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**Biogeosciences**
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**Technical note**
28 Aug 2019

**Technical note** | 28 Aug 2019

Technical Note: Isotopic corrections for the radiocarbon composition of CO_{2} in the soil gas environment must account for diffusion and diffusive mixing

^{1}Department of Earth Sciences, Dalhousie University, Halifax, Nova Scotia, B3H 4R2, Canada^{2}School of Biological Sciences, University of Utah, Salt Lake City, Utah 84112, USA^{3}Department of Earth Sciences, St. Francis Xavier University, Antigonish, Nova Scotia, B2G 2W5, Canada

^{1}Department of Earth Sciences, Dalhousie University, Halifax, Nova Scotia, B3H 4R2, Canada^{2}School of Biological Sciences, University of Utah, Salt Lake City, Utah 84112, USA^{3}Department of Earth Sciences, St. Francis Xavier University, Antigonish, Nova Scotia, B2G 2W5, Canada

**Correspondence**: Jocelyn E. Egan (jocelyn.egan@dal.ca)

**Correspondence**: Jocelyn E. Egan (jocelyn.egan@dal.ca)

Abstract

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Earth system scientists working with radiocarbon in
organic samples use a stable carbon isotope (*δ*^{13}C) correction
to account for mass-dependent fractionation, but it has not been evaluated
for the soil gas environment, wherein both diffusive gas transport and
diffusive mixing are important. Using theory and an analytical soil gas
transport model, we demonstrate that the conventional correction is
inappropriate for interpreting the radioisotopic composition of CO_{2}
from biological production because it does not account for important gas
transport mechanisms. Based on theory used to interpret *δ*^{13}C of
soil production from soil CO_{2}, we propose a new
solution for radiocarbon applications in the soil gas environment that fully
accounts for both mass-dependent diffusion and mass-independent diffusive
mixing.

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How to cite.

Egan, J. E., Bowling, D. R., and Risk, D. A.: Technical Note: Isotopic corrections for the radiocarbon composition of CO_{2} in the soil gas environment must account for diffusion and diffusive mixing, Biogeosciences, 16, 3197–3205, https://doi.org/10.5194/bg-16-3197-2019, 2019.

1 Introduction

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Radiocarbon allows us to measure the age of soil-respired CO_{2} (CO_{2} diffusing from
the soil surface to the atmosphere, also called soil flux as in Cerling
et al., 1991), but the traditional reporting convention for radiocarbon was
not established for soil gas-phase sampling; rather, it was established for solid (organic
matter) sample analysis. The validity of this convention has never been
explicitly tested for soil-respired CO_{2}.

The traditional radiocarbon reporting convention, Δ^{14}C
(Stuiver and Polach, 1977), uses a mass-dependent correction based
on the isotopic composition of wood. Its purpose is to correct for
biochemical fractionation against the radiocarbon isotopologue
(^{14}CO_{2}) abundance during photosynthesis, which is assumed to be
twice as strong as for ^{13}CO_{2} based on their respective departures
in molecular mass from ^{12}CO_{2}. The classical reference describing
these conventional calculations is Stuiver and Polach (1977).

In the soil gas environment, researchers have different implicit
expectations for fractionation processes. They generally assume that
^{14}C of CO_{2} is not biochemically fractionated in the gas phase
between the points of CO_{2} production (biological production of CO_{2} by soil
organisms and roots) and measurement (subsurface or flux chamber samples).
This assumption is reasonable based on the short residence time of CO_{2} (minutes to days) in the soil profile before emission to the atmosphere.
However, soil gas isotopic signatures depart in predictable ways from the
signature of production because of physical fractionation. It has been
recognized for decades that *δ*^{13}C of CO_{2} at any point in
the soil profile will never equal the isotopic signature of production
because of transport fractionations that alter produced CO_{2} before it
is measured (Cerling et al., 1991). This theory readily extends
to ^{14}C.

We argue here that in the case of soil pore space ^{14}C in which mixing of
air masses occurs, the assumption that mass-dependent fractionation is
twice as large for ^{14}C as *δ*^{13}C by biochemical and physical
processes no longer holds. Using theory and the physical modeling of soil gas
transport to illustrate the issue with the current reporting convention
correction, we propose an alternative approach for specific use cases.

