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Isotope Geology Claude J. (cambridge)

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second is that, conversely, a solid atordinary tem-
peratures retains its isotopic composition over timewithoutbecoming homogeneouswith
itssurroundings.This iswhy rocksare reliable isotoperecords.Thispropertyisadirectcon-
sequenceofthe di¡usionpropertiesofnatural isotopes in liquids andsolids.
The theory of di¡usion, that is, the spontaneous motion of atoms in£uenced by di¡er-
ences in concentration,provides anapproximatebutadequateformula:
x �
ffiffiffiffiffiffi
Dt
p
wherex is thedistancetraveledby the element, t is time in seconds, andD thedi¡usion coef-
¢cient (cm2 s�1).
Exercise
In a liquid silicate at 1200K the diffusion coefficient for elements like Rb, Sr, or K is
D¼ 10�6 cm2 s�1. In solid silicates heated to 1200K, D¼ 10�11 cm2 s�1.
How long does it take for two adjacent domains of 1 cmdiameter to become homogeneous:
(1) within a silicate magma?
(2) between a silicate magma and a solid, which occurs during partial melting when 10% of
the magma coexists with the residual solid?
Answer
(1) In a silicate magma if D¼ 10�6 cm2 s�1, t� x2/D¼ 106 s, or about 11 days.
If it takes 11 days for the magma to homogenize on a scale of 1 cm, on a 1-km scale
(¼ 105 cm), it will take t� x2/D¼ 1010¼ 1016 s, or close to 3 � 108 years, given that
1 year� 3 � 107 s.
In fact, homogenization at this scale would not occur by diffusion but by advection or
convection, that is, a general motion of matter, and so would be much faster.
(2) In the case of a magma impregnating a residual solid with crystals of the same dimen-
sions (1 cm), t� x2/D¼ 1011 s or about 3 � 105 years, or 300 000 years, which is rather fast
in geological terms. For 1-mm crystals, which is more realistic, t� 10�2/D¼ 3 � 103 years,
or 3000 years. So isotope equilibrium is established quite quickly where amagma is in the
presence of mineral phases.
Asecond importantquestion iswhether rocks atordinary temperatures can retain their
isotope compositions without being modi¢ed and without being re-homogenized. To
14 Isotopes and radioactivity
answer this, it must be remembered that the di¡usion coe⁄cient varies with temperature
by theArrhenius law:
D ¼ D0 exp �DH
RT
� �
where�H is the activation energy, which is about 40 kcal per mole,R is the ideal gas con-
stant (1.987 calperKpermole), andT theabsolute temperature (K). IfD¼ 10�11cm2 s�1in
solids at1300 8C,what is thedi¡usion coe⁄cientDoratordinary temperatures?
Dor
D1300
¼ exp �DH
R
I
Tor
� I
T1300
� �� �
¼ hBi
fromwhichDor ¼ Bh i D1300.
Calculate Bh i to¢ndthat itgives1.86 � 10�25, thereforeDor¼ 2 � 10�36 cm2 s�1.
Tohomogenizea1-mmgrainatordinary temperatures takes
t ¼ 10
�2
2 � 10�36 ¼ 0:5 � 10
34s � 1:5 � 1026 years;
which is in¢nitelylong toall intents andpurposes.
Important remark
Rocks therefore retain the memory of their history acquired at high temperatures. This is the prime
reason isotope geology is so incredibly successful and is the physical and chemical basis of isotope
memory. The phenomenonmight be termed isotopic quenching, by analogy with metal which, if it is
immersedwhen hot in coldwater, permanently retains the properties it acquired at high temperature.
1.3.3 A practical application of isotopic exchange:
isotopic dilution
Supposewewish tomeasure the rubidium contentofa rock.Rubidiumhas two isotopes, of
mass 85 and 87, in the proportion 85Rb/87Rb¼ 2.5933. (This is the value foundwhen mea-
suring theRb isotope compositionofnatural rocks.)The rock is dissolvedwithamixtureof
acids.To the solution obtained,we add a solutionwith aknownRb contentwhich hasbeen
arti¢cially enriched in 85Rb (spike), whose 85Rb/87Rb ratio in the spike is known.The two
solutionsmixandbecomeisotopicallybalanced.Onceequilibriumisreached, theabsolute
Rb contentof the rockcanbe determinedbysimplymeasuring the isotope composition of
a fractionofthemixture.
Writing
85Rb
87Rb
� �
¼ 85
87
	 
gives:
85
87
� �
mixture
¼ ð
85RbÞrock þ ð85RbÞspike
ð87RbÞrock þ ð87RbÞspike
¼
85
87
	 
spike
þ 8587
	 
rock
87Rbrock
87Rbspike
h i
1þ 87Rbrock87Rbspike
h i :
Topandbottomare dividedby 87Rb,bringingout 8587
	 
