Geoscience Reference
In-Depth Information
The equation of isotope dilution
Let us set out to measure the number of moles of rubidium in a particular sample. Natural
Rb has two isotopes,
85
Rb (72.16%) and
87
Rb (27.84%). A commercial supplier sells a
87
Rb
spike containing 2.0%
85
Rb and 98.0%
87
Rb (numbers are atomic abundances). Note that,
contrary to butterfly species, commercial spikes are never isotopically pure. Let us dissolve a
sample, containing, by definition, natural Rb, and add a Rb spike to the solution. Let us call
n
the number of moles of any isotope in the mixture with atomic mass as superscript and use
“nat” and “sp” to refer to the Rb contributed to the mixture (“mix”) by the sample and the
spike, respectively. Mass balance conditions on each isotope read:
n
8
mix
=
n
8
nat
+
n
85
sp
n
8
mix
=
n
8
nat
+
n
87
sp
(E.44)
Dividing the second equation by the first and then dividing again by
n
85
sp
and using isotopic
ratios, we get:
87
Rb
85
Rb
nat
n
8
nat
/
87
Rb
85
Rb
sp
87
Rb
85
Rb
n
85
/
sp
+
/
mix
=
(E.45)
n
8
nat
/
n
85
sp
+
1
By collecting the terms containing the mole ratios, we get:
87
Rb
85
Rb
sp
−
87
Rb
85
Rb
mix
/
/
n
8
nat
n
85
sp
=
87
Rb
85
Rb
mix
−
87
Rb
85
Rb
nat
/
/
87
Rb
85
Rb
mix
98.0
/
2.0
−
/
=
87
Rb
85
Rb
mix
−
(E.46)
/
72.16
/
27.84
If we know
n
85
sp
, i.e., how many moles of
85
Rb from the spike were added to the mixture
(typically by weighing an aliquot of a calibrated spike solution), the determination of the
isotope composition of the mixture (
87
Rb
85
Rb)
mix
with a mass spectrometer yields the
number of moles
n
8
nat
of natural Rb present in the sample and therefore its Rb concentration.
For better precision, (
87
Rb
/
85
Rb)
mix
must be kept different enough from both the natural
ratio (no under-spiking) and from the spike ratio (no over-spiking).
/
counting butterflies on a very small island, named Eye-Dee (
Fig. E.3
). Eye-Dee's forests
are populated by red butterflies unfortunately not known by their pretty Latin name. We
take advantage of a nearby island, which happens to be populated only by blue butterflies.
Let us harvest a few buckets of blue butterflies, which we count, then quickly release
them on Eye-Dee. A few wing flaps later, we collect a sample of the mixed population:
the proportion of red and blue in our catch clearly gives us the number of red butterflies
on Eye-Dee. The blue butterflies are the spike. If
85
Rb and
87
Rb are substituted for the
red and blue butterflies, we have the measurement of rubidium concentrations by isotopic
dilution. By measuring the proportion of each isotope, we gain access to the number of
atoms of each of them initially present. Isotopic spikes are produced by specialist firms.
A remarkable property of this method is that the size of the sample collected by the net
