Theoretical Establishment of the Mass Balance Equation and Determination of the Proportion (S)ff of Fossil Fuels as an Indicator of the Suess Effect

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1. Introduction

The conventional ^{14}C method, among its many applications, can also be used for the determination of ^{14}C isotopic ratio values; ^{14}C/C ratio expressed in
$\Delta {}^{14}C$ [1]. With this method one can get an idea of the degree of pollution and to be able to estimate CO_{2} emissions from fossil fuels [1] [2] [3]. The estimation of fossil fuel CO_{2}, the main cause of the Suess effect, cannot be determined directly from a radiocarbon age measurement [4] but can be estimated using a mathematical model for the global carbon cycle.

This discussion focuses on a proposed mathematical approach to establish the mass balance equations used by several authors in the determination of fossil CO_{2} using the carbon-14 (C14 or ^{14}C) as a tracer.

The natural production of carbon-14 is generated by solar rays of cosmic and galactic sources. These solar rays contain protons that react with molecules in the air to release thermalized neutrons [5] with very high energy. These neutrons formed in the upper atmosphere then collide with air molecules such as nitrogen (preferentially) and oxygen. They are decelerated due to the many neutron-molecule collisions and attain the thermal energy of the gases. As a result, they react with the atomic nuclei of the air, present in the atmosphere and troposphere, to form carbon-14. The production of C14 is brought about by the capture of neutrons by nitrogen (80% in the atmosphere according to [6] is by far the most frequent reaction ^{14}N (n, p) ^{14}C [7].

2. Approach and Method

The properties of the same chemical element vary because of the difference in their masses. This difference is at the origin of isotopic fractionation, that is to say processes that vary the isotopic composition of organic or inorganic compounds.

For a given sample, its isotopic composition can be expressed either as the relative abundance of isotopes (in %); or according to the isotope ratio (without unit) or according to the isotope ratio in relation to a standard (noted *δ* and expressed ‰)._{ }

In this approach, we establish the equation of conservation of the isotopic composition of carbon in general and in particular C14 for a given sample of material.

The isotopic fractionation process can be described mathematically by comparing the isotopic ratios of the two equilibrium compounds (*a* Û *b*) or the two compounds before and after a physical or chemical transition process. For example the isotopic fractionation factor, named
${\epsilon}_{a/b}$ :_{ }

${\epsilon}_{a/b}=\left(\frac{{R}_{b}}{{R}_{a}}-1\right)$ (2.1)

*R* is an isotopic ratio. For example, the isotopic ratio of carbon can be defined by:

${}^{13}R=\frac{13C}{12C}$ , for the isotope ^{13}C (2.2)

And
${}^{14}R=\frac{14C}{12C}$ for the ^{14}C isotope. (2.3)

It is the ratio of the least abundant and the most abundant isotope.

As far as literature is concerned, the three most abundant carbon isotopes are: ^{12}C (98.89%, the most abundant isotope), ^{13}C (1.1%) and ^{14}C (about 10^{−12} %).

Judging from the relations (2.2) and (2.3), the number of ^{12}C, ^{13}C and ^{14}C isotopes in a sample is obtained respectively.

${}^{12}C=\frac{{C}_{G}}{1+{}^{13}R+{}^{14}R}$ (2.4)

${}^{13}C=\frac{{}^{13}R{C}_{G}}{1+{}^{13}R+{}^{14}R}$ (2.5)

${}^{14}C=\frac{{}^{14}R{C}_{G}}{1+{}^{13}R+{}^{14}R}$ (2.6)

Since the compound has interacted with other sources of carbon, the carbon global (*C _{G}*) contained in one of the compounds is considered to come from several sources.

Each carbon source contributes to the overall carbon mass balance. Thus each source contains *C _{a}*,

${}^{14}C{}_{G}={}^{14}C{}_{a}+{}^{14}C{}_{b}+{}^{14}C{}_{c}+\cdots $ (2.7.a)

${}^{13}C{}_{G}={}^{13}C{}_{a}+{}^{13}C{}_{b}+{}^{13}C{}_{c}+\cdots $ (2.7.b)

${}^{12}C{}_{G}={}^{12}C{}_{a}+{}^{12}C{}_{b}+{}^{12}C{}_{c}+\cdots $ (2.7.c)

*C _{a}*,

Carbon reacts with atmospheric oxygen to give carbon dioxide CO_{2}.