2 Theory

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To understand why the mass-dependent correction used in the Stuiver and
Polach (1977) radiocarbon reporting convention may be a poor fit for soil
gas studies, we can look at our current understanding of the stable isotopic
composition, *δ*^{13}C, of soil CO_{2} (pore space CO_{2}, mole
fraction with respect to dry air). We use delta notation to present the
stable isotopic composition of CO_{2}:

$$\begin{array}{}\text{(1)}& {\mathit{\delta}}^{\mathrm{13}}\mathrm{C}=\left({\displaystyle \frac{{R}_{\mathrm{s}}}{{R}_{\mathrm{VPDB}}}}-\mathrm{1}\right)\mathrm{1000},\end{array}$$

where *δ*^{13}C is the isotopic composition in per mill (see Table 1 for a full list of abbreviations),
*R*_{s} is the ^{13}C∕^{12}C ratio of the sample, and *R*_{VPDB} is the
^{13}C∕^{12}C ratio of the international standard, Vienna Pee Dee
Belemnite.

From foundational work done by Cerling et al. (1991) we know that the isotopic
composition of soil CO_{2} is different from that of soil-respired
CO_{2}. Any change in *δ*^{13}C of soil CO_{2} with depth is
influenced by (1) mixing of atmospheric and biological (or biogeochemical)
sources of isotopically distinct CO_{2}, which may occur via diffusion (no
bulk gas flow; referred to as diffusive mixing for the remainder of the
paper) or advection (bulk gas flow), and (2) kinetic fractionation by
diffusion. The effect of these is illustrated in Fig. 1 using a simulated
soil gas profile. In panel (a) two depth profiles of *δ*^{13}C of
CO_{2} that were modeled in a steady-state environment are shown (the
model will be described in Sect. 3). The profiles differ only in soil
diffusivity; all other characteristics were held constant, including rates
of production, *δ*^{13}C of CO_{2} in the atmosphere (−8 ‰; circle), and biological production (−25 ‰; square with dashed line). In the resultant depth
profile with higher soil diffusivity in panel (a), the *δ*^{13}C of soil CO_{2} ranges from −8 ‰ to −15.1 ‰. In the depth profile representing a soil with lower
diffusivity, the *δ*^{13}C of soil CO_{2} ranges from −8 ‰ to
−20.6 ‰. We stress again that these two isotopic depth
profiles differ only due to differences in transport as a result of their
varying soil diffusivities. In the depth profile with lower soil
diffusivity, atmospheric CO_{2} does not penetrate downwards as readily,
so the profile shape is much steeper near the soil–atmosphere boundary and
is more reflective of the production source composition, −25 ‰, at depth. In the depth profile with higher soil
diffusivity, atmospheric air of −8 ‰ more readily mixes
from the surface downward by diffusion, so the near-surface isotopic
composition will be more reflective of the atmosphere due to diffusive
mixing of these end-members near the soil surface.

Importantly, the soil CO_{2} never equals the *δ*^{13}C of
production (−25 ‰) at any depth in either profile in
Fig. 1a. It is not possible to directly measure *δ*^{13}C of
production in situ because diffusion and diffusive mixing alter the character of
CO_{2} immediately after its production. From the site of production in
the soil, ^{12}CO_{2} diffuses somewhat faster through the soil than
^{13}CO_{2} because the former has a lower mass. This diffusive
difference leads to isotopic fractionation and results in depth profiles of
*δ*^{13}C of soil CO_{2} that are isotopically enriched (less
negative) compared to the source of production. Work by Cerling (1984)
and later by Cerling et al. (1991) demonstrated that the mass differences
between the two isotopologues led to a difference in the diffusion rate of each
in air, amounting to a fractionation of 4.4 ‰ (note that
this applies only to binary diffusion of CO_{2} in air and will differ if
CO_{2} diffuses in other gases). As a result, the *δ*^{13}C of
soil CO_{2} measured at any depth will be enriched by a minimum of 4.4 ‰ relative to the biological production CO_{2} source.
However, the *δ*^{13}C of soil-respired CO_{2} can be
considerably more enriched than 4.4 ‰ relative to
production due to diffusive mixing with the atmosphere as shown in Fig. 1a.

