleigh equation:

R_s / R_so = ƒ^(a-1) (Eq. 2.23)

Readers of this book and articles dealing with isotope fractionations must be careful: both fractionation and enrichment factors are defined in various ways by different authors, especially in the biological literature. Kinetic fractionation factors are typically described in terms of enrichment or discrimination factors, using such symbols as b, e, or D. In particular, the enrichment factor is sometimes defined in reverse (i.e., e_s-p), and some researchers define a "discrimination factor" D_s/p = (a_s/p -1)1000, where s/p denotes "substrate relative to products." Good discussions of fractionations associated with biological processes include Hübner (1986) and Fogel and Cifuentes (1993).

A good example of the complexities of kinetic reactions is given by the fractionation between CO₂ and photosynthetic organic carbon. The fractionation can be described by the model (Fogel and Cifuentes, 1993):

D = A + (C_i / C_a)(B - A) (Eq. 2.24)

2.4 Sample Collection, Analysis, and Quality Assurance

2.4.1 Sampling guidelines

Considerable field effort is often required to collect a sample that adequately represents the average composition of the medium being sampled, at the time it is sampled. For small streams, this can be as simple as collecting water as it flows over a weir or rock ledge. For large rivers, lakes, soils, and organisms, mass-integrated composites may be required. Adequate coverage of this vital topic is beyond the scope of this chapter. The reader is advised to look at the references given in subsequent chapters, or consult colleagues who routinely collect such samples. Other useful sources of information include: Clark and Fritz (1997; chapter 10: "Field methods for sampling"), Mazor (1997), and the Web pages of various isotope laboratories.

Below is a potpourri of guidelines and suggestions related to collecting, bottling, and preserving samples for analysis of the most commonly-used environmental isotopes. The reader should keep in mind that the optimum methods often depend on the laboratory chosen for analysis and their preferred preparation methods, and should always inquire before planning the field campaign. Collection of duplicates is alway advisable -- in case of breakage of samples during transport and to use as checks of the reproducibility of the laboratory (i.e., submit 5-10% of these as "blind duplicates," with different sample ID numbers than their duplicates).

d¹⁸O/ ²H of water
Natural waters are easy to collect. The water sample is put in a clean dry bottle, which is filled almost completely to the top, and capped tightly. The main objective is to protect the sample from evaporation and exchange with atmospheric water vapor. Samples should not be filtered unless they contain oil (e.g., mineral oil added to rain collectors to help prevent evaporation) or contain abundant particulate matter. Bottle rinsing, chilling, and addition of preservatives are unnecessary. Freezing does not affect the composition of the water but can break the bottles in transit; for this reason, many users prefer plastic bottles. Our experience suggests that caps with conical plastic inserts (e.g., "poly-seal" caps) are the most reliable, followed by teflon-lined caps. For extended storage, use of glass bottles and waxing of the caps is advisable. Sample-size is lab-dependent; typical volumes range from 10-60 mL. In some laboratories, samples as small as a few µL can be analyzed.

Determinations of both hydrogen and oxygen isotope ratios are usually made on the same bottle of water. It is wise to collect many more samples than one can afford to analyze at the present; samples have a long shelf life if bottled correctly, and can be archived for future analysis. One should make sure that the laboratory chosen to analyze the samples normalizes their values according to IAEA guidelines (Coplen, 1994), and reports values relative to VSMOW. If the samples are saline, one should check whether the lab is preparing samples by an equilibration or quantitative-conversion method (see below). Waters with high contents of volatile organic matter may require distillation.

For many purposes, especially hydrograph separations (see Chapter 1), analysis for all samples for both oxygen and hydrogen isotopes is unnecessary because of the high correlation coefficient between these isotopes (see Chapter 3). A sensible alternative is to have some smaller percentage analyzed for both isotopes, either initially or after the data for the first isotope are evaluated. For hydrograph studies in arid environments or studies that involve evaporated water in ponds or wetlands, analysis of samples for both isotopes is probably advisable. Because most labs have fewer problems analyzing waters for d¹⁸O than for d²H, if the samples are not analyzed in duplicate and will only be analyzed for one isotope, it is usually better to choose d¹⁸O.

Solid and vapor samples are more difficult to collect for d¹⁸O and d²H. Snow and ice samples can be collected in tightly sealed bags or jars, melted overnight, and then poured into bottles. Plant and soil samples should be collected in air-tight containers matched to the sample size. Common procedures include waxing of soil cores, use of heat-sealed bags, or insertion into tiny tree-core-size vials. Water vapor samples are collected by pumping vapor through a cold-trap where the vapor is quantitatively retained. For more information on various sampling procedures, see Chapters 3-6.