For a selected sample, for example, if it is a plant sample, or an oceanic sample etc. the amount of carbon dioxide measured
$C{O}_{2}{}_{{}_{Sample}}$ is obtained with the contribution of at least three other CO_{2} sources. There is the naturally (background) occurring source of
$C{O}_{2}{}_{{}_{bg}}$ , the fossil-fuel source
$C{O}_{2}{}_{{}_{ff}}$ and other sources of biogenic origin, and unknown origin
$C{O}_{2}{}_{{}_{other}}$ .

We may write:

${}^{12}C{O}_{2}{}_{{}_{Sample}}={}^{12}C{O}_{2}{}_{{}_{bg}}+{}^{12}C{O}_{2}{}_{{}_{ff}}+{}^{12}C{O}_{2}{}_{{}_{other}}$ (2.8.a)

${}^{13}C{O}_{2}{}_{{}_{Sample}}={}^{13}C{O}_{2}{}_{{}_{bg}}+{}^{13}C{O}_{2}{}_{{}_{ff}}+{}^{13}C{O}_{2}{}_{{}_{other}}$ (2.8.b)

${}^{14}C{O}_{2}{}_{{}_{Sample}}={}^{14}C{O}_{2}{}_{{}_{bg}}+{}^{14}C{O}_{2}{}_{{}_{ff}}+{}^{14}C{O}_{2}{}_{{}_{other}}$ (2.8.c)

By Liquid Scintillation Counting (LSC) or by accelerator mass spectrometry (AMS), we are interested in ^{14}C. The ^{13}C and ^{12}C are stable isotopes, their composition does not vary.

(2.6) and (2.8.c) provide this relationship:

$\frac{{}^{14}R{}_{Sample}\times C{O}_{2}{}_{{}_{Sample}}}{1+{}^{13}R{}_{Sample}+{}^{14}R{}_{Sample}}=\frac{{}^{14}R{}_{bg}\times C{O}_{2}{}_{{}_{bg}}}{1+{}^{13}R{}_{bg}+{}^{14}R{}_{bg}}+\frac{{}^{14}R{}_{ff}\times C{O}_{2}{}_{{}_{ff}}}{1+{}^{13}R{}_{ff}+{}^{14}R{}_{ff}}+\frac{{}^{14}R{}_{other}\times C{O}_{2}{}_{{}_{other}}}{1+{}^{13}R{}_{other}+{}^{14}R{}_{other}}$ (2.9)

The isotopic abundances of carbon 13 on carbon 12 or carbon 14 on carbon 12 are very low in front of one. So they can be ignored in front of one.

Taking this approximation into account, relation (2.9) can produce the following equation:

${}^{14}R{}_{Sample}\times C{O}_{2}{}_{{}_{Sample}}={}^{14}R{}_{bg}\times C{O}_{2}{}_{{}_{bg}}+{}^{14}R{}_{ff}\times C{O}_{2}{}_{{}_{ff}}+{}^{14}R{}_{other}\times C{O}_{2}{}_{{}_{other}}$ (2.10)

We specify that ^{12}CO_{2}, ^{14}CO_{2}, ^{13}CO_{2} are included in each of the following components:
$C{O}_{2}{}_{{}_{Sample}}$ ,
$C{O}_{2}{}_{{}_{bg}}$ ,
$C{O}_{2}{}_{{}_{ff}}$ and
$C{O}_{2}{}_{{}_{other}}$ . It can be expressed as a relationship of this form:

$C{O}_{2}={}^{12}C{O}_{2}+{}^{13}C{O}_{2}+{}^{14}C{O}_{2}$ (2.11)

According to [8] [9] [10], the isotope fractionation on carbon 13 is:

$\delta {}^{13}C{}_{S}={10}^{3}\left(\frac{{}^{13}R{}_{sample}}{{R}_{VPDB}}-1\right)$

Then from that, we can have this:

${}^{13}R{}_{sample}={R}_{VPDB}\left(1+\frac{\delta {}^{13}C{}_{S}}{{10}^{3}}\right)$ (2.12)

We have VPDB = Vienna Pee Dee Belemnite which is used for the modern standard.