A convenient theoretical formulation for correcting *δ*^{13}C for
both diffusion fractionation and diffusive mixing was introduced by Davidson (1995), following on the work of Cerling (1984) and
Cerling et al. (1991). This approach allows one to combine
measurements of CO_{2} and its isotopic composition within the soil and
the air above it to infer the isotopic composition of CO_{2} produced in
the soil. This only applies when transport within the soil is purely by
diffusion (no bulk air movement). The Davidson (Davidson, 1995)
solution uses the difference between the diffusion coefficients for ^{12}C
and ^{13}C as follows:

$$\begin{array}{}\text{(2)}& {\mathit{\delta}}_{J}^{\mathrm{13}}={\displaystyle \frac{{C}_{\mathrm{s}}\left({\mathit{\delta}}_{\mathrm{s}}^{\mathrm{13}}-\mathrm{4.4}\right)-{C}_{\mathrm{a}}({\mathit{\delta}}_{\mathrm{a}}^{\mathrm{13}}-\mathrm{4.4})}{\mathrm{1.0044}({C}_{\mathrm{s}}-{C}_{\mathrm{a}})}},\end{array}$$

where ${\mathit{\delta}}_{J}^{\mathrm{13}}$ is the *δ*^{13}C composition of
CO_{2} from soil production (biological respiration within the soil),
*C*_{s} and ${\mathit{\delta}}_{\mathrm{s}}^{\mathrm{13}}$ are the mole fraction and isotopic
composition of soil CO_{2}, and *C*_{a} and ${\mathit{\delta}}_{\mathrm{a}}^{\mathrm{13}}$
are the mole fraction and isotopic composition of CO_{2} in the air just
above the soil. In Fig. 2a the mole fraction and isotopic composition of
soil CO_{2} at a 40 cm depth and of the air just above the soil were
“sampled” from model-generated soil depth profiles and the (unrounded)
values were used to calculate the isotopic composition of production using
Davidson's equation (*C*_{s}= 14 780 ppm, ${\mathit{\delta}}_{\mathrm{s}}^{\mathrm{13}}=-\mathrm{20.3832}$ ‰, *C*_{a}=380 ppm, and ${\mathit{\delta}}_{\mathrm{a}}^{\mathrm{13}}=-\mathrm{8}$ ‰). The resulting ${\mathit{\delta}}_{J}^{\mathrm{13}}$ (e.g., Eq. 2) at this depth equals the true isotopic
composition of production (see inset box, Fig. 2a). However, because the
Davidson approach accounts for diffusion and diffusive mixing, at any given
soil depth, not just 40 cm, the modeled values of *C*_{s} and
${\mathit{\delta}}_{\mathrm{s}}^{\mathrm{13}}$ in Figs. 1a and 2a will always yield (via Eq. 2) the true isotopic composition of production, ${\mathit{\delta}}_{J}^{\mathrm{13}}=-\mathrm{25}$ ‰ (dashed line). If *δ*^{13}C of soil CO_{2} were (erroneously) interpreted to represent the
*δ*^{13}C of soil-respired CO_{2}, the error could be as large as
the absolute value of (${\mathit{\delta}}_{\mathrm{a}}-{\mathit{\delta}}_{J}^{\mathrm{13}})-\mathrm{4.4}$ ‰. This Davidson (1995) ${\mathit{\delta}}_{J}^{\mathrm{13}}$ approach has been shown to be robust when applied to field data from natural soils (Breecker et al., 2012; Bowling
et al., 2015; Liang et al., 2016).

While ^{14}C is a radioactive isotope and thus decays with time, the
half-life is sufficiently long so that ^{14}CO_{2} behaves similarly to
stable isotopes on the timescales at which diffusion occurs in a soil gas
system. In this way, *δ*^{13}C diffusive
fractionation theory can be applied to the radiocarbon isotopic composition,
*δ*^{14}C, so long as we account for the mass difference. The larger
mass of ^{14}C means that the diffusion fractionation factor is calculated
to be 8.8 ‰ based on the atomic masses of
^{14}CO_{2}, ^{12}CO_{2}, and bulk air (Southon, 2011).