Tritium
The amount of water needed for tritium analysis depends on the age of the water (old waters contain little tritium) and the sensitivity of analysis needed. Typical sample sizes range from 10 mL to 1L. Samples are collected in unrinsed glass or high-density polyethylene bottles and should not be filtered. The bottles should then be sealed and returned to the laboratory for analysis. The collection date should be noted on the bottle to obtain an accurate determination of the tritium concentration for the time of collection.

d¹³C and ¹⁴C of dissolved inorganic carbon
There are two main methods in common use for the collection of DIC (dissolved inorganic carbon) for the measurement of ¹³C or ¹⁴C, depending on which of two laboratory preparation methods is being used: gas stripping or carbonate precipitation. Both preparation methods insure quantitative removal of the DIC and provide a d¹³C or ¹⁴C value for total DIC. Analysis for d¹³C generally requires 10-100 µM of carbon. Analysis of ¹⁴C by conventional beta-counting methods requires as much as 1 g of C; analysis by AMS usually requires about 1 mg of C.

For laboratories that use a gas-stripping method to extract the CO₂, samples are usually collected in sample-rinsed glass bottles with septa-caps, or in vessels with stopcocks or valves. Such samples should be filtered to remove particulate carbon, and perhaps poisoned (using mercuric chloride, acid, or organic biocide) to prevent biological activity; the bottles should be kept chilled until analyzed to prevent biological fractionations.

The alternative technique is the precipitation method. Samples should be pre-filtered if there might be suspended carbonate particulate material in the water. The carbonate is precipitated by adding a strongly basic solution of strontium or barium chloride (Gleason et al., 1969) to the sample in a sample-rinsed bottle. The base increases the pH to 10-11 where all the inorganic carbon is CO3-2, and the Ba or Sr precipitates all the DIC in the water. This reagent and the treated samples must be protected against contamination by atmospheric CO2. Glass bottles are best because CO2 diffuses through most plastic bottles. Bottles should have poly-seal caps that are taped securely. Bottles should be individually wrapped in bubble paper and shipped in insulated boxes or coolers filled with artificial "peanuts" to insure against breakage.

d¹⁵N of dissolved inorganic nitrogen
A number of different preparation methods are in common use; inquire what collection method is preferred by the contract laboratory for their particular preparation method. In particular, it is important to verify that the laboratory is accustomed to analyzing natural abundance samples. Laboratories that primarily analyze agricultural samples often use methods that are appropriate for labeled (¹⁵N-spiked) samples but have unacceptable analytical precisions for natural abundance studies. Check that the laboratory has a good track record for natural samples. Samples can be analyzed for the d¹⁵N of ammonium and/or nitrate; analysis of total nitrogen is probably worthless. Generally, samples are filtered through 0.1 micron filters, put in rinsed glass bottles, poisoned (with sulfuric acid, mercuric chloride, or chloroform), chilled or frozen, wrapped in insulating packing material, and sent to the laboratory in ice chests. Sample-size requirements are in the range of 10-100 µM of N. Nitrate samples can also be analyzed for d¹⁸O in a few laboratories.

An alternate method is to concentrate the NO₃ or NH₄ on anion or cation exchange resins (Garten, 1992; Silva et al., submitted; Chang et al., in review). Collection of nitrate on anion exchange resins eliminates the need to send large quantities of chilled water back to the laboratory, eliminates the need for hazardous preservatives, makes it easier to archive samples, and allows analysis of extremely low-nitrate waters.

d³⁴S of dissolved sulfate
Depending on the sulfate concentration, samples are filtered directly into glass bottles or are first pre-concentrated on an exchange resin. Sulfate from dilute waters should be collected on ion exchange resin in the field if the concentration of sulfate in the water is believed to be less than 20 mg/L. Similar to collection methods for NO₃ or NH₄ on ion exchange resins, collection of sulfate on exchange resins avoids problems of incomplete precipitation of BaSO₄ in dilute samples, eliminates the need to send large quantities of chilled water back to the lab, eliminates the need for hazardous preservatives, makes it easier to archive samples, and allows analysis of extremely low-sulfate waters.

Low-sulfate water samples are first acidified before passing through ion exchange columns. The sulfate is then eluted from the resin using a relatively small volume of concentrated barium chloride solution. The final volume of the solution is much less than that of the original water sample and the sulfate from the sample is thus concentrated in this much smaller volume (generally 10-500 µM of SO₄ is required). The solution is reacidified and sulfate is precipitated by adding BaCl_2. BaSO₄ is then collected by filtration and analyzed for d³⁴S. Sulfate can also be analyzed for d¹⁸O in some laboratories. Large quantities of sulfate can also be analyzed for ³⁵S, a natural radioisotope with a half-life of 87 days, using liquid scintillation counting.