It should also be noted that we can well assume this approximation ${}^{14}R\approx {\left({}^{13}R\right)}^{2}$ .

Thus (2.10) become:

$\begin{array}{l}{}^{14}C{O}_{2}{}_{{}_{Sample}}\times \delta {}^{13}C{}_{Sample}\\ ={}^{14}C{O}_{2}{}_{{}_{bg}}\times \delta {}^{13}C{}_{bg}+{}^{14}C{O}_{2}{}_{{}_{ff}}\times \delta {}^{13}C{}_{ff}+{}^{14}C{O}_{2}{}_{{}_{other}}\times \delta {}^{13}C{}_{other}\end{array}$ (2.13)

The fractionation for ^{14}C is almost exactly double that for ^{13}C [11] [12] so we can write:

$\begin{array}{l}{}^{14}C{O}_{2}{}_{{}_{Sample}}\times \delta {}^{14}C{}_{Sample}\\ ={}^{14}C{O}_{2}{}_{{}_{bg}}\times \delta {}^{14}C{}_{bg}+{}^{14}C{O}_{2}{}_{{}_{ff}}\times \delta {}^{14}C{}_{ff}+{}^{14}C{O}_{2}{}_{{}_{other}}\times \delta {}^{14}C{}_{other}\end{array}$ (2.14)

If $\delta {}^{13}C=-25\u2030$ , we have $\Delta {}^{14}C=\delta {}^{14}C$ and if not, we can write $\Delta {}^{14}C=\alpha \delta {}^{14}C+\beta $ .

(By this formula below: $\Delta {}^{14}C=\delta {}^{14}C-2\left(\frac{\delta {}^{14}C}{1000}+1\right)\left(25+\delta {}^{13}C\right)$ [13] )

To the difference assumed or not introduced by the isotope fractionation on the sample, we can have:

$\begin{array}{l}{}^{14}C{O}_{2}{}_{{}_{Sample}}\times \Delta {}^{14}C{}_{Sample}\\ ={}^{14}C{O}_{2}{}_{{}_{bg}}\times \Delta {}^{14}C{}_{bg}+{}^{14}C{O}_{2}{}_{{}_{ff}}\times \Delta {}^{14}C{}_{ff}+{}^{14}C{O}_{2}{}_{{}_{other}}\times \Delta {}^{14}C{}_{other}\end{array}$ (2.15)

3. Approach of Mass Balance Equation

For a sample of vegetable or an atmospheric sample, we can assume that the carbon dioxide CO_{2} is formed:

Of a component that represents the value of the CO_{2} concentration in a certain level of the atmosphere (the troposphere) or background component free from any anthropogenic source. It is noted
${\left(C{O}_{2}\right)}_{bg}.$

Of another component that represents the value of fossil fuel CO_{2} concentration. This value is derived from the burning of fossil fuels. This is the fossil fuel CO_{2} component. It is noted
${\left(C{O}_{2}\right)}_{ff}.$

Of other large components that represent the value of the CO_{2} concentration in the biosphere or exchanges between systems. It is noted
${\left(C{O}_{2}\right)}_{bio}.$

And an obscure (obs) component that represents the value of unknown and uncontrolled sources of CO_{2} concentration
${\left(C{O}_{2}\right)}_{obs}.$

These assumptions can be translated from the following equations:

${\left(C{O}_{2}\right)}_{Sample}={\left(C{O}_{2}\right)}_{bg}+{\left(C{O}_{2}\right)}_{ff}+{\left(C{O}_{2}\right)}_{bio}+{\left(C{O}_{2}\right)}_{obs}$ (3.1)

And

$\begin{array}{c}{\left(C{O}_{2}\right)}_{Sample}\times \Delta {}^{14}C{}_{Sample}={\left(C{O}_{2}\right)}_{bg}\times \Delta {}^{14}C{}_{bg}+{\left(C{O}_{2}\right)}_{ff}\times \Delta {}^{14}C{}_{ff}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\left(C{O}_{2}\right)}_{bio}\times \Delta {}^{14}C{}_{bio}+{\left(C{O}_{2}\right)}_{obs}\times \Delta {}^{14}C{}_{obs}\end{array}$ (3.2)