We can show that the ^{14}CO_{2} distribution in soils will be like that
of ^{13}CO_{2} if we model its distribution through depth in the same
synthetic soil gas environment. In Fig. 1b we present a modeled soil
environment with defined atmospheric and production source CO_{2} isotopic
composition boundary conditions for *δ*^{14}C, the ^{14}C
equivalent to *δ*^{13}C (Stuiver and Polach, 1977):

$$\begin{array}{}\text{(3)}& {\mathit{\delta}}^{\mathrm{14}}\mathrm{C}=\left({\displaystyle \frac{{A}_{\mathrm{s}}}{{A}_{\mathrm{abs}}}}-\mathrm{1}\right)\mathrm{1000},\end{array}$$

where *δ*^{14}C is the isotopic composition in
per mill, *A*_{s} is the measured activity of the sample, and
*A*_{abs} is the activity of the oxalic acid standard (both unitless). As in
Fig. 1a, in panel (b), the profile with lower soil diffusivity, the
downward penetration of atmospheric CO_{2} into the soil profile is
reduced, and as a consequence the isotopic depth profile more closely
reflects (but does not equal) the composition of production (−50 ‰; dashed line). When the diffusion rate is high and
transport is rapid, the atmospheric source is more readily able to penetrate
the profile and mix with the production source. In both profiles, the
measured value of soil CO_{2} at a given depth will *not* equal
the isotopic production value of −50 ‰ because of
diffusion and diffusive mixing. Similar profiles of *δ*^{14}C of
soil CO_{2} with depth, highlighting the diffusive effects, have been
presented by Wang et al. (1994).

Since *δ*^{14}C transport of soil CO_{2} is like that of *δ*^{13}C, it follows that we should apply corrections for *δ*^{14}C
like those in Eq. (2) in order to calculate the isotopic composition of
production. The *δ*^{14}C reformulation of Davidson's
${\mathit{\delta}}_{J}^{\mathrm{13}}$ equation is as follows:

$$\begin{array}{}\text{(4)}& {\mathit{\delta}}_{J}^{\mathrm{14}}={\displaystyle \frac{{C}_{\mathrm{s}}\left({\mathit{\delta}}_{\mathrm{s}}^{\mathrm{14}}-\mathrm{8.8}\right)-{C}_{\mathrm{a}}({\mathit{\delta}}_{\mathrm{a}}^{\mathrm{14}}-\mathrm{8.8})}{\mathrm{1.0088}({C}_{\mathrm{s}}-{C}_{\mathrm{a}})}},\end{array}$$

where ${\mathit{\delta}}_{J}^{\mathrm{14}}$ is the *δ*^{14}C composition of soil
production, *C*_{s} and ${\mathit{\delta}}_{\mathrm{s}}^{\mathrm{14}}$ are the mole fraction and *δ*^{14}C composition of the soil CO_{2}, and *C*_{a} and ${\mathit{\delta}}_{\mathrm{a}}^{\mathrm{14}}$ are the mole fraction and *δ*^{14}C composition of
CO_{2} in the air just above the soil. This Davidson reformulation for
*δ*^{14}C, ${\mathit{\delta}}_{J}^{\mathrm{14}}$, was applied to a model-generated
profile of soil *δ*^{14}C at a 40 cm depth in Fig. 2b, like in
panel (a) for *δ*^{13}C (*C*_{s}= 14 780 ppm, ${\mathit{\delta}}_{\mathrm{s}}^{\mathrm{14}}=-\mathrm{39.3989}$ ‰, *C*_{a}=380 ppm, and
${\mathit{\delta}}_{\mathrm{a}}^{\mathrm{14}}=\mathrm{45.5276}$ ‰; see inset
box, Fig. 2b). As was the case for *δ*^{13}C in Fig. 2a, the
modeled values of *C*_{s} and ${\mathit{\delta}}_{\mathrm{s}}^{\mathrm{14}}$ at any depth will yield
the true isotopic composition of production, −50 ‰ (dashed line), because this approach accounts for diffusion and diffusive
mixing.

The typical approach that has been used for interpreting the ^{14}C
composition of soil CO_{2} and soil-respired CO_{2} (e.g.,
Trumbore, 2000) differs from the *δ*^{14}C example above
because a *δ*^{13}C correction is applied to account for
the mass-dependent isotopic fractionation of biochemical origin, ultimately
converting *δ*^{14}C to a variant called Δ^{14}C
(Stuiver and Polach, 1977). The derivation of the mass-dependent
correction is provided in Stuiver and Robinson (1974), wherein
observations are normalized to an arbitrary baseline value of −25 ‰ for *δ*^{13}C (a value for terrestrial wood),
and the ^{13}C fractionation factors are squared to account for the
^{14}C∕^{12}C fractionation factor as follows:

