On the conversion of tritium units to mass fractions for hydrologic applications

1. Introduction

Tritium, a radioactive isotope of hydrogen, is a useful tracer in hydrologic and oceanographic studies Citation1–3. Tritium is integral to the water molecule and detectable at exceedingly low levels Citation4. Natural cosmogenic processes produce tritium in the upper atmosphere Citation5. Rutherford et al. were the first to produce anthropogenic tritium – although initially they did not recognise it as such Citation6. Anthropogenic production completely overwhelmed cosmogenic production during the period of atmospheric nuclear-weapons testing (1945–1980) Citation7. Nuclear power generation and fuel reprocessing continue to augment natural production Citation4. At sufficiently high levels, tritium is a regulated contaminant Citation8.Footnote

Rigorous modelling of tritiated groundwater and soil–water transport by advection–diffusion processes require the conversion of analytical tritium units (TU) to mass-fraction (or mole fraction) units Citation9 (or at minimum a demonstration of their linear interdependence). Estimating amounts of tritium in contaminant plumes or fluxes of tritium across compliance and other boundaries of interest – the land–atmosphere interface, for example – requires similar conversions to mass (or mole) quantities of tritium Citation10. The conversion of analytic to mass units is unobvious, nonlinear, and dependent on assumptions that are rarely specified. The full range of activities in pristine to highly contaminated systems requires a general approach. To be useful, the approach must be practical yet rigorous, with well-defined limitations. A thorough review of the literature did not locate any such conversions. This note documents the derivation of a broadly applicable conversion from analytical units to mass-fraction units for tritiated water in the terrestrial hydrosphere.

2. Derivation

We denote the tritium subset of hydrogen atoms by ³H, the group of protium (¹H) and deuterium (²H) atoms by ^{1, 2}H (for our purposes constituting the ‘non-tritium’ subset of hydrogen atoms), and the complete set of relevant hydrogen atoms by ^{1, 2, 3}H. Heavier isotopes of hydrogen have been synthesised, but are assumed unimportant in hydrologic contexts as their half-lives are shorter than 10⁻²¹ s. Similarly, we denote the complete set of environmentally persistent oxygen atoms, ¹⁶O, ¹⁷O, and ¹⁸O (see discussion), by ^{16, 17, 18}O or, for simplicity, O. We thus assume that tritiated water (either ^{1, 2}HO³H or ³H₂O) plus non-tritiated water (^{1, 2}H₂O) form the complete set of water molecules (^{1, 2, 3}H₂O). We perform calculations for a representative sample of water from the system under consideration, for which the relative abundances of the non-tritium atoms making up the water are evaluated using their standard atomic weights and then further evaluated according to the range of values reported for terrestrial hydrologic samples.

The SI unit of activity is the Becquerel (Bq), a hypothetical number of atoms having an instantaneous decay rate of one per second. Concentrations of tritium from environmental analytical laboratories are most commonly reported in TU Citation11, where one TU is defined as a particular (SI) number fraction (X) unit (specifically, a particular isotope number fraction) equal to one tritium atom per 10¹⁸ hydrogen atoms Citation11 Citation12. The TU arose from Libby et al.’s estimate of the natural abundance of tritium in Norwegian surface water, which was taken to approximate secular equilibrium between cosmogenic production and radioactive decay Citation13. On a mass-of-water basis, 1 TU is approximately equal to 0.118 Bq kg⁻¹. The conversion is imprecise due to uncertainty in the decay rate of tritium (4500±8 d) Citation14. In contrast, the conversion of tritium concentration to mass fraction carries with it the relatively high precision of atomic mass determinations.

The derivation presented here is in terms of atomic quantities, which will not change with further refinements in decay rate. Discrepancies between the relative abundance of the non-tritium isotopes in the water sampled versus those assumed for calculations represent the outstanding source of uncertainty in analysis that follows, unless these abundances are independently assessed. In this latter case, the independently determined abundances of the non-tritium isotopes can be substituted for the terrestrial hydrospheric-water values in the equations that follow. The following derivation is for water only, meaning that the mass of solutes is either negligible or separately taken into account (see discussion). Additionally, measurement errors in any analytically determined quantity amounts (bulk sample mass, for example) are not considered.

