6.4 Calculation of Tracer Age
The calculation of CFC tracer ages is similar to that of [latex]{SF}_{6}[/latex]. The procedure involves comparing a measured concentration in a water sample with historical concentrations in the atmosphere. Henry’s Law is used to relate the measured aqueous concentration of a sample to the equilibrium gas‑phase concentration. Because the Henry Coefficient is temperature dependent, and because the partial pressure of CFCs in the atmosphere depends on total pressure, both the recharge temperature and pressure (i.e., elevation) are needed for age dating.
If groundwater contains excess air and, therefore, some excess CFCs above equilibrium solubility, the amount of excess air could be subtracted from the measured concentration. However, because CFCs are much more soluble than tracers such as [latex]{SF}_{6}[/latex], corrections for excess air are seldom made. For example, the calculated age of a sample that recharged in 1975 with 5 cc/kg—∆[latex]Ne[/latex] of 45 percent, which is moderately large—would change by less than one year for CFC‑11, CFC‑12, and CFC‑113.
The temperature and salinity dependence of the Henry Coefficient was investigated by Warner and Weiss (1985) and Bu and Warner (1995) who provided empirical equations for calculating the equilibrium constant ([latex]K_{h}[/latex]) for the solubility form of Henry’s Law expressed by Equation (23) in Section 5.4.
Values for [latex]K_{h}[/latex] and e at various temperatures for fresh water are shown in Table 7. If the recharge temperature (T), salinity (S), and elevation—hence P—are known, Equation (26) [i.e., [latex]C = K_{h}(T,S)x\left( P - e(T) \right)\rbrack[/latex]can be solved for the dry air mole fraction (x) and then x can be computed for a measured aqueous concentration (C) as shown in Equation (26). The dry air mole fraction can then be compared to the history of the CFCs in the atmosphere to obtain a recharge date as illustrated in Figure 25.
| [latex]x = \frac{C}{K_{h}(T,S)\left( P - e(T) \right)}[/latex] | (26) |
|---|
Table 7 ‑ Values of Kh for CFC‑11, CFC‑12, CFC‑113, and water vapor (e) at various temperatures for fresh water (salinity = 0).
| Temperature (°C) | Kh CFC‑11
mol kg-1atm-1 |
Kh CFC‑12
mol kg-1atm-1 |
Kh CFC‑113
mol kg-1atm-1 |
e
atm |
|---|---|---|---|---|
| 0 | 0.03871 | 0.00942 | 0.01263 | 0.0060 |
| 10 | 0.02113 | 0.00548 | 0.00653 | 0.0121 |
| 20 | 0.01288 | 0.00353 | 0.00378 | 0.0230 |
| 30 | 0.00865 | 0.00247 | 0.00241 | 0.0418 |
| 40 | 0.00633 | 0.00187 | 0.00168 | 0.0726 |
Kh for CFC‑11 and CFC‑12 from Warner and Weiss (1985); Kh for CFC‑113 from Bu and Warner (1995); e from Antoine (1888) and Equation (6) of this book.
A general procedure for calculating a tracer age for CFCs is as follows:
- Measure the aqueous concentration (C) in a groundwater sample.
- Estimate the recharge elevation (Z) of the sample.
- Compute the recharge pressure (P) using Equation (5) in Section 2.3 of this book.
- Estimate the recharge temperature for the sample. This could use noble gas thermometry, if available, or could be taken as the average annual temperature of the recharge zone for the sample.
- Estimate the recharge salinity for the sample. In many cases, this is assumed to be zero for fresh water in the recharge area.
- Compute the equilibrium constant (Kh) for the temperature and salinity of the recharge area. This could be computed using the equations of Warner and Weiss (1985) and Bu and Warner (1995), or could be approximated from the values given in Table 7.
- Compute the vapor pressure of water (e) at the recharge temperature. This could be computed using empirical equations such as Antoine (1888) that is presented in Equation (6) of Section 2.3 of this book, or could be interpolated from the values given in Table 7.
- Compute the mole fraction (x) of CFCs in the atmosphere that would have resulted in the measured aqueous concentration using Equation (26).
- Compare the computed mole fraction—which is essentially equivalent to the mixing ratio—to the atmospheric history (Figure 24) to derive the estimated year the sample recharged.
- Compute the age as sample date minus recharge year.