C, H, N, O, and S isotopes of solid samples
Solid organic and inorganic samples (e.g., animals, plants, minerals, and soils) and liquids (such as oils) can also be analyzed for their isotopic composition. Particulate matter in water can be captured on fiberglass filters and processed similar to methods used for other solid samples. Requirements for solid samples are similar to the requirements for solute samples of the same element (i.e., 1-100 µM of the element of interest). Biologically labile samples (e.g., leaves, fish, manure) should be kept cold until processed. Freeze-drying is an ideal means for preserving the samples; air-drying results in loss of volatile organic matter and probably some isotopic fractionation.

Lithogenic (metals and semi-metals) isotopes
The sample size is dependent on the species being analyzed. Analysis of Sr, Li, or B requires a minimum of 1 µg; Pb and Nd require a minimum of 0.1 µg. Aqueous samples should be filtered; 0.1 micron filters are best for Nd and Pb, and 0.45 micron filters are best for Sr, Li, and B (Thomas D. Bullen, pers. comm. 1997). Aqueous samples are collected in rinsed plastic bottles and acidified to pH ~ 2 using Ultrex HNO₃. Blanks should be sent to the laboratory along with your samples, including the triple distilled water used for filtering and the clean water run through the processing equipment. One must be careful about possible contamination with lithium grease, borate soaps or detergents, and strontium chloride reagents.

2.4.2 Analytical methods and instrumentation

Stable isotopes are analyzed either on gas- or solid-source mass spectrometers, depending on both the masses of the isotopes and the existence of appropriate gaseous compounds stable at room temperature. Radioisotopes can be analyzed by counting the number of disintegrations per unit time on gamma ray or beta particle counters, or analyzed on mass spectrometers.

Gas-source mass spectrometers
Many methods are used to prepare gases for C, H, N, O, and S (CHNOS) stable isotope content, but in all the cases the basic steps are the same. Sample preparation involves the quantitative conversion or production of pure gas from solely the compound of interest, cryogenic or chromatographic purification of the gas, introduction of the gas into the mass spectrometer, ionization to produce positively charged species, dispersion of different masses in a magnetic field, impaction of different masses on different collector cups, and measurement of the ratios of the isotopes in the ionized gas. In general, hydrogen is analyzed as H₂, oxygen and carbon are both analyzed as CO₂, nitrogen is analyzed as N₂, and sulfur is usually analyzed as SO₂. The analytical precisions are small relative to the ranges in values that occur in natural earth systems. Typical one standard deviation analytical precisions for oxygen, carbon, nitrogen, and sulfur isotopes are in the range of 0.05 to 0.2‰; typical precisions for hydrogen isotopes are poorer, from 0.2 to 1.0‰, because of the lower ²H:¹H ratio.

Although the topic is rarely discussed, the activity coefficients of isotopic species are not all equal to 1 (i.e., the isotope concentration of a sample is not necessarily equal to the isotope activity). The activity coefficient for a particular isotope can be positive or negative, depending on solute type, molality, and temperature. The isotopic compositions of waters and solutes can be significantly affected by the concentration and types of salts because the isotopic compositions of waters in the hydration spheres of salts and in regions farther from the salts are different (see Horita (1989) for a good discussion of this topic). In general, the only times when it is important to consider isotope activities is for low pH, high SO₄, and/or high Mg brines because the activity and concentration d values of these waters (d_a and d_c) are significantly different. For example, the difference (dD_a- dD_c) between the activity and concentration d values for sulfuric acid solutions in mine tailings is about +16‰ for 2 molal solutions. For normal saline waters (e.g., seawater), the activity coefficients for d¹⁸O and d²H are essentially equal to 1.

Virtually all laboratories report d¹⁸O activities (not concentrations) for water samples. The d²H of waters may be reported in either concentration or activity d values, depending on the method used for preparing the samples for analysis. Methods that involve quantitative conversion of the H in H₂O to H₂, produce d_c values. Methods that equilibrate H₂O with H₂ (or H₂O with CO₂) produce d_a values. "Equilibrate" in this case means letting the liquid and gas reach isotopic equilibrium at a constant, known temperature. To avoid confusion, laboratories and research papers should always report the method used.

