Trace Element
Geochemistry
Lecture 24
Geochemical Classification
The Rare Earth Elements
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The rare earths and Y are strongly
electropositive. As a result, they form
predominantly ionic bonds, and behave as
hard charged spheres.
The lanthanide rare earths are in the +3
valence state over a wide range of
oxygen fugacities.
In the transition metals, the s orbital of the
outermost shell is filled before filling of lower
electron shells is complete so the
configuration of the valence electrons is
similar in all the rare earth, hence all exhibit
similar chemical behavior.
Ionic radius, which decreases progressively
from La3to Lu3+ (93 pm) governs their
relative behavior.
Because of their high charge and large
radii, the rare earths are incompatible
elements.
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The degree of incompatibility of the lanthanides varies
with atomic number. Highly charged U and Th are highly
incompatible elements, as are the lightest rare earths.
However, the heavy rare earths have sufficiently small
radii that they can be accommodated to some degree in
many common minerals such as Lu for Al in garnet. Eu2+
can substitute for Ca in plagioclase.
Rare Earth Diagrams
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The systematic variation in lathanide rare
earth behavior is best illustrated by plotting
the log of the relative abundances as a
function of atomic. Relative abundances
are calculated by dividing the
concentration of each rare earth by its
concentration in a set of normalizing values,
such as the concentrations of rare earths in
chondritic meteorites.
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Rare earths are also refractory elements, so that their
relative abundances are the same in most primitive
meteorites - and presumably (to a first approximation) in
the Earth.
Why do we use relative abundances? The
abundances of even-numbered elements
in the solar system are greater than those of
neighboring odd-numbered elements and
abundances generally decrease with
increasing atomic number, leading to a
saw-toothed abundance pattern.
Normalizing eliminates this.
Abundances in chondritic meteorites are
generally used for normalization. However,
other normalizations are possible: sediments
(and waters) are often normalized to
average shale.
Partition Coefficients
The Partition Coefficient
• Geochemists find it convenient to define a partition or
distribution coefficient of element i between phases α and β:
Dia - b
Cia
= b
Ci
• Where one phase is a liquid, the convention is the solid is
placed on top:
s
s-ℓ
i
D
Ci
= ℓ
Ci
• Incompatible elements are those with Ds/l ≪ 1. Compatible
elements are those with Ds/l ≥ 1. These terms refer to
partitioning between silicate melts and phases common to
mantle rocks (peridotite). It is this phase assemblage that
dictates whether lithophile trace elements are concentrated
in the Earth’s crust, hence the significance of these terms.
Thermodynamic Basis
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The chemical potentials of element i in phases α and β are
µia = µia o + RT ln lia Xia
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At equilibrium:
µib = µib o + RT ln lib Xib
µia = µib
lia Xia
µi - µi = RT ln b b
li X i
ao
o
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bo
Since trace elements obey Henry’s Law, we can replace the activity coefficient with h.
The left hand side is ∆G˚, so that
Xia hib -∆ G˚/ RT
= ae
b
Xi
hi
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hib -∆ G˚/ RT
Di = C a e
hi
where C is simply a constant converting concentration units (usually
ppm) to mole fraction.
Thus the distribution coefficient is a kind of equilibrium constant.
and
a -b
Relationship among
distribution coefficients
• In a system with three phases, α, β, and γ, if α and β
are in equilibrium and α and γ are in equilibrium,
then β and γ must also be in equilibrium. It follows
that:
Dα-β=Dα-γ/Dα-β
• This relationship has practical use. For example, if
we can determine the partition coefficient for an
element between pyroxene and melt and between
garnet and pyroxene, we can then calculate the
garnet–melt partition coefficient for this element.
Temperature and Pressure
Dependence
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In ideal solutions, the temperature dependence of the partition coefficient is the same as that of
the equilibrium constant:
a -b
-∆ G˚/ RT
i
∆G can be expanded into entropy and enthalpy terms:
D
=e
∆ H˚+(P - P˚)DV
æ ¶ln Di ö
çè
÷ø =
¶T P
RT 2
In ideal solution, and assuming again that ∆V is independent of temperature and pressure, the
pressure dependence is also the same as that of the equilibrium constant:
-∆ V
æ ¶ln Di ö
çè
÷ø =
¶P T
RT
We would predict a strong pressure dependence when the ionic radius of an element differs
greatly from that of the available crystal lattice site. Thus, for example, we would predict the
partition coefficient for K between pyroxene and melt would be strongly pressure-dependent
since the ionic radius of K is 150 pm and is much larger that the size of the M2 site in
clinopyroxene, which is normally occupied by Ca, with a radius of about 100 pm. Conversely,
where the size difference is small (e.g., Mn (83 pm) substituting for Fe (78 pm)), we would expect
the pressure dependence to be smaller.
In non-ideal solutions, the T and P dependencies will be more complex because the activity
coefficients (the Henry’s Law coefficients) will also depend on T and P.
