r = HC^ C2 - KbC2/bz (K = Kc + Kd (+ K ))
the isotope separation and the enrichment can be calculated as a function of time.
This is simple in the initial stage of the enrichment and when the final equilibrium has been reached, but complicated in the intermediate stages.
0-
L-'VcJoHL
[yTj/ru]
Fig. 2.2. Thermal diffusion column with reservoirs.
V_, Vv VQ — volume of storage reservoir, enrich-ment reservoir and column respectively,
T, 7"c — temperature in the reservoirs and average temperature in the column,
m_, m+, \xL — mass of gas in the storage reservoir, the enrichment reservoir and the column of length L, respectively.
The transport equation merely gives the mass transport of the isotopic molecules considered. The isotopic composition of the gas in different parts of the system is determined by the time the transport has lasted and by the amounts of gas in the different volumes. We consider a system represented by fig. 2.2. Since the amount of gas used for the enrichment is finite, an enrichment of one isotope in V+ neces-sarily is accompanied by a depletion of this isotope in V_. It is therefore practical to introduce two quantities:
the separation factor (Jones and Furry, 1946)
Q = :
isotope ratio in enrichment volume V+
isotope ratio in reservoir V
and the enrichment
concentration in V obtained by enrichment Q = concentration prior to enrichment
Though q is closely related to the TD properties of the gas and to the TD column, q' will be the quantity of practical interest for radiocarbon dating.
We first treat the simple problems of the initial stage of the enrichment and the equilibrium situation and then discuss a practical approximation describing the transient behaviour during enrichment.
To start with, we have a homogeneous gas mixture. Consequently dC2/dz = 0 and the initial transport
T2 = HC^C2. (2.6)
As long as the concentration gradient can be neglected, the heavy isotope is trans-ported at a constant rate through the column and is concentrated linearly in V+,
dC
2/6t = M^T
2/M
2m
+. (2.7)
Here we assumed that C2 « C1 so p ^ p1.
In the dynamic equilibrium situation the net transport is zero, dC2
T0 = HC,C0 -K = 0. (2.8a)
2 1 2 6z
For a binary mixture C1 = 1—C2, so
dC2 H
dz, and
C9( 1 - C9) K
C
7(z)ICAz) H
= e x p - (z-z ). (2.8b)
C
2b
Q)IC^z
o) K
The equilibrium separation factor then is:
C
2(L)IC
y(L) H
q = = e x p - Z . = e x p 2 y U , (2.9)
e
C
2(0)/C,(0) K
where A = H/2K.qe is constant for a certain column and certain operating conditions and independ-ent of the volumes used. Although time consuming, the thermal diffusion constants can be determined from measurements of qe (De Vries, 1956; Saviron et al., 1975, 1976).
The transient behaviour of the isotope separation and the enrichment must be derived from the transport equation (2.3) and the equation of continuity for species 2 (c.f. eq. 2.1):
a ( p2C2) / 3 f = - d i v U2) = - d i v (p2C2v - p2D grad C2 -p2DaC^C2 grad In 7").
(2.10) The exact solution for a practical system is complex and has not been given.
Solutions under certain simplifying assumptions are given by Bardeen (1940) and Jones and Furry (1946).
In a binary mixture with C2«C^, so C-j = 1— C2 « 1, r2 is linearly related to C2. The solution presenting the concentration of the heavy isotope as a function of time and place in the column is a series expansion in terms of the final equilibrium value and a number of transient terms with rapidly decreasing time constants. In practice, often one transient term gives a sufficient approximation. Because of the complicated form of the exact solution, an approximate formula is to be preferred in most cases.
