Lecture 3
A. Partition Coefficient (read “Terminology for trace-element partitioning”,
Beattie et al., Geochim. Cosmochim. Acta, 57, 1603-1606, 1993)). The partition coefficient is the concentration ratio of an element (i) between two phases ( α and
β ) is
D i
α / β
= C /C i
Within the region of Henry’s Law behavior, the partition coefficient is independent of the concentration of “i” but it is dependent on temperature, pressure and phase composition.
Nomenclature: for applications to igneous rocks where partitioning of elements between solid phases (s) and melt ( ) are important, most papers follow the convention of
D i s/
= C /C i
( D i s /
) the partition coefficient is also known as the distribution coefficient).
Note that D is commonly used for partition coefficient but if kinetic considerations are combined with studies of trace element partitioning, D has precedence for designating Diffusion Coefficient. Also K designates equilibrium constant, and K
D
or K
C
are used to designate compound partition coefficients. A common example is =
(Fe/Mg)olivine
(Fe/Mg)melt
; that is, K
D
or K
C
are ratios of two partition coefficients.
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B. Relationship between partition coefficient and equilibrium constant
Consider partitioning of Ni between forsterite solid (Mg
2
SiO
4
) and a melt of the same composition:
K = a s
Ni
/a
Ni
= x s
Ni
γ s
Ni x
Ni
γ
Ni
Therefore D = x s
Ni
/x
Ni
= K eq
( γ
Ni
/ γ s
Ni
) i.e., the partition coefficient is related to the equilibrium constant by a ratio of activity coefficients ( γ ).
C. Formation Reaction
The partitioning of Ni between Mg
2
SiO
4
solid and melt can also be described by a formation reaction (see Wood and Fraser, Chapter 6).
2NiO
( l )
+ SiO
2 (l)
= Ni
2
SiO
4 (s)
(a a ol
Ni
2
SiO
4(s)
NiO
)
2
(a
SiO
2
)
=
( γ ol
Ni
( γ
2
SiO
NiO x
4
)(x
NiO
) ol
Ni
2
SiO
2 a
SiO
4
2
)
Since D = X ol
Ni
2
SiO
4
/X
NiO
it is clear that D is sensitive to a
SiO
2
.
(We will see later that mineral/melt D’s vary with the SiO
2
content of melts).
D.
Exchange Reaction
The same partitioning process can be described by
2MgO
( l )
+ Ni
2
SiO
4(s)
= 2NiO
(l)
+ Mg
2
SiO
4(s)
(a
NiO
2 s
Mg
K eq
=
(a
MgO
2 s
Ni
2
SiO
2
SiO
4
4
)
) and
2

( γ
K eq
=
( γ
Ni
2
SiO
4
/ γ )
2
/ γ
Mg
2
SiO
4
)
(x
NiO
/x
MgO
(x s
Ni
2
SiO
4
/ x s
Mg
2
)
2
SiO
4
)
This is a compound partition coefficient involving the ratio of two partition coefficients (Ni and Mg), i.e. K
D
= (X
Mg
2
SiO
4
/X SiO )
2
4 s /(X
MgO
/X
NiO
) l so that
K eq
= K
D
times a ratio of activity coefficients. If Ni and Mg are non-ideal to similar extents, it might be expected that the ratio of activity coefficients for NiO/MgO is near unity. As a result K
D
may be more nearly constant than D. Also note that the a
SiO
2 term has cancelled out.
E.
Geothermometer/Geobarometer
Since equilibrium constants vary with pressure and temperature, partition coefficients are also a function of pressure and temperature; hence there is the potential of using variations in TE partition coefficients to infer P and T.
Δ G ˚ = Δ H ˚ -T Δ S ˚ = Δ U ˚ +P Δ V ˚ -T Δ S ˚ =-RTln K eq
So
Ln K eq
=
−Δ H
RT
+
Δ S
R
(
∂ ln Keq
∂ T
)P =
Δ H
RT
2
or
(
∂ ln Keq
∂ 1/T
)P =
−Δ H
R
so a plot of ln K eq
(lnD) versus 1/T yields a straight line. Similarly a geobarometer for pressure is
(
δ lnK
∂ P
)T =
−Δ V o
RT
3

(relatively little is known about D variations versus pressure).
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12.479 Trace-Element Geochemistry
Spring 2013
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