Bottom line: partition coefficient is temperature and pressure dependent.
Importance of Ionic Size and
Charge
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Ionic radius (picometers) vs. ionic charge
contoured for clinopyroxene/liquid partition
coefficients. Cations normally present in
clinopyroxene M1 and M2 sites are Ca2+, Mg2+, and
Fe2+, shown by ✱ symbols. Elements whose charge
and ionic radius most closely match that of the
major elements have the highest partition
coefficients
Much of the interest in trace elements in
igneous processes centers on the elements
located in the lower left portion of the
periodic table (K, Rb, Cs, Sr and Ba; the rare
earths, Y, Zr, Nb, Hf and Ta).
One reason for this focus of attention is that
these elements are all lithophile and
therefore present at relatively high
abundance in the Earth’s crust and mantle.
There is another reason, however: their
chemical behavior is comparatively simple:
their behavior in igneous systems is mainly
(not entirely) a function of ionic size and
charge.
The other trace elements that receive the
most attention from igneous geochemists
are the first transition series elements.
Though their electronic structures and
bonding behavior are considerably more
complex, charge and size are also
important. Many of these elements,
particularly Ni, Co, and Cr, have partition
coefficients greater than 1 in many Mg–Fe
silicate minerals. Hence the term
“compatible elements” often refers to
these elements.
Quantitative Treatment
• Consider the substitution reaction:
Mℓ + CaMgSi2O6 ⇄ Caℓ + MMgSi2O6
• The Gibbs free energy change of this reaction can be
expressed as:
M -Ca
Di
∆ Gr = ∆ Gexchange
- ∆ Gmelting
• The first term is ∆G for transferring an M2+ ion from the
melt to the crystal lattice and simultaneously transferring
a Ca2+ ion from the lattice site to the liquid.
• The second term is the ∆G associated with the melting of
diopside, and governs the distribution of Ca between
diopside and the liquid. The distribution coefficient for
element M then depends on these two components of
free energy:
Di/
M
D
Di
æ ∆ Gmelting
- ∆ GexM -Ca ö
= exp ç
÷
RT
è
ø
Substitution & Lattice Strain
• If we substitute an
atom that larger than
one normally
occupying a site in a
crystal lattice, for
example a Ba2+ for a
Mg2+ in olivine, the
lattice will strain to
accommodate that
atom.
• Strain is a form of stored
energy.
• We can express this as?
Olivine Lattice
Strain Theory
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According to the lattice strain energy theory, ∆Gexchange is dominated by
the energy associated with the lattice strain resulting from M2+ being a
different size than Ca2+. Because the melt (at least at low pressure) has a
far less rigid structure and is more compressible than the solid, any strain
in the melt is essentially negligible compared with the strain in the solid. In
other words:
∆ GexM -Ca @ ∆ Gstrain
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The strain energy, ∆Gstrain, may be calculated as:
∆ Gstrain
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1
é r0
ù
2
= 4p EN A ê (rm - r0 ) + (rm - r0 )3 ú
3
ë2
û
where r0 is the optimal radius of the lattice site, rM is the ionic radius of M,
NA is Avogadro’s Number, and E is Young’s modulus, which is the ratio of
stress applied to the resulting strain (change in dimension) and has units
of pressure. It is a property of the material and is related to
compressibility.
Blundy & Wood Model
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We can now substitute this expression into
Di/
M
D
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and obtain
Di/
DMDi/ = DCa
Di
æ ∆ Gmelting
- ∆ GexM -Ca ö
= exp ç
÷
RT
è
ø
1
ì
é r0
2
3ùü
-4
p
EN
(r
r
)
+
(r
r
)
A
m
0
m
0
ïï
êë 2
úû ïï
æ -∆ Gstrain ö
3
exp ç
=
exp
í
ý
è RT ÷ø
RT
ï
ï
ïî
ïþ
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where
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Where charge is different from the ion normally occupying the
site, we much consider the coupled substitution. For example, we
can balance charge by substituting both La3+ and Na+ into the
Ca2+ site in diopside. In this case, our relationship is:
Di/
Ca
D
=e
Di
-∆ Gmelt
/RT
Di
La-Ca
Na-Ca
∆ Gmelt
-∆ Gstrain
-∆ Gex
RT
•
Di/
DLa
=e
The point is we can build a theoretical framework to predict how
partitioning will depend on ionic size and charge (and the T and
P dependency).
Blundy & Wood Model
We still depend heavily on experimental determination of partition
coefficients. But by experimentally determining a few, we can use
the Blundy & Wood model to predict others (as well as pressure
dependence).
Dependence on Composition
Ol-liquid partition coefficient
for Zn parameterized as a
function of non-bridging
oxygens in the melt and T.
pyx-liquid partition coefficient
for Sm as a function of liquid
and solid composition
Rare Earth Partition
Coefficients