Further assuming (see discussion) that

	(1) all tritium exists as monotritiated water, i.e. as the molecule 1, 2H–16, 17, 18O–3H (1, 2HO3H),
	(2) deuterium and protium are uniformly distributed throughout all water molecules, i.e. the probability that a given non-tritium hydrogen is either protium or deuterium does not depend on the presence or absence of tritium as the other hydrogen atom in the molecule,
	(3) the relative abundance of the three oxygen isotopes is similarly independent of the distribution of tritium,
	(4) the molar mass of 1, 2H in the modelled sample S, M(1, 2H)S, is equal to that of hydrogen in water of the terrestrial hydrosphere T, M(1, 2H)T, which – corresponding to the 1999 standard atomic weight compilation – has a ‘standard’ value of 1.00794 10−3 kg mol−1, but ranges at least from 1.007904 to 1.008003 10−3 kg mol−1 [15 ,
	(5) the molar mass of 16, 17, 18O in the modelled sample S, M(O) S, is equal to that of oxygen in water of the terrestrial hydrosphere T, M(O) T, which has a ‘standard’ value of 15.9994 10−3 kg mol−1 but ranges at least from 15.99906 to 15.99946 10−3 kg mol−1 [15 , and
	(6) the molar mass of 3H, M(3H), is equal to 3.016049 10−3 kg mol−1 [16 ;

the analysis can proceed as follows:

The molar mass of monotritiated water (^{1, 2}HO³H) is

or, applying the ‘standard’ values cited above Similarly, the molar mass of non-tritiated (^{1, 2}H₂O) water is or, again applying the ‘standard’ values,

From its definition, the concentration (isotope number fraction) known as TU equals exactly one tritium atom per 10¹⁸ hydrogen atoms. Thus, if the water sample S has a tritium concentration c(³H)_S of 1 TU, one can write:

where N( ⁱ E)_S denotes the number of entities (here atoms of specific isotopes or isotope groups) of element E having mass number i (protons plus neutrons) in sample S Citation12.

Regardless of isotopic species, each water molecule in sample S contains two hydrogen atoms. Thus,

and in which N(Q)_S specifies the number of atoms or molecules in the specified isotope group or isotopologue group Q (E∈Q ) in sample S.

Then, per mole (abbreviated mol) of sample water (containing both tritiated and non-tritiated isotopologues) with tritium isotope number fraction X(³H)_S, the mass of tritiated water, M(^{1, 2}HO ³H)_S, is:

where N _A is the number (the Avogadro constant, tying atomic to macroscopic quantities) equal to the number N of atomic entities per mole Citation12. The first term on the right-hand side of Equation(3) is X(³H)_S, the tritium isotope number fraction pertaining to the sample. The second and third terms are statements of the number of the different types of hydrogen isotopes (isotope groups) in their respective water isotopologues (by EquationEquation (1), for example). The fourth states the equivalency relationship between the Avogadro constant and the mole. The fifth gives the molar mass of the tritiated water in the sample.

Absent sample-specific data, the molar mass of the tritiated water in the sample is approximated using standard values for molar masses of non-tritium isotopes as reported for water in the terrestrial hydrosphere, together with the molar mass of tritium:

Using the values given above, with tritium concentration c expressed in TU, EquationEquation (3) (with Equation(4)) gives the molar mass of tritiated water in the sample:

In the same mole of water, the number fraction of non-tritiated water is the complement of the number fraction of tritiated water, i.e. X(^{1, 2}H ₂O)=1−X(^{1, 2}HO ³H). Using this fact, the molar mass of non-tritiated water is similarly given by

Assuming the previously shown approximation derived from standard terrestrial values, EquationEquation (6) expressed in TU gives for the molar mass of non-tritiated water in the sample:

The mass fraction of tritiated water to total mass of water w(^{1, 2}HO ³H)_S is, by definition

Substituting from EquationEquations (7) and Equation(5), and simplifying, provides the general quantity equation result:

The conversion depends nonlinearly on the tritium concentration. However, for tritium concentrations (number fractions X(³H)) greatly less than X(³H)=1=10¹⁸ TU, the mass fraction of tritiated water in sample S, w(^{1, 2}HO ³H)_S is approximated by

which is accurate to 1 ppm for c≤4.5×10¹² TU (5.3×10¹¹ Bq kg⁻¹).

3. Natural isotopic variation

The relative abundances of the stable isotopes constituting water in the hydrologic cycle are not constants of nature, but instead vary both systematically from location to location across landscapes around the globe (and through time) as well as in response to local geochemical conditions Citation15. As already mentioned, the relative abundances of the non-tritium isotopes in a given sample of water can be directly measured and substituted in the final terms of EquationEquations (3) and Equation(6), which generally will result in different, more accurate values for the numerical coefficients appearing in the numerators on the right-hand sides of EquationEquations (5) and Equation(7), and hence Equation(9) and Equation(10). In the derivation of EquationEquations (9) and Equation(10), estimates of the relative abundances of hydrogen and oxygen isotopes in water of the hydrologic cycle served as surrogates in the absence of sample-specific data.