Most conventional CHNOS mass spectrometers are dual inlet machines that have both a sample and a standard inlet or introduction port. In such instruments, the ratios of the isotopes of interest (e.g., ¹³C/¹²C) in the sample gas are measured relative to the same ratios in a gaseous standard that is analyzed more-or-less simultaneously. Such instruments usually have either "double collectors" or "triple collectors," meaning that either two or three masses of the ionized gas can be measured simultaneously. For example, N₂ contains three species: ¹⁴N¹⁴N, ¹⁴N¹⁵N and ¹⁵N¹⁵N (i.e., masses 28, 29, and 30). A triple-collecting mass spectrometer would measure the abundances of all these species relative to the abundances in a gaseous standard introduced through the "standard" inlet. A double-collecting mass spectrometer would only measure the 28 and 29 masses (actually m/e is measured since the molecules are ionized, with positive charges).

Another type of stable isotope mass spectrometer is the so-called continuous flow mass spectrometer. Such instruments may lack a dual inlet, and usually have triple collectors. These instruments represent a "marriage" of chromatography and mass spectrometry, and are similar to conventional organic mass spectrometers in that gas samples are introduced into the mass spectrometer within a stream of helium gas, usually from an automated sample preparation unit (e.g., an elemental analyzer or gas chromatograph). In general, the analytical precision available for continuous flow mass spectrometers is slightly poorer than with conventional methods, but this may change in the next few years. The main advantage of the continuous flow method is that such instruments are very easily combined with various on-line preparation systems, dramatically lowering the manpower cost of isotope analyses. For an exceptionally thorough discussion of modern stable isotope mass spectrometry see Barrie and Prosser (1996).

Solid-source mass spectrometers
Elements analyzed as solids (e.g., strontium, lithium, boron, lead, etc.) are prepared by precipitating selected compounds on wire filaments, loading the filaments into the source of a thermal ionization (solid-source) mass spectrometer, ionizing the compounds to produce gases (negative or positive charged), and measuring the abundances of selected isotopes in the gas on multiple collectors. Some light-mass solids (e.g., boron and lithium) are reported in the standard d units. Generally, the heavier-mass elements are reported in terms of the relative abundances of two isotopes (e.g., ²⁰⁷Pb/²⁰⁶Pb); however, strontium isotope abundances (⁸⁷Sr/⁸⁶Sr) are occasionally reported in d notation relative to some arbitrary standard. Solid-source mass spectrometry has been shown to give a more accurate analysis of certain radium and uranium isotopes that conventionally were measured by decay counting methods.

Gas and liquid scintillation counters
Radioactive isotopes can be measured by a number of methods, depending on the mass, abundance, type of decay involved, accuracy desired, and money available. Some, of course, can be analyzed on solid source mass spectrometers (e.g., uranium-series isotopes). Otherwise, radioisotopes are analyzed on liquid scintillation counters (LSC) and gas proportional counters (both with enrichment), and on accelerator mass spectrometers (see below). Liquid scintillation and gas proportional systems are the most common systems used for light isotopes with beta decays. Gas proportional counting usually requires that the isotope being analyzed form a suitable counting gas, so that elements with high electronegativities, such as chlorine and sulphur, are not suitable for this type of analysis. The two isotopes most commonly used in hydrology, tritium and ¹⁴C, have generally been analyzed using liquid scintillation or gas proportional counting. Radon is analyzed either by gas Geiger or proportional counting in the field, or sent to a laboratory for liquid scintillation counting, depending on the accuracy desired. For isotopes that decay by gamma and alpha emission, and beta emissions where the target isotope cannot be reduced to a suitable chemical form for LCS or gas counting, the use of solid scintillation-counting using crystals, or more advanced systems like lithium-germanium drift counters, have been utilized.

Accelerator mass spectrometry
Accelerator mass spectrometers (AMS), sometimes called "tandem" accelerators, are very large (> 10m), expensive, high-resolution, mass spectrometers (with either gas or solid-sources) that accelerate charged particles through very high (mega-volt) electrical fields to separate different isobars and isotopes (Figure 2.1). These instruments can analyze some radioactive species more rapidly, with greater accuracy, and/or with much smaller sample sizes (e.g., mg rather than g samples) than previous counting methods. For example, tritium can now be analyzed using the helium ingrowth method, although it frequently requires long delays (6 months) to accrue enough ³He to obtain an accurate analysis. AMS has become the method of choice for some isotopes, such as ¹⁴C, ³⁶Cl and ¹²⁹I. It will give accuracies close to those obtained by traditional methods, and samples can be analyzed much more rapidly by AMS.

