Thermobarometry
Lecture 12
We now have enough thermodynamics to put it
to some real use: calculating the temperatures
and pressures at which mineral assemblages
(i.e., rocks) equilibrated within the Earth.
Some theoretical
considerations
• We have seen that which phase assemblage is stable
and the composition of those phases depends on ∆Gr,
which we use to calculate K
o We also know ∆Gr depends on T and P.
• Reactions that make good geothermometers are those
that depend strongly on T.
-∆ H ∆ S
•
ln K =
+
o
r
RT
R
o What would characterize a good geothermometer?
• Similarly, a good geobarometer would be one strongly
depending on P
o
∆ VT,Pref
æ ¶ ln K ö
çè
÷ø =
¶P T
RT
• A good geothermometer will have large ∆H; a good
geobarometer will have large ∆V.
Univariant Reactions
• Univariant (or invariant)
reactions provide
possible
thermobarometers.
• There are 3 phases in
the Al2Si2O5 system.
o When two coexist, we need
only specify either T or P, the
other is then fixed.
o All three can coexist at just one
T and P.
o First is rare, second is rarer.
Garnet Peridotite
Geobarometry
•
•
•
•
•
•
•
Garnet becomes the high
pressure aluminous phase in the
mantle, replacing spinel.
Aluminum also dissolves in the
orthopyroxene (also
clinopyroxene)
We can write the reaction as:
Mg2Si2O6+MgAl2SiO6 =
Mg3Al2Si3O12
l.h.s. is the opx solid solution - Al
end member does not exist as
pure phase.
Significant volume change
associated with this reaction (but
also depends on T).
Other complexities arise from Ca,
Fe, and Cr in phases.
Original approach of Wood
and Banno generally assumed
ideal solution
Garnet Peridotite
Geobarometry
• Subsequent refinements
used asymmetric solution
model to match
experimental data.
• Recognize two distinct
sites in opx crystal:
o Smaller M1: Al substitutes here
o Larger M2: Ca substitutes here
• P given by
-C2 C22 + C3C1
P=
C3
• where C3 is constant and
other parameters
depend on K, T, and
composition.
Gt 3
(1- XCa
)
K = M1 M 2 M1 M1
X Mg,Fe X Mg,Fe X Mg,Fe X Al
Solvus Equilibria
• Another kind of
thermobarometer is based
on exsolution of two phases
from a homogenous single
phase solution.
• This occurs when the excess
free energy exceeds the
ideal solution term and
inflections develop, as in the
alkali feldspar system.
• Because it is strongly
temperature dependent
and not particularly pressure
dependent, this makes a
good geothermometer.
Temperature in
Peridotites
Ca2+ 
• Temperatures calculated from compositions of
co-existing orthopyroxene (enstatite) and
clinopyroxene (diopside) solid solutions, which
depend on T.
Exchange Reactions
• There are a number of
common minerals where
one or more ions substitutes
for others in a solid solution.
o
The Fe2+–Mg2+ substitution is
common in ferromagnesian
minerals.
• Let’s consider the exchange
of Mg and Fe between
olivine and a melt
containing Mg and Fe.
o
o
This partitioning of these two ions
between melt and olivine depends
on temperature.
We can use a electron microprobe
to measure the composition of
olivine and co-existing melt
(preserved as glass).
Olvine-Melt
Geothermometer
• Reaction of interest can be written as:
MgOol + FeOl = MgOl + FeOol
o (note, this does not involve redox, so we write it in terms of oxides since
these are conventionally reported in analyses. We could write it in terms of
ions, however.)
• Assuming both solid and liquid solutions are ideal,
the equilibrium constant for this reaction is:
Ol
X FeO
X MgO
K=
Ol
X FeO X MgO
• Unfortunately ∆H for the reaction above is small, so
it has weak temperature dependence.
Roeder & Emslie
Geothermometer
• Roeder & Emslie (1970)
decided to consider
two separate reactions:
• MgOliq –> MgOOl and
FeOliq –> FeOOl
• Based on empirical
data, they deduced
the temperature
dependence as:
See Example 4.3 for how to do the
calculation - biggest effort is simply
converting wt. percent to mole fraction.
Ol
X MgO
3740
log K D = log l =
-1.87
X MgO
T
• and
Ol
XFeO
3911
log K D = log l =
- 2.50
XFeO
T
Buddington and Lindsley
Oxide Geothermometer
Recall this diagram from
Chapter 3
• Things get interesting in real systems
containing Ti, because both
magnetite and hematite are solid
solutions.
• Partition of Fe and Ti between the
two depends on T and ƒO2.
Buddington and Lindsley
Oxide Geothermometer
• The reaction of interest is:
yFe2TiO4 + (1-y)Fe3O4 + ¼O2 = yFeTiO3 + (3/2 -y)Fe2O3
magnetite s.s. hematite s.s.
• The equilibrium constant for this reaction is
K=
y
3/2-y
aFeTiO
a
Fe
3
2O3
y
1-y
aFe
a
ƒO2
TiO
2
4 Fe3O4
• The reaction can be thought of as a combination of an
exchange reaction:
Fe3O4 + FeTiO3 = Fe3TiO4 + Fe2O3
magnetite + illmenite = ulvospinel + hematite
• plus the oxidation of magnetite to hematite:
4Fe3O4 + O2 = 6Fe2O3
Computing Temperature
and Oxygen Fugacity
• The calculation is complex
because the system cannot
be treated as ideal (except
titanomagnetite above
800˚C). Equilibrium constant
is: ∆ G é X (1- X ) ù é l l ù
2
-
• and
= ln ê
2
êë (1- XUsp )X Ill
Usp
RT
-
Ill
2
Hem
2
Mt Ill
ú + ln ê
úû
ë l l
Usp
ú
û
12
é X 12 ù
é l Hem
ù
∆G
= ln ê Hem
+
ln
- ln ƒO2
ê
4 ú
4 ú
RT
ë X Mt û
ë lMt û
• Must calculate λ’s using
asymmetric solution model
(using interaction
parameters), then solve for T
and ƒO2. Example 4.4 shows
how.