Coplen et al. Citation15 placed the ‘standard’ values (and standard ranges) of molar masses for the isotopes of oxygen, hydrogen, and other elements in the context of the reported range of variations in the terrestrial hydrosphere. These ranges will likely expand as data accrue from additional locations. The ‘standard’ values that were used to arrive at the coefficients presented here lie within, but not necessarily at the centre of, the ranges for terrestrial hydrospheric water. Substituting minimum and maximum values (as cited in the previous section) for the ‘standard’ values results in a minimum corresponding range in the coefficient (on the right-hand side of EquationEquation (10)) expressed in bracket form as [nominal minimum; nominal maximum] of [2.22287×10⁻¹⁸; 2.22300×10⁻¹⁸]. This represents a range in the coefficient (relative to its midpoint) of 58 ppm.

4. Discussion

Given three isotopic species of hydrogen (¹H, ²H, ³H) and three environmentally persistent isotopes of oxygen (¹⁶O, ¹⁷O, ¹⁸O), there are 18 distinct isotopic species of water, half of which contain tritium. (In addition to the heavier hydrogen isotopes already mentioned, 10 isotopes of oxygen other than ¹⁷O exist, but are also relatively short-lived, with half-lives ranging from ∼10⁻²¹ to ∼ 122 s.) Rather than accounting for all nine (thirty-nine) tritium-containing species, the problem is made tractable by assuming that all aqueous tritium exists as ¹⁶O-, ¹⁷O-, and ¹⁸O-variants of monotritiated water (assumption Equation(1)). Neglecting minor isotopic variations in hydrogen–oxygen bond dynamics, and assuming that isotopic exchange has proceeded to equilibrium, the fraction of bitritiated water is given by the joint probability that both hydrogen atoms in a given water molecule are tritium, which, to a first approximation is (c×10⁻¹⁸)×(c×10⁻¹⁸)=c ²×10⁻³⁶. This is small enough to be ignored.

The assumption Equation(4) that the naturally occurring blend of hydrogen isotopes has a molar mass unaffected by tritium is justified by tritium's exceedingly low natural abundance. The estimated total natural inventory of tritium in the hydrosphere (here restricted to oceans plus terrestrial water) and the atmosphere is ∼2.4 and ∼4.2 kg, respectively Citation17 Citation18. These values represent a balance between rates of cosmogenic production in the upper atmosphere, mixing into the rest of the hydrologic cycle (principally the oceans), and radioactive decay. The hydrologic inventory of non-tritium hydrogen is greater than that of tritium, both natural and anthropogenic, by more than 16 orders of magnitude Citation7 Citation18.

The general expression Equation(9) breaks down at sufficiently high tritium activities. In practice, even the most highly contaminated environments will rarely violate assumption Equation(1). For example, groundwater near the Hanford plutonium–uranium extraction facility, which operated from 1956 to 1990, had a maximum sampled tritium concentration in 1998 of ∼ 1.2×10⁶ TU (reported as 3,870,000 pCi l⁻¹) Citation19. The neglect of bitritiated water is justified, as its fractional abundance is only about 1.4×10⁻²⁴ (=c ²×10⁻³⁶). A potentially more serious issue is the neglect of hydrogen atoms in OH⁻ and H⁺ ions. Most groundwaters are naturally buffered to nearly neutral conditions Citation20. At neutral pH, the fraction of tritium in the two ions can be shown to be about c×1.8×10⁻⁹. This fraction, negligible for neutral solutions, grows exponentially as pH deviates from neutrality. Disposed effluents at Hanford included highly concentrated sodium hydroxide and nitric acid solutions Citation21. Neglect of these ions would not be justified where such wastes entered the ground. This is in addition to the considerations of solute masses mentioned above.

Three points merit emphasis. First, the relevant mass fraction in hydrologic transport problems is the mass ratio of tritium-bearing solute (here, tritiated water) to total solution (overwhelmingly non-tritiated water) by which the transport takes place. Second, one molecule of water, the transporting substance, contains two hydrogen atoms. Third, the TU is defined on the basis of atoms, not molecules. Mistaking tritiated water transport for tritium atom transport results in a fractional error equal to the mass ratio of monotritiated water to a single atom of tritium, or roughly 20÷ 3=6 ⅔. Overlooking the second and third points can produce a factor-of-two error. Compounding the errors can cause transport calculations to be off by a factor of ∼13 ⅓. Such mistakes are commonly made, with order-of-magnitude consequences.

The conversion from measured specific activities (Bq kg⁻¹) to tritium concentrations (TU) carries with it the uncertainty in the half-life of tritium. The currently recommended value Citation14 is uncertain to more than 1 part in 600. The uncertainty in applying the linear approximation (EquationEquation (10)) is therefore not limiting.

5. Conclusions

Careful accounting and due attention to assumptions, definitions, and isotopic speciation leads to a general equation for converting between tritium concentrations and mass fractions for hydrologic applications. The assumptions are reasonable for a broad range of environmental conditions. While the general relationship is nonlinear, its linear approximation is accurate to one part per million and applicable to most situations likely to be encountered in practice. The natural variation in the relative abundances of non-tritium isotopes making up hydrologic water increases the uncertainty by about a factor of at least fifty-eight.