2.4.3 Quality assurance of contract laboratories

How does one find a good contract laboratory for analyzing samples? Choices include university laboratories, private commercial companies, and government laboratories that can accept contract (or collaborative) work. A primary selection criterion should be that the laboratory has been making the desired type of analysis for several years on a routine basis (e.g., samples submitted to some university laboratories may be analyzed by temporary student help, who do not perform analyses on a routine basis). Make inquiries among colleagues about the long-term track record of the laboratory. Good laboratories have active QA/QC programs, with documentation generally available on request. In our opinion, the laboratory should analyze about 5-15% of the samples in duplicate, as an internal verification that "everything" is operating correctly. Furthermore, laboratories with automated preparation systems and computer-controlled data management systems probably produce better and more reliable data on a long-term basis than laboratories where everything is done manually. The reader is cautioned to beware of bargains (caveat emptor); quality work usually costs more than the average price. Furthermore, the potential long-term cost of wrong interpretations, due to bad data, should be factored into the total cost of the analyses when evaluating laboratory choices.

One should also consider collecting duplicates in case the sample bottle is broken or lost in transit. Most laboratories routinely analyze each sample only once; if high precision data are required, either request duplicate analysis of each sample (and triplicates if the duplicates do not agree within some predetermined range) or send in "blind" duplicates. Sending in 10-15% blind duplicates is advisable, in any case. If any result seems questionable, immediately request a repeat. Most laboratories keep analyzed samples for a couple months before discarding them and will reanalyze modest numbers of samples at no additional cost.

For water samples, immediately plot the data on a dD vs. d¹⁸O diagram; outliers, especially ones that plot appreciably above the GMWL (the line defined by "dD = 8 d¹⁸O + 10" -- see Figure 2.7), should be viewed with skepticism and possibly reanalyzed. Few natural processes produce waters that plot significantly above the GMWL; exceptions include methanogenesis in landfills (Baedecker and Back, 1979) and silicate hydrolysis.

2.5 Applications of Isotope Tracers in Catchment Hydrology

The applications of environmental isotopes as hydrologic tracers in low temperature (< 40ºC) systems fall into two main categories:

• tracers of the water itself: water isotope hydrology
• tracers of the solutes in the water: solute isotope biogeochemistry.

Chapters 10 - 22 provide an overview of some of the myriad applications of environmental isotopes to catchment hydrobiogeochemistry. Most of the chapters focus on a particular type of catchment and how isotopes can be used to understand the functioning of the catchment, or on specific kinds of uses of isotopes in catchments (e.g., on determining flowpaths or obtaining climatic information). For general information on uses of isotopes of some particular element, especially for applications "beyond the catchment," useful Web sources of information include:

http://www.iaea.or.at/ The Web site for the IAEA (International Atomic Energy Agency). This page contains information on IAEA publications, how to order isotope reference materials, and how to access the IAEA isotope databases.

http://www.nist.gov/ The Web site for the National Institute of Standards and Technology (formerly NBS) provides information on ordering isotope reference materials.

http://wwwrcamnl.wr.usgs.gov/isoig/ The Web site of the USGS Isotope Interest Group (IsoIG). This page contains a variety of links to isotope-related resources, including short notes on isotope fundamentals and applications, information about isotope reference standards, links to several search engines for finding publications, and a link to the Web site for the ISOGEOCHEM email discussion group. The ISOGEOCHEM listserver primarily focuses on the stable isotope community, contains links to many isotope laboratories, and contains an archive of previous emails with a full search engine. If any of the other URLs listed here have changed, check the IsoIG Web site for updated links.

http://wwwrcamnl.wr.usgs.gov/isoig/period/ This Web site contains a "clickable" periodic table that provides information about many isotopes, including lists of noteworthy publications, and descriptions of the uses of these isotopes to hydrology, geology, and biology; it contains a search engine.

http://wwwrcamnl.wr.usgs.gov/isoig/isopubs/ The Web site for this book.

The sections below are intended as a brief introduction to the many uses of environmental isotopes in catchment hydrology, for readers who might be unfamiliar with what various isotopes have to offer, and a lead-in to the more thorough discussions in succeeding chapters.

2.5.1 Water isotope hydrology

Isotope Hydrology addresses the application of the measurements of isotopes that form water molecules: the oxygen isotopes (oxygen-16, oxygen-17, and oxygen-18) and the hydrogen isotopes (protium, deuterium, and tritium). These isotopes are ideal tracers of water sources and movement because they are integral constituents of water molecules, not something that is dissolved in the water like other tracers that are commonly used in hydrology (e.g., dissolved species such as chloride). Water isotopes can sometimes be useful tracers of water flowpaths, especially in groundwater systems where a source of water with a distinctive isotopic composition forms a "plume" in the subsurface (see Chapter 18 or Bullen et al., 1996).

