Partition
Coefficients
Lecture 26
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 liquid is
placed on top:
Di
-s
Ci
= s
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
•
The chemical potentials of element i in phases α and β are
µia = µia o + RT ln lia Xia
•
At equilibrium:
µib = µib o + RT ln lib Xib
µia = µib
lia Xia
µi - µi = RT ln b b
li X i
ao
o
•
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
•
•
•
lib -∆ G˚/ RT
Di = C a e
li
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
•
•
•
•
•
•
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
•
•
•
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
Strain Theory
•
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
•
The strain energy, ∆Gstrain, may be calculated as:
∆ Gstrain
•
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
•
We can now substitute this expression into
Di/
M
D
•
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
•
where
•
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 bot 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
Crystal Field Theory
•
•
•
Because transition metal outer
electron orbitals are directional,
partitioning depends on site
geometry. Furthermore, they are
not completely filled, so there is a
choice as to which orbit an
electron will go.
There is a trade of in energy costs
between placing 2 electrons in 1
orbit and placing them in
geometrically unfavorable orbits
for a given lattice site geometry.
It can be quantified as a site
preference energy, the
difference between the Crystal
Field Stabilization Energy (CFSE)
of octahedral and tetrahedral
sites.
Crystal Field Theory
Transition metal partitioning depends on the distribution of octahedral and
tetrahedral sites in the mineral and the liquid (and well as charge and size) Thus
partitioning is more complex.