.
A littlemanipulationgives:
87Rbrock ¼ 87Rbspike 85
87
� �
spike
85
87
	 
spike
� 8587
	 
mixture
85
87
	 
mixture
� 85
87
	 
rock
" #
:
15 Isotopy
Becauseweknowthe87Rbcontentofthespike, theRbcontentoftherockcanbeobtainedby
simply measuring the isotope ratio of the mixture, without having to recover all the Rb by
chemical separation.
Suppose, say, the isotopic composition of the spike ofRb is 85Rb/87Rb¼ 0.12.The natu-
rally occurring 85Rb/87Rb ratio is 2.5933.We dissolve1g of rock and add to the solution1g
ofspike containing3 ngof 87Rbpercubic centimeter.After thoroughlymixing the solution
containing the dissolved rockand the spike solution,measurementofa fraction of themix-
tureyields a ratio of 85Rb/87Rb¼ 1.5.TheRb contentofthe rockcanbe calculated.We sim-
plyapply the formula:
87Rbrock ¼ 87Rbspike 0:12� 1:5
1:5� 2:59
� �
¼ 1:266;
or
87Rbrock ¼ 1:266� 87Rbspike ¼ 1:266 � 3 � 10�9 g ¼ 3:798 � 10�9 g:
As we took a sample of 1g of rock, C87Rb ¼ 3:798 ng g�1 ¼ 3:798 ppb. Therefore
C87total ¼ C87Rbð1þ 2:5933Þ ¼ 13:42 ppb.
This method can be used for all chemical elements with several stable isotopes for which
spikeshavebeenpreparedthatarearti¢ciallyenrichedinoneormoreisotopesandforelements
with a single isotope, provided it is acceptable to use a radioactive isotope (and so potentially
dangerousforwhoeverconductstheexperiment)asaspike.Themethodhasthreeadvantages.
First,aftermixingwiththespike,chemicalseparationmethodsneednotbeentirelyquanti-
tative. (The yields of the various chemical operations during analysis do not count.) Isotope
ratiosalonematter,aswell asanycontamination,whichdistorts themeasurement,ofcourse.
Then, as themass spectrometer makes very sensitive andvery precise measurements of
isotope ratios, isotopic dilution may be used to measure the amounts of trace elements,
eventhetiniesttraces,withgreatprecision.
Isotope dilutionwas invented for theneedsof laboratoryanalysisbutmaybe extended to
natural processes. As shall be seen, variable isotope ratios occur in nature. Mixes of them
canbe used to calculate proportions bymass of the geochemical elements involved just by
simplemeasurementsof isotope ratios.
As canbe seen, isotope dilution is an essential method in isotope geochemistry. But just
howprecise is it?This exercisewillallowustospecify theerror (uncertainty) in isotopedilu-
tionmeasurement.
Exercise
The isotope ratios of the spike, sample, and mixture are denoted RT, RS, and RM, respectively.
We wish to find out the quantity X of the reference isotope Cj in the sample. To do this,
quantity Y of spike has been mixed and the isotope ratios (Ci/Cj)¼ R of the spike, sample, and
mix have been measured. What is the uncertainty of the measurement?
Answer
Let us begin with the formula
X ¼ Y RT � RM
RM � RS
����
����
16 Isotopes and radioactivity
which may be written RMX – RSX¼ RTY – RMY or RM (Xþ Y)¼ RTYþ RSX, or alternatively
RM ¼ RT Y
X þ Y þ RS
X
X þ Y :
Weposit XX þ Y ¼W and Y¼ 1�W. Let us calculate the logarithmic derivative and switch to�
(finite difference):
DX
X
¼ DY
Y
þ DðRT � RMÞðRT � RMÞ þ
DðRM � R SÞ
ðRM � R SÞ :
We can transform (RT – RM) and (RM – RS) as a function of (RT – RS), from which:
DX
X
¼ DY
Y
þ 2DRMðRT � RSÞ
1
W
þ 1
1�W
� �
¼ DY
Y
þ 2DRMðRT � R SÞ
1
W ð1�W Þ
� �
:
Neglecting the uncertainty on RT and RS, which are assumed to be fully known, uncertainty is
minimum when:
� RT – RS is maximum. A spikemust therefore be prepared whose composition is as remote as
possible from the sample composition.
� 1/[w(1�w)] ismaximum for given values of RT and RS, that is,