In most low-temperature environments, stable hydrogen and oxygen isotopes behave conservatively in the sense that as they move through a catchment, any interactions with oxygen and hydrogen in the organic and geologic materials in the catchment will have a negligible effect on the ratios of isotopes in the water molecule. Although tritium also exhibits insignificant reaction with geologic materials, it does change in concentration over time because it is radioactive and decays with a half-life of about 12.4 years. The main processes that dictate the oxygen and hydrogen isotopic compositions of waters in a catchment are: (1) phase changes that affect the water above or near the ground surface (evaporation, condensation, melting), and (2) simple mixing at or below the ground surface.

Stable oxygen and hydrogen isotopes can be used to determine the contributions of old and new water to a stream (and to other components of the catchment) during periods of high runoff because the rain or snowmelt (new water) that triggers the runoff is often isotopically different from the water already in the catchment (old water). Chapters 3-7 discuss the sources of variability in the isotopic compositions of water in rain, snow, soil water, plants, and groundwater (respectively) and explain why the old and new water components often have different isotopic compositions. Tritium (³H) is an excellent tracer for determining time scales for the mixing and flow of waters, and is ideally suited for studying processes that occur on a time scale of less than 100 years (see Chapters 3, 7, and 9). Chapters 10-14 explore how isotopes can be used to investigate hydrologic processes in various catchment types (rain-dominated temperate and tropical catchments, snowmelt-dominated catchments, arid basins, and lake-dominated systems, respectively).

2.5.2 Solute isotope biogeochemistry

Isotope Biogeochemistry addresses the application of isotopes of constituents that are dissolved in the water or are carried in the gas phase. Isotopes commonly used in solute isotope biochemistry research include the isotopes of: sulfur (Chapter 15), nitrogen (Chapter 16), and carbon (Chapters 17 and 18). Less commonly applied isotopes in geochemical research include those of: strontium, lead, uranium, radon, helium, radium, lithium, and boron (see Chapters 8, 9, 18, 19, and 20).

Unlike the isotopes in the water molecules, the ratios of solute isotopes can be significantly altered by reaction with biological and/or geological materials as the water moves through the catchment. Although the literature contains numerous case studies involving the use of solutes (and sometimes solute isotopes) to trace water sources and flowpaths, such applications include an implicit assumption that these solutes are transported conservatively with the water. In a strict sense, solute isotopes only trace solutes. Solute isotopes also provide information on the reactions that are responsible for their presence in the water and the flowpaths implied by their presence.

As discussed above, water isotopes often provide relatively unambiguous information about residence times and relative contributions from different water sources; these data can then be used to make hypotheses about water flowpaths. Solute isotopes can provide an alternative, independent isotopic method for determining the relative amounts of water flowing along various subsurface flowpaths. However, the least ambiguous use of solute isotopes in catchment research is tracing the relative contributions of potential solute sources to groundwater and surface water. Although there has been extensive use of carbon, nitrogen, and sulfur isotopes in studies of forest growth and agricultural productivity, solute isotopes are not yet commonly used for determining weathering reactions and sources of solutes in catchment research. This book attempts to remedy that situation.

2.5.3 Mixing

Isotopic compositions mix conservatively. In other words, the isotopic compositions of mixtures are intermediate between the compositions of the endmembers. Despite the awkward terminology (i.e., the notation and units of ‰) and negative signs, the compositions can be treated just like any other chemical constituent (e.g., chloride content) for making mixing calculations. For example, if two streams with known discharges (Q₁, Q₂) and known d¹⁸O values (d¹⁸O₁, d¹⁸O₂) merge and become well mixed, the d¹⁸O of the combined flow (Q_T) can be calculated from:

Q_T = Q₁ + Q₂ (Eq. 2.25)

d¹⁸O_TQ_T = d¹⁸O₁Q₁ + d¹⁸O₂Q₂ . (Eq. 2.26)

What is not so obvious is that on many types of X-Y plots, mixtures of two endmembers will not necessarily plot along lines but instead along hyperbolic curves (Figure 2.9a). This is explained very elegantly by Faure (1986) using the example of ⁸⁷Sr/⁸⁶Sr ratios. The basic principle is that mixtures of two components that have different isotope ratios (e.g., ⁸⁷Sr/⁸⁶Sr or ¹⁵N/¹⁴N) and different concentrations of the element in question (e.g., Sr or N) form hyperbolas when plotted on diagrams with coordinates of isotope ratios versus concentration. As the difference between the elemental concentrations of two components (endmembers) approaches 0, the hyperbolas flatten to lines. The hyperbolas are concave or convex depending on whether the component with the higher isotope ratio has a higher or lower concentration than the other component. Mixing hyperbolas can be transformed into a straight lines by plotting isotope ratios versus the inverse of concentration (1/C), as shown in Figure 2.9b.

