Diploma thesis in Physics submitted by Florian Freundt born in ...

2.1. Noble gas temperatures 2 Theory
modeled and measured data. The quality of this matching is assessed by using the χ 2 method
that weights the deviation between the measured and the predicted concentrations with the
measurement accuracy σi:
UA, PR and MR model
χ 2 = �
i
�
Ci,m − Cmodel �2 i
σ2 i
(2.12)
The first model created to account for the presence of an excess air component was the unfractionated
air model, or UA model. It is based on the assumption that enclosed air bubbles are
completely dissolved in the water at or below the ground water table, leading to an atmospheric
composition of the excess air component of the noble gas concentration without any signs of
fractionation, as described by
C UA
i = Ci,eq + A · Ci,atm (2.13)
where Ci,atm is the unfractioned excess air component and A is the fraction of air entrapped in
the water volume, a scaling parameter of this model. Using Henry’s law (Equation 2.1), this
relation is expressed as
C UA
i = Ci,eq(1 + A · Hi) (2.14)
This basic assumption of complete dissolution did not prove to be realistic though, as Stute et al.
[1995] showed excess air components in ground water to be enriched in the heavier noble gases.
They introduced a modification of the UA model, called partial re-equilibration model, or PR
model, assuming diffusive gas loss affecting the completely dissolved excess air component. The
mass and temperature dependent diffusion coefficients of the individual noble gases [Jähne et al.,
1987] would then account for the occurring fractionation. Inter-isotopic fractionation effects are
not observed however [Peeters et al., 2003], on which the PR model’s applicability was criticized.
Whether this effect should actually be expected at all to occur in a manner similar to internoble-gas
fractionation has been questioned though [Bourg and Sposito, 2008]. The PR model
can be expressed as [Aeschbach-Hertig et al., 2008]:
�
�
� ��
β
C PR
i
= Ci,eq
1 + A · Hi · exp
−FPR
� Di
DNe
(2.15)
where FPR is a parameter characterizing the amount of excess air loss, Di are the noble gas
diffusion coefficients and 0.5 ≤ β ≤ 1 is a model parameter from gas transfer theory. Since
this single step excess air intrusion and degassing leads to initial Ne excess amounts requiring
unrealistic physical conditions in some studies [Kipfer et al., 2002] and failed to describe certain
noble gas records [Ballentine and Hall, 1999], a variation of the PR model was introduced as the
multi-step partial re-equilibration model, or MR model, by Kipfer et al. [2002]. The MR model
assumes a physically more realistic occurrence of multiple (n) excess air intrusions and degassing
steps, containing the PR model as the special case n = 1:
�
�
� ��
β
C MR
i
= Ci,eq
1 + A · Hi ·
n�
exp
k=1
20
−k · FPR
� Di
DNe
(2.16)