$$\begin{array}{}\text{(5)}& \begin{array}{rl}{A}_{\mathrm{SN}}& =\phantom{\rule{0.125em}{0ex}}{A}_{\mathrm{s}}{\left[{\displaystyle \frac{{R}_{\mathrm{s}}\left(-\mathrm{25}\right)}{{R}_{\mathrm{s}}}}\right]}^{\mathrm{2}},\\ & =\phantom{\rule{0.125em}{0ex}}{A}_{\mathrm{s}}{\displaystyle \frac{{\left[\left(\mathrm{1}-\frac{\mathrm{25}}{\mathrm{1000}}\right)\times \phantom{\rule{0.125em}{0ex}}{R}_{\mathrm{VPDB}}\right]}^{\mathrm{2}}}{{\left[\left(\mathrm{1}+\frac{{\mathit{\delta}}^{\mathrm{13}}\mathrm{C}}{\mathrm{1000}}\right)\times {R}_{\mathrm{VPDB}}\right]}^{\mathrm{2}}}},\\ & ={A}_{\mathrm{s}}{\displaystyle \frac{{\left[\left(\mathrm{1}-\frac{\mathrm{25}}{\mathrm{1000}}\right)\right]}^{\mathrm{2}}}{{\left[\left(\mathrm{1}+\frac{{\mathit{\delta}}^{\mathrm{13}}C}{\mathrm{1000}}\right)\right]}^{\mathrm{2}}}},\end{array}\end{array}$$

where *A*_{SN} is the normalized sample activity, *A*_{s} is the sample
activity, and *δ*^{13}C is the isotopic composition of the sample
(soil CO_{2} in our case). As explained in Stuiver and Robinson (1974),
the 0.975 term sometimes used in forms of *A*_{SN} is equivalent to $\left(\mathrm{1}-\frac{\mathrm{25}}{\mathrm{1000}}\right)$, which we will retain for clarity. The equation
for Δ^{14}C, the *δ*^{13}C-corrected variant of *δ*^{14}C, can then be created from Eq. (5) by substituting in delta
notation for Δ^{14}C of Δ^{14}C = (*A*_{SN}∕*A*_{abs}
-1) × 1000 following Stuiver and Robinson (1974):

$$\begin{array}{}\text{(6)}& {\mathrm{\Delta}}^{\mathrm{14}}\mathrm{C}=\left[\left(\mathrm{1}+{\displaystyle \frac{{\mathit{\delta}}^{\mathrm{14}}\mathrm{C}}{\mathrm{1000}}}\right){\displaystyle \frac{{\left(\mathrm{1}-\frac{\mathrm{25}}{\mathrm{1000}}\right)}^{\mathrm{2}}}{{\left(\mathrm{1}+\frac{{\mathit{\delta}}^{\mathrm{13}}\mathrm{C}}{\mathrm{1000}}\right)}^{\mathrm{2}}}}-\mathrm{1}\right]\mathrm{1000}.\end{array}$$

Combining Eqs. (3) and (6) leads to

$$\begin{array}{}\text{(7)}& {\mathrm{\Delta}}^{\mathrm{14}}{\mathrm{C}}_{\mathrm{old}}=\left[\left({\displaystyle \frac{{A}_{\mathrm{s}}}{{A}_{\mathrm{abs}}}}\right){\displaystyle \frac{{\left(\mathrm{1}-\frac{\mathrm{25}}{\mathrm{1000}}\right)}^{\mathrm{2}}}{{\left(\mathrm{1}+\frac{{\mathit{\delta}}^{\mathrm{13}}\mathrm{C}}{\mathrm{1000}}\right)}^{\mathrm{2}}}}-\mathrm{1}\right]\mathrm{1000}.\end{array}$$

For more information on the derivation of Eqs. (6) and (7) see Stuiver and
Robinson (1974), page 88. In Eq. (7) we have added the subscript “old” to
highlight that this is the common approach used to date for soil gas
applications – we will introduce a “new” method with Eq. (15). The terms on
the left-hand side of Eqs. (6) and (7) are identical. Note that *A*_{abs} in
our notation is equivalent to *A*_{O} in Stuiver and Robinson (1974).