Graphical methods are commonly used for determining whether the data support an interpretation of mixing of two potential sources or fractionation of a single source. Implicit in such efforts is often the idea that mixing will produce a "line" connecting the compositions of the two proposed endmembers whereas fractionation will produce a "curve." However, as shown in Figure 2.10a, both mixing and fractionation (in this case, denitrification) can produce curves (Mariotti et al., 1988), although both relations can look linear for small ranges of concentrations. However, the equations describing mixing and fractionation processes are different and under favorable conditions, the process responsible for the curve can be identified. This is because Rayleigh fractionations are exponential relations (Equation 2.13), and plotting d values versus the natural log of concentration will produce a straight line (Figure 2.10b). If an exponential relation is not observed and a straight line is produced on a d vs 1/C plot (like Figure 2.9b), this supports the contention that the data are produced by simple mixing of two endmembers.
 
2.5.4 Isotopically labeled materials

Man-made materials with isotopic compositions that are not observed in nature are called "spiked" or isotopically labeled materials. There are many commercial suppliers of isotopically labeled liquids, gases, and solids -- some with multiple-labeled atoms (e.g., water with unusual ¹⁸O/¹⁶O and ²H/¹H ratios, or organic molecules with various percentages of the elements of specific functional groups labeled with uncommon isotopic compositions). The most common watershed use of spiked tracers is for agricultural studies of plant uptake of nutrients. Other applications include whole-catchment experiments where labeled NH₄, NO₃, or SO₄ is sprinkled in artificial rain (see Chapters 15 and 16), and plot studies where labeled H₂O is applied to the land surface to make it easier to trace to movement of "new" water into the subsurface.

Materials can be enriched in either the common or less common isotope. Advantages of the former include low price, ready availability, and absence of potential contamination problems. The main disadvantage is that the lowest possible value for a material is -1000‰. In contrast, materials enriched in the less common isotope with values greater than +10 · 10⁶ ‰ are commonly available. Why the the lower limit of the permil scale is -1000‰ is illustrated by the following example for a water with no deuterium (i.e., all the hydrogen is protium):

d²H = [(²H / ¹H)_x / (²H / ¹H)_s - 1)] · 1000 (Eq. 2.27)

d²H = [(0/¹H)_x / (²H / ¹H)_s - 1)] · 1000 = (0 - 1) · 1000 = -1000‰ . (Eq. 2.28)

d²H = [(95/5) / (156 · 10^-6) - 1)] · 1000 = +122 · 10⁶ ‰ . (Eq. 2.29)

F_TC_Tn_T = F₁C₁n₁ + F₂C₂n₂ (Eq. 2.30)

2.5.5 Stable isotopes in geochemical modeling

In chemical reaction modeling, usually several reaction models can be found that satisfy the data. For each model reaction path, calculations are used to predict the chemical and isotopic composition of the aqueous phase as well as the amounts of minerals dissolving or precipitated along a flow path. The power of the stable isotope technique in groundwater modeling lies in the fact that we have added one more thermodynamic component to our system for each isotope ratio that is measured (Plummer et al., 1983). These isotopic compositions can be used along with chemical data in geochemical mass balance and reaction path models (e.g., BALANCE, PHREEQE, NETPATH, etc.) to deduce geochemical processes, test hypotheses on hydrology and geochemical mechanisms, and eliminate possible reaction paths (Plummer et al., 1991).

For example, the d¹³C of total dissolved inorganic carbon (DIC) is generally a function of the d¹³C of the rocks and extent of reaction with the rocks in a system. Thus, d¹³C can be a good indicator of which geochemical reactions are occurring (Chapter 18). Sulfur is similar to carbon in this respect, and changes in d¹³C along a flowpath sometimes can reflect reactions that also cause changes in d³⁴S (e.g., progressive calcite precipitation along a flowpath in response to degassing of CO₂ causes gypsum to dissolve). Changes in ¹⁴C content along a flowpath are useful for indicating changes in residence time. On the other hand, there is little change in dD and d¹⁸O of water during reactions with minerals along shallow, low-temperature flowpaths. Therefore, sulfur and carbon isotope data along a flowpath can sometimes be used to eliminate one or more plausible reaction models developed from chemical data, by comparing the observed changes in isotopic compositions with reaction progress (Figure 2.8) along a flowpath. Other useful stable isotope tracers include d¹⁵N and d¹⁸O of nitrates (Chapter 16) and d³⁴S and d¹⁸O of sulfates (Chapter 15). Useful radiogenic isotopes include carbon-14, strontium-87, and various uranium-series isotopes (Chapter 7-9, 18, and 20).