${\left(C{O}_{2}\right)}_{Sample}$ ,
${\left(C{O}_{2}\right)}_{bg}$ ,
${\left(C{O}_{2}\right)}_{ff}$ ,
${\left(C{O}_{2}\right)}_{bio}$ and
${\left(C{O}_{2}\right)}_{obs}$ are respectively the global CO_{2} concentration obtained from the sample, the background CO_{2} concentration, fossil fuels CO_{2} concentration, biogenic CO_{2} concentration and unknown or/and uncontrolled (obscure) sources of CO_{2} concentration.

$\Delta {}^{14}C{}_{Sample}$ ,
$\Delta {}^{14}C{}_{bg}$ ,
$\Delta {}^{14}C{}_{ff}$ ,
$\Delta {}^{14}C{}_{bio}$ and
$\Delta {}^{14}C{}_{obs}$ are respectively ^{14}C level of the sample, the background ^{14}C level, fossil fuels ^{14}C level, sources of biogenic ^{14}C level and unknown or/and uncontrolled (obscure) sources ^{14}C level.

Finally, we will use these relations for the model of the equation that will allow us to find the local Suess effect caused by fossil fuels, as several authors such as [4] [14] [15] [16].

By (3.1) and (3.2) equations we can obtain the fossil-fuel CO_{2} component
${\left(C{O}_{2}\right)}_{ff}$ that we can, without any approximation write in this way:

$\begin{array}{c}{\left(C{O}_{2}\right)}_{ff}={\left(C{O}_{2}\right)}_{bio}\frac{\Delta {}^{14}C{}_{bio}-\Delta {}^{14}C{}_{Sample}}{\Delta {}^{14}C{}_{Sample}-\Delta {}^{14}C{}_{ff}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\left(C{O}_{2}\right)}_{bg}\frac{\Delta {}^{14}C{}_{bg}-\Delta {}^{14}C{}_{Sample}}{\Delta {}^{14}C{}_{Sample}-\Delta {}^{14}C{}_{ff}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\left(C{O}_{2}\right)}_{obs}\frac{\Delta {}^{14}C{}_{obs}-\Delta {}^{14}C{}_{Sample}}{\Delta {}^{14}C{}_{Sample}-\Delta {}^{14}C{}_{ff}}\end{array}$ (3.3)

To establish the mass balance equation, we will consider these approximations used by several authors like [4] [14] [16] [17] [18] [19] [20] and available in literature. These approximations make perfect sense.

1) Fossil carbon dioxide ${\left(C{O}_{2}\right)}_{ff}$ is free of significant amounts of radiocarbon, as fossil materials are several billion years old, a period of time long enough for the total amount of radiocarbon contained in these fossil organic materials to be removed.

Taking into account the relationship $\Delta {}^{14}C=1000\left(\frac{{\left({}^{14}C/C\right)}_{Sample}}{{\left({}^{14}C/C\right)}_{Std}}-1\right).$

We can assume that $\Delta {}^{14}C{}_{ff}=-1000\u2030$ (3.a)

2) Other sources of carbon dioxide have their $\Delta {}^{14}C$ supposed equal to that of the background atmosphere. C-14 biogenic concentration $\Delta {}^{14}C{}_{bio}$ and C-14 obscure source concentration $\Delta {}^{14}C{}_{obs}$ are equal to the C-14 concentration in the background $\Delta {}^{14}C{}_{bg}$ . (3.b)

Taking into account, these two approximations we can have:

${\left(C{O}_{2}\right)}_{ff}={\left(C{O}_{2}\right)}_{bg}\frac{\Delta {}^{14}C{}_{bg}-\Delta {}^{14}C{}_{Sample}}{\Delta {}^{14}C{}_{bg}+1000}$ (3.4)

We will use this formula, in the following to determine the Suess effect of each sample used.

This formula is a universal one; no matter how many different components make up the overall carbon dioxide concentration measured in a sample
${\left(C{O}_{2}\right)}_{sample}$ , as long as we assume that the ^{14}C concentration of the various components is taken to be equal to the background ^{14}C concentration.

