Geoscience Reference
In-Depth Information
of each ratio is the same (or in a constant ratio). This is why the conservation relations in
ternary petrological diagrams, whose coordinates are normalized to the sum of the three
variables depicted (as in the famous AFM plots), are linear. By contrast, plots involving
fractional parameters, such as the magnesium number mg#
=
Mg/(Fe
+
Mg), which is
widely used in petrology, must be handled with care.
2.2 Elemental fractionation
It is very useful to introduce partition coefficients when studying the substitution of minor
elements or trace elements in the lattice of minerals in equilibrium with magmatic fluids or
natural solutions from which they precipitate. When an element
i
is in solution in two co-
existing phases
j
and
J
(e.g.
j
stands for seawater and
J
for a carbonate that precipitates
out), the Nernst law can be written:
x
J
x
j
k
0
exp
−
G
0
RT
K
J
/
j
(
=
T
,
P
,
x
)
=
(2.10)
where
x
i
j
is the molar proportion of element
i
in phase
j
,
R
is the gas-law constant, and
G
0
a measure of the energy of exchange of this element between the two phases
j
and
J
. The partition (or distribution) coefficient
K
J
/
j
depends on the temperature
T
,pres-
sure
P
, and composition of the phases. The pre-exponential factor
k
0
is a measure of the
non-ideality of the solutions. After a simple adjustment that takes molecular weights into
account, the concentration ratios between phases may also be described by partition coef-
ficients. The ratios of trace element concentrations (typically less concentrated than 1000
parts per million) in two phases under comparable conditions of temperature, pressure,
and aggregate composition of the system (acid, basic, aqueous) are usually constant and
referred to as partition coefficients. It is a common usage in geochemistry to restrict the
term partition coefficients to mineral/liquid coefficients, but this choice is arbitrary and
occasionally misleading. A liquid/mineral partition coefficient is a perfectly valid concept.
Figure 2.5
gives a few examples of mineral/liquid partition coefficients in magmas.
The term “reasonably” justifies the activity of many experimental and analytical labora-
tories. To a fairly good degree of approximation, the dependence of
K
on temperature and
pressure can be described by:
H
RT
2
d
T
+
V
RT
d
P
dln
K
=
(2.11)
(
Appendix C
) where
V
measure differences in the enthalpy and molar volume of
element
i
between the two phases. The dependence of
K
on composition is more complex:
a satisfactory model for solids is Onuma's elastic model, in which energies are elastic ener-
gies required by the substituting element to replace that element which normally makes up
the mineral. In order to insert an atom of ionic radius
r
0
into a crystallographic site, which
we consider as a spherical cavity of radius
r
, work must be done against the electrostatic
forces
F
. As a first approximation, these forces can be taken as proportional to the con-
traction or expansion of the cavity radius from
r
to
r
0
, i.e.
F
H
and
≈−
k
(
r
0
−
r
)
(Hooke's law).