Equation (7) uses *δ*^{13}C as an input parameter to make a mass-dependent
correction to obtain Δ^{14}C, but the profiles of *δ*^{13}C and *δ*^{14}C of soil CO_{2} (Fig. 1) highlight the fact that
*both* vary within the soil because of diffusion and diffusive
mixing. This makes it unclear what form of *δ*^{13}C should actually
be used in the place of the mass-dependent correction in the soil gas
environment (*δ*^{13}C of the soil CO_{2} is measured but *δ*^{13}C of biological production is not) as diffusive mixing is not a
mass-dependent process. When Δ^{14}C_{old} is modeled through
depth like *δ*^{13}C and *δ*^{14}C in Figs. 1 and 2 it also
varies with depth as shown in Fig. 2c. However, using a Δ^{14}C
variant of Davidson's *δ*_{J} (as for *δ*^{14}C in Fig. 2b) at the same 40 cm depth does not correctly reproduce the
specified model value for the Δ^{14}C of production of −50 ‰ like it did for *δ*^{13}C and *δ*^{14}C (*C*_{s}= 14 780 ppm, ${\mathrm{\Delta}}_{\mathrm{s}}=-\mathrm{48.4319}$ ‰, *C*_{a}=380 ppm, and Δ_{a}=10 ‰; see inset box, 2c). We therefore adapted the
mass-dependent correction in Δ^{14}C_{old} using Davidson's (1995) theory to demonstrate how and why it should be used for Δ^{14}C soil gas applications.

3 Methods – model description

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We used an analytical gas transport model to simulate a range of natural
soil profiles of ^{12}CO_{2}, ^{13}CO_{2}, and ^{14}CO_{2} in
order to present soil gas transport theory. The model is based on Fick's
second law of diffusion:

$$\begin{array}{}\text{(8)}& \mathit{\theta}{\displaystyle \frac{\partial \mathrm{Conc}}{\partial t}}={\displaystyle \frac{\partial}{\partial z}}\left(D\phantom{\rule{0.125em}{0ex}}\left(z,t\right){\displaystyle \frac{\partial \mathrm{Conc}}{\partial z}}\right)+P\left(z,t\right),\end{array}$$

where *θ* is the soil air-filled pore space, Conc is the concentration,
*t* is time, *D*(*z*,*t*) is the soil gas diffusion function, and *P*(*z*,*t*) is the biological
production function, with the latter two dependent on both depth *z* and time
*t*.

The model was run in steady state,

$$\begin{array}{}\text{(9)}& {\displaystyle \frac{\partial \mathrm{Conc}}{\partial t}}=\mathrm{0},\end{array}$$

and both diffusion and production rates were constant with depth:

$$\begin{array}{}\text{(10)}& {\displaystyle}& {\displaystyle}D\left(z\right)=D,\text{(11)}& {\displaystyle}& {\displaystyle}P\left(z\right)=P.\end{array}$$

The following boundary conditions were used:

$$\begin{array}{}\text{(12)}& {\displaystyle}& {\displaystyle}C\left(z=\mathrm{0}\right)={\mathrm{Conc}}_{\mathrm{atm}},\text{(13)}& {\displaystyle}& {\displaystyle \frac{\partial C}{\partial z}}{\mathrm{|}}_{z=L}=\mathrm{0},\end{array}$$

where Conc_{atm} is the concentration of CO_{2} in air just above the soil
and *L* is the model lower spatial boundary, the point below which no
production or diffusion occurs. Equation (8) is solved analytically to yield the
following equation:

$$\begin{array}{}\text{(14)}& \mathrm{Conc}\left(z\right)={\displaystyle \frac{P/L}{D}}\left(L\times z-{\displaystyle \frac{{z}^{\mathrm{2}}}{z}}\right)+{\mathrm{Conc}}_{\mathrm{atm}}.\end{array}$$

In the model, isotopologues of CO_{2} are treated as independent gases,
with their own specific concentration gradients and diffusion rates
(Cerling et al., 1991; Risk and Kellman, 2008; Nickerson and Risk, 2009). We assume total CO_{2} to be ^{12}CO_{2}
because of its high abundance. The error associated with this
assumption is less than 0.01 % (Amundson et al., 1998). Equation (14) is thus
applied for ^{13}CO_{2} and ^{14}CO_{2}. For the full derivation see
Nickerson et al. (2014) Sect. 2.3.

