Thermodynamics
Begin with a brief review of Chapter 5
Natural systems tend toward states of minimum energy
Energy States
• Unstable: falling or rolling

Stable: at rest in lowest
energy state

Metastable: in low-energy
perch
Figure 5.1. Stability states. Winter (2010) An Introduction to Igneous
and Metamorphic Petrology. Prentice Hall.
Gibbs Free Energy
Gibbs free energy is a measure of chemical energy
Gibbs free energy for a phase:
G = H - TS
Where:
G = Gibbs Free Energy
H = Enthalpy (heat content)
T = Temperature in Kelvins
S = Entropy (can think of as randomness)
Thermodynamics
DG for a reaction of the type:
2A + 3B =C +4D
DG = S (n G)products - S(n G)reactants
= GC + 4GD - 2GA - 3GB
The side of the reaction with lower G will be more stable
Thermodynamics
For other temperatures and pressures we can use the equation:
(ignoring DX for now)
where V = volume and S = entropy (both molar)
dG = VdP - SdT
We can use this equation to calculate G for any phase at any T and P
by integrating
GT
2
P2
- GT P =
1
1
z z
P2
P1
VdP -
T2
T1
SdT
If V and S are constants, our equation reduces to:
GT2 P2 - GT1 P1 = V(P2 - P1) - S (T2 - T1)
Now consider a reaction, we can then use the equation:
dDG = DVdP - DSdT
(again ignoring DX)
DG for any reaction = 0 at equilibrium
Worked Problem #2 used:
dDG = DVdP - DSdT
and G, S, V values for albite, jadeite and quartz to
calculate the conditions for which DG of the reaction:
Ab + Jd = Q
is equal to 0
Method:



Table 27-1. Thermodynamic Data at 298K and
0.1 MPa from the SUPCRT Database
Mineral
S(J)
G (J)
V
3
(cm /mol)
Low Albite
Jadeite
Quartz
207.25
133.53
41.36
-3,710,085
-2,844,157
-856,648
100.07
60.04
22.688
From Helgeson et al. (1978).
from G values for each phase at 298K and 0.1 MPa calculate DG298, 0.1 for the
reaction, do the same for DV and DS
DG at equilibrium = 0, so we can calculate an isobaric change in T that would
be required to bring DG298, 0.1 to 0
0 - DG298, 0.1 = -DS (Teq - 298)
(at constant P)
Similarly we could calculate an isothermal change
0 - DG298, 0.1 = -DV (Peq - 0.1)
(at constant T)
NaAlSi3O8 = NaAlSi2O6 + SiO2
P - T phase diagram of the equilibrium curve
How do you know which side has which phases?
Figure 27.1. Temperature-pressure
phase diagram for the reaction:
Albite = Jadeite + Quartz
calculated using the program TWQ
of Berman (1988, 1990, 1991).
Winter (2010) An Introduction to
Igneous and Metamorphic
Petrology. Prentice Hall.
pick any two points on the equilibrium curve
dDG = 0 = DVdP - DSdT
dP DS
=
Thus
dT DV
Figure 27.1. Temperature-pressure
phase diagram for the reaction:
Albite = Jadeite + Quartz
calculated using the program TWQ
of Berman (1988, 1990, 1991).
Winter (2010) An Introduction to
Igneous and Metamorphic
Petrology. Prentice Hall.
Gas Phases
z
Return to dG = VdP - SdT, for an isothermal process:
GP - GP =
2
1
P2
P1
VdP
For solids it was fine to ignore V as f(P)
For gases this assumption is shitty
You can imagine how a gas compresses as P increases
How can we define the relationship between V and P for a gas?
Gas Pressure-Volume Relationships
Ideal Gas
– As P increases V decreases
– PV=nRT Ideal Gas Law
•
•
•
•
•
P = pressure
V = volume
T = temperature
n = # of moles of gas
R = gas constant
= 8.3144 J mol-1 K-1
P x V is a constant at constant T
Figure 5.5. Piston-and-cylinder apparatus to
compress a gas. Winter (2010) An Introduction to
Igneous and Metamorphic Petrology. Prentice Hall.
Gas Pressure-Volume Relationships
Since
GP - GP =
2
1
z
P2
P1
VdP
z
we can substitute RT/P for V (for a single mole of gas), thus:
GP - GP =
2
1
P2
P1
RT
dP
P
z
and, since R and T are certainly independent of P:
G P - G P = RT
2
1
P2
P1
1
P
dP
Gas Pressure-Volume Relationships
And since
z
1
dx = ln x
x
GP2 - GP1 = RT ln P2 - ln P1 = RT ln (P2/P1)
Thus the free energy of a gas phase at a specific P and T, when
referenced to a standard atate of 0.1 MPa becomes:
GP, T -
o
GT =
RT ln (P/Po)
G of a gas at some P and T = G in the reference state (same T and 0.1 MPa)
+ a pressure term
Gas Pressure-Volume Relationships
The form of this equation is very useful
o
GP, T - GT = RT ln (P/Po)
For a non-ideal gas (more geologically appropriate) the same
form is used, but we substitute fugacity ( f ) for P
where f = gP
g is the fugacity coefficient
Tables of fugacity coefficients for common gases are available
At low pressures most gases are ideal, but at high P they are not
Dehydration Reactions
• Mu + Q = Kspar + Sillimanite + H2O
• We can treat the solids and gases separately
GP, T - GT = DVsolids (P - 0.1) + RT ln (P/0.1) (isothermal)
• The treatment is then quite similar to solid-solid reactions, but
you have to solve for the equilibrium P by iteration
Dehydration Reactions
(qualitative analysis)
dP = DS
dT DV
Figure 27.2. Pressure-temperature
phase diagram for the reaction
muscovite + quartz = Al2SiO5 + Kfeldspar + H2O, calculated using
SUPCRT (Helgeson et al., 1978).
Winter (2010) An Introduction to
Igneous and Metamorphic Petrology.
Prentice Hall.
Solutions: T-X relationships
Ab = Jd + Q was calculated for pure phases
When solid solution results in impure phases
the