If the nature of the samples used to quantify the fossil-fuel CO_{2} concentration are not plants, this proposed methodology can be modified (oceanic samples for example).

To better quantify the contribution of fossil fuels to the level of carbon dioxide in the atmosphere, we will determine the proportion of CO_{2} derived from fossil fuels, which will be called
${\left(S\right)}_{ff}$ for the following.

If we propose to work with samples, after 1950 (year zero in calendar ^{14}C), we can have this formula, available in the literature:

$\Delta {}^{14}C=1000\left(F{\text{e}}^{\mu \left(y-1950\right)}-1\right)$ (3.5)

*F* is the fraction modern of component in the mixture; *y* is the year of sample collection and
$\mu =\frac{\mathrm{ln}2}{5568}$ is the radioactive constant decay of ^{14}C and 5568 years is the conventional half-life of Libby.

We want to use Equation (3.4) and (3.5) and then propose a new alternative equation that will be applicable regardless of the nature of the sample, vegetable etc.

(3.4) becomes:

${\left(S\right)}_{ff}\left(\%\right)=100\left(1-\frac{{F}_{Sample}}{{F}_{bg}}\right)$ (3.6)

The relation is independent of the time parameter.

${F}_{bg}$ and
${F}_{Sample}$ is respectively the fraction of modern carbon for background and for the sample.
${\left(S\right)}_{ff}\left(\%\right)$ is the proportion of fossil-fuel derived CO_{2} of the sample.

Now we will apply it to a particular case with results that we have already obtained with the general method. At first, we give general information for our samples in Table 1.

Now we will tabulate the F-isotope fractionation values, in Table 2, that we found from the different samples we used. The method used is the same as we did in our previous work [21], *i.e.* Liquid Scintillation Counting (Code Laboratory DK) or by AMS-Arizona (Code Laboratory AA). By the way, due to the shortage of material available and more complete equipment, and in order to extend our measures, we have taken values from this article to have a continuous chronology until a more recent date.

In Table 3, we summarize the results of ${F}_{bg}$ , ${F}_{Sample}$ and ${\left(S\right)}_{ff}$ . The proportion of fossil-fuel is calculated by the Formula (3.6).

$\Delta {}^{14}C{}_{bg}$ Values provide at [22] [23].

After calculating the proportion (*S _{ff}*) of fossil fuels, to be seen in the Table 3, we have compared the values found:

In wooded area: Mbao Forest (1960, 1966 and 2007) and UCAD Botanic Garden (1970, 1979, 1990, 2000).

And in not wooded area often industrial sites: Highway (1961), SAR Factory (1964, 1968, 2005), Fass District (1976), Beach (1981, 1987, 1993, 1996, 1998 and 2009), Soumbédioune Market (2003) and Airport Runway (2010).

We notice that the values of the proportion of fossil fuels obtained in wooded area is more important that the values founded in not wooded area, with an average of 2.31% against 1.39%.

Sites where trees or botanical gardens grow and which are located in urban areas have higher value of proportion of fossil fuel
${\left(S\right)}_{ff}$ . This is thought to be due to the enormous amount of anthropogenic CO_{2} absorbed by the plants or trees.

But globally we can note, by Figure 1, that the values we found are small positive values. The largest value is 4.29% (in 1960 at wooded area) and the smallest value is 0.06% (in 1968 at not wooded area).

Figure 1. (*S*)* _{ff}* values in wooded area and in not wooded area.

Table 1. General information for samples.

Table 2. *F _{Sample}* values in our samples.

Table 3. Fossil fuel CO_{2} results (in ppm).

4. Conclusion

At first, the objective of this paper is to exploit a simple method for establishing the mass balance equation. For the determination of the values of the fraction of modern carbon *F* of our samples, we use the ^{14}C conventional method. Next, in order to estimate the Suess effect indicator, we have determined the fossil fuel fraction *S _{ff}* based on the mass balance equations for CO

Acknowledgements

This work was supported by the Radiocarbon laboratory of IFAN Cheikh Anta Diop of Dakar (Cheikh Anta Diop University). We express our thanks to all the team of the botanic laboratory IFAN Cheikh Anta Diop de Dakar for having made available the leaves samples. In particular, we would like to thank Alpha Oumar Diallo for the important contributions made to this research article.

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