Felber and Pak (1974) developed an approximation method under the assump-tion that C2« C < | for the practical case of a finite, large V_, while Vc and V+ are of the same magnitude. In their paper the treatment of Jones and Furry (1946) for
V»VC and V_ = oo is followed. The actual situation is accounted for by a correc-tion of the mass m+ in the positive reservoir both for the amount of enriched material present in the column and for the depletion of the column gas adjacent to
V_. This leads to the following equations for the transient behaviour of r, q' and q:
T{t) =HC2oexp(-t/tr). (2.11)
q'it) = q'e-(q'e-V exp (-t/tr); (2.12)
Q' -lq'e - V exp (-t/tr)
q(t) = a - 5 1 L . (2.13)
^ + ( ^e- ^ e) e x p (-f / fr)
Here
M0 m"
f = — — ( < 7 ' - D (2.14)
r M^H e
is the time in which the initial transport would yield the equilibrium concentration of the rare heavy isotope in V and m" is the fictitious corrected mass consisting of the mass in V and a term taking into account the enrichment in the column near
V+ and the depletion near V/_at equilibrium:
m [(q - 1)/ln q - 1] -m+ [q - ( a - 1)/ln q]
m" = m++iJiL e ! + e e L . (2.15)
m_(qe-1)+ViL[qe-(qe-V/\nqe]
From a consideration of mass balance follows
q'=q(m_+m'+')/(m_+qm'+'). (2.16)
Felber and Pak (1974) found excellent agreement between the results of the ap-proximation formulas 2.11—2.16 and those of the formulas derived more exactly.
The important question now is whether these equations describe the experi-mental results with sufficient accuracy to be used to predict the enrichment obtained. Comparing the theoretical and the experimental results Felber and Pak (1972) found considerable discrepancies. The experimental values of H and Kc were up to 40% smaller than the theoretical ones, that of tr was about 40% higher. The enrichment, however, was fairly close to the theoretical value, indicating that the ratio VjV_ in this case probably has more influence on q'e than H and K.
The apparatus was shown to work reproducibly. This means that, provided the experimental conditions (p, V, Ty, T2) are kept constant, a few calibration runs can give the experimental parameters necessary to predict the enrichment.
If a more variable system is used (for instance a volume V_ depending on the amount of sample available), one can either make calibrations for a whole range of experimental conditions or determine the enrichment by measuring the simulta-neous enrichment of a closely related molecuie (De Vries, 1956; Grootes et al., 1975). This also has the advantage of being independent of the reproducibility of the column behaviour, provided a constant relation between the separation factors or enrichments of the two isotopic species exists.
The time dependence of the enrichment is given by eq. 2.12 in which both q'e and tr strongly depend on V_. In eq. 2.13, which gives the separation factor as a function of time, qe is volume independent (q'e is only present in a correction factor).
In an earlier paper (Grootes et al., 1975) we used a simple approximate formula for the time dependence of the isotope separation,
(<7-1) = (<7e-1)(1 - e -f / tr ) ,
and showed that the relation between the separation factors of different isotopic species could be given by a polynomial. In this way the influence of V_ and the
de-ft
pletion of the abundant molecule C1 = 1— 2 C. appears explicitly in the calcula-tion of the enrichment. /=2
The present more detailed treatment yields a relation for the enrichment of the same form (eq. 2.12).
From eq. 2.12 the relation between the enrichments of two different isotopic species I and II can be derived by elimination of t:
After rearrangement a series expansion gives
qU' - 1 = a, (qV - 1 ) + a2{qr - 1 )2 + a3(<7r - 1 )3 +
where a. = ^ er-1 ) rr < * " n
1
toi'-Dr!
1m""H
l'
I uW
K - 1 w -v < "
((7; - i r 2!(f
,r1)^ 2!(m;
Mr/y'
ft"'-WjftJ
1-rj)(2fj
,-f|)
(gr^-1)
33!(^
1)
3m'
(2.17)
(2.18)
(2.19)
m' yll/^l' i l i ^ H '
//"to; -1) " X -1)
m H 3\(m''")3H]
m
„\\
m,„\
JhZr
y l / ^ H '" ' X -
1 )A / X ' - D
2m"'
A77./^X-1) / / V ' - l ) !
Apparently the first coefficient a-| only slightly depends (through m^) on the size of V_ , as opposed to a2, a3, . . . (through q'e). If the values of H and /C are known for both isotopic molecules, the coefficients of the calibration relation can be calculated for each volume \/_ used. These values (sect. 2.1.4) are to be compared with the experimental results (sect. 2.4).