The analytical gas transport model was applied across a range of soil
diffusivity ($\mathrm{1}\times {\mathrm{10}}^{-\mathrm{7}}$, $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{6}}$, and $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{5}}$ m^{2} s^{−1}) and
soil production rates (0.5, 1, 2, and 4 µmol CO_{2} m^{−3} s^{−1}). The specific soil diffusivity and production rates used to
generate each profile are stated in the figure caption of that profile. We
used a *δ*^{13}C of biological production (−25 ‰) and atmospheric CO_{2} (*δ*_{a}; −8‰) and Δ^{14}C of biological production (−50 ‰) and atmospheric CO_{2} (Δ_{a}; 10 ‰) to represent realistic conditions found in
nature. The other model boundary conditions were as follows: *L*=0.8 m, *z*=0.025 m, and Conc_{atm}= 15 833 µmol m^{−3} (∼380 ppm). The output of the model under these applied conditions were profiles
of ^{12}CO_{2}, ^{13}CO_{2}, and ^{14}CO_{2} for each depth
(*z*) down to the bottom boundary (*L*).

4 Results

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Based on Davidson's (1995)
theory and what we demonstrated with Fig. 2c, rather than using the
*δ*^{13}C soil pore space as a mass-dependent correction, we suggest
instead using the value ${\mathit{\delta}}_{J}^{\mathrm{13}}$ (Eq. 2), the
biological production of *δ*^{13}C, in its place in the denominator
of Eq. (7) as follows:

$$\begin{array}{}\text{(15)}& {\mathrm{\Delta}}^{\mathrm{14}}{\mathrm{C}}_{\mathrm{new}}=\left[\left({\displaystyle \frac{{A}_{\mathrm{s}}}{{A}_{\mathrm{abs}}}}\right){\displaystyle \frac{{\left(\mathrm{1}-\frac{\mathrm{25}}{\mathrm{1000}}\right)}^{\mathrm{2}}}{{\left(\mathrm{1}+\frac{{\mathit{\delta}}_{J}^{\mathrm{13}}}{\mathrm{1000}}\right)}^{\mathrm{2}}}}-\mathrm{1}\right]\mathrm{1000}.\end{array}$$

Values of Δ^{14}C_{new} through depth represent
transport-fractionation-corrected soil CO_{2} values of radiocarbon, and
in comparison to Δ^{14}C_{old}, they are corrected for
mass-independent diffusive mixing.

A depth profile of Δ^{14}C_{new} is presented in Fig. 3 (dashed
line). To generate this soil depth profile we used the numbers from the
simulated profiles in Fig. 2 and inserted them into Eq. (2) to determine
${\mathit{\delta}}_{J}^{\mathrm{13}}$ at each depth. These values were then used in
Eq. (15) to obtain Δ^{14}C_{new} of soil CO_{2} through depth.
The Δ^{14}C_{new} profile (dashed line) is more isotopically
enriched compared to the Δ^{14}C_{old} profile (solid line) in
Fig. 3. Values sampled from the Δ^{14}C_{old} profile (the
same as the one presented in Fig. 2c) were not able to reproduce the
specified model value for the Δ^{14}C of production of −50 ‰ using a Δ^{14}C variant of Davidson's
*δ*_{J}. To demonstrate that Δ^{14}C_{new} does correct
for gas transport fractionations, it can be placed into ${\mathrm{\Delta}}_{J}^{\mathrm{14}}$,
a Δ^{14}C adaption of Davidson (1995) for ^{14}C (Eq. 4), as
follows:

$$\begin{array}{}\text{(16)}& {\mathrm{\Delta}}_{J}^{\mathrm{14}}={\displaystyle \frac{{C}_{\mathrm{s}}\left({\mathrm{\Delta}}^{\mathrm{14}}{\mathrm{C}}_{\mathrm{new}}-\mathrm{8.8}\right)-{C}_{\mathrm{a}}\left({\mathrm{\Delta}}_{\mathrm{a}}^{\mathrm{14}}-\mathrm{8.8}\right)}{\mathrm{1.0088}({C}_{\mathrm{s}}-{C}_{\mathrm{a}})}},\end{array}$$

where ${\mathrm{\Delta}}_{J}^{\mathrm{14}}$ is the Δ^{14}C composition of soil
production, *C*_{s} and Δ^{14}C_{new} are the mole fraction
and Δ^{14}C composition of the soil CO_{2}, and *C*_{a} and
${\mathrm{\Delta}}_{\mathrm{a}}^{\mathrm{14}}$ are the mole fraction and Δ^{14}C composition
of CO_{2} in the air just above the soil.