2.5.6 Use of a multi-isotope approach for the determination of flowpaths

Flowpaths are the individual pathways contributing to surface flow in a catchment (see Chapter 1). These result from runoff mechanisms that include, but are not limited to, saturation-excess overland flow, Hortonian overland flow, near-stream groundwater ridging, hillslope subsurface flow through the soil matrix or macropores, and shallow organic-layer flow. Knowledge of hydrologic flowpaths in catchments is critical to the preservation of public water supplies and the understanding of the transport of point and non-point source pollutants (Peters, 1994). The need to incorporate flowpath dynamics is recognized as a key ingredient in producing reliable chemical models (Robson et al., 1992). In other words, if the model used gets the hydrology wrong, it is unlikely to correctly predict the geochemical response.

Stable isotopes such as ¹⁸O and ²H are shown throughout this book to be an improved alternative to traditional non-conservative chemical tracers because waters are often uniquely labeled by their isotopic compositions (Sklash and Farvolden, 1979), often allowing the separation of waters from different sources (e.g., "new" rain vs. "old" pre-storm water). However, studies have shown that flowpaths commonly cannot be identified to a high degree of certainty using d¹⁸O or dD data and simple hydrograph separation techniques because waters within the same flowpath can be derived from several different sources (Ogunkoya and Jenkins, 1991). Furthermore, an underlying theme of many of the chapters in Part 2 of this book is that the isotopic composition of rain, throughfall, meltwater, soil water, and groundwater are commonly variable in time and space. If such variability is significant at the catchment scale (i.e., if hillslope waters that are variable in composition actually reach the stream during the storm event) or if transit times are long and/or variable, then simple two- and three-component, constant composition, mixing models may not provide realistic interpretations of the system hydrology.

One solution is to include alternative, independent isotopic methods for determining the relative amounts of water flowing along different subsurface flowpaths into hydrologic models. Reactive solute isotopes such as ¹³C, ³⁴S, and ⁸⁷Sr can provide valuable information about flowpaths (not water sources) useful for geochemical and hydrologic modeling precisely because they can reflect the reactions characteristic of and taking place along specific flowpaths (see Bullen et al., 1996; Chapter 18). In many instances, the waters flowing along mineralogically distinctive horizons can be distinctively labeled by their chemical composition and by the isotopic compositions of solute isotopes like ¹³C, ⁸⁷Sr, ³⁴S, ¹⁵N, etc. For example, waters flowing through the soil zone often have d¹³C values that are depleted in ¹³C relative to deeper groundwaters because of biogenic production of carbonic acid in organic soils; these same shallow waters can also have distinctive Pb and Sr isotopic compositions.

2.6 Summary

The dominant use of isotopes in catchment research in the last few decades has been to trace sources of waters and solutes. Generally, such data were evaluated with simple mixing models to determine how much was derived from either of two (sometimes three) constant-composition sources. The world does not seem this simple anymore. With the expansion of the field of isotope hydrology in the last decade, made possible by the development and increased availability of automated preparation and analysis systems for mass spectrometers, we have documented considerable heterogeneity in the isotopic compositions of rain, soil water, groundwater, and solute sources. We are still grappling with how to deal with this heterogeneity in our hydrologic and geochemical models. A major challenge is to use the variability as signal, not noise, in our models (Kendall et al., 1995); the isotopic and chemical compositions are providing very detailed information about sources and reactions in shallow systems, if only we can develop appropriate models to use the data. This integration of chemical and isotopic data with complex hydrologic models constitutes an important frontier of catchment research.

Acknowledgments

Much of this chapter is the result of many years of teaching Isotope Hydrology at the USGS National Training Center and at short-courses at GSA (Geological Society of America) meetings and elsewhere by C.K., who would like to thank the co-instructors and the many students of these classes for helping to refine her understanding of isotope geochemistry. Both authors would also like to thank Joel Gat for his contributions to the first draft of the chapter, and Neil Ingraham, Carl Bowser, and Jim O'Neil for their careful reviews of early versions.

References

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