Unlike in the case of Δ^{14}C_{old} demonstrated in the inset
box in Fig. 2c, using the same 40 cm depth from the Δ^{14}C_{new} profile, we were able to reproduce the specified model
value for the Δ^{14}C of production of −50 ‰ (*C*_{s}= 1 780 ppm, ${\mathrm{\Delta}}_{\mathrm{s}}=-\mathrm{39.3989}$ ‰,
*C*_{a}=380 ppm, and Δ_{a}=45.5276 ‰;
see inset box, Fig. 3).

In the soil gas environment, Δ^{14}C_{new} convention should be
used to properly interpret soil-respired CO_{2} from soil CO_{2}, as it
corrects for all related transport fractionations. For researchers who have
soil CO_{2} data previously interpreted using Δ^{14}C_{old},
the following steps will help correct for transport fractionations: (1) use
${\mathit{\delta}}_{\mathrm{s}}^{\mathrm{13}}$ and Δ^{14}C_{old} to back out the
activity of the sample (*A*_{s}); (2) calculate the isotopic composition of
production for *δ*^{13}C using Eq. (2), ${\mathit{\delta}}_{J}^{\mathrm{13}}$;
(3) use ${\mathit{\delta}}_{J}^{\mathrm{13}}$ and *A*_{s} in Eq. (7) to calculate
Δ^{14}C_{new}; and finally (4) determine the radiocarbon isotopic
composition of production using Eq. (16), ${\mathrm{\Delta}}_{J}^{\mathrm{14}}$.

Going forward, several changes to best practice are recommended. On a lab
level, for new soil CO_{2} data, we propose that laboratories report
radiocarbon using Eq. (3) for *δ*^{14}C, the uncorrected radiocarbon
variant, so that the first step above (use ${\mathit{\delta}}_{\mathrm{s}}^{\mathrm{13}}$
and Δ^{14}C_{old} to back out the activity of the sample; *A*_{s}) can be avoided, and researchers can proceed with steps 2–4. We
also suggest that researchers measure *δ*^{13} alongside Δ^{14}C so that they are not dependent on the AMS-measured *δ*^{13} for potential back corrections.

The Davidson (1995) *δ*_{J} method was the gradient approach we used
in our study, but alternative gradient approaches, such as those presented
for *δ*^{13}C by Goffin et al. (2014) and Nickerson et al. (2014) and for Δ^{14}C by Phillips et al. (2013), would likely be similarly successful
in producing depth-dependent compositions of production. They are, however,
not quite as straightforward as the *δ*_{J} method.

5 Conclusions

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This traditional Δ^{14}C solution, which uses *δ*^{13}C of soil CO_{2} as a mass-dependent correction, is not
appropriate for the soil gas environment, as it does not account for
mass-independent mixing processes. We propose a new best practice for
Δ^{14}C work in the soil gas environment that accounts for gas
transport fractionations and produces true estimates of Δ^{14}C of production.

Author contributions

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Author contributions.

JEE, DRB, and DAR conceptualized the theory and method for proving the new solution for radiocarbon applications in the soil gas environment. JEE carried out the modeling, validation, visualization, and writing of the original draft. DRB, DAR, and JEE reviewed and edited the draft.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

Thanks to Thure Cerling for helpful discussions on the paper and to anonymous reviewers whose input was important in shaping the final presentation of this material.

Financial support

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Financial support.

Jocelyn E. Egan is grateful for support from a Research-in-Residence Award from the Inter-University Training in Continental-scale Ecology Project, National Science Foundation, Directorate for Biological Sciences (grant no. EF-1137336). Jocelyn E. Egan was also funded by the Natural Sciences and Engineering Research Council of Canada (NSERC). This research was also supported by the US Department of Energy, Office of Science (grant no. DE-SC0010625).

Review statement

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Review statement.

This paper was edited by Dan Yakir and reviewed by three anonymous referees.

References

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Short summary

Traditionally a mass-dependent correction is made when measuring the radiocarbon composition in organic samples. This correction has not been evaluated for the soil gas environment where gas transport processes are important. Here, we show using theory that this traditional correction is not appropriate for estimating the radiocarbon composition of soil biological production. We also propose a new solution that accounts for soil gas transport processes.

Traditionally a mass-dependent correction is made when measuring the radiocarbon composition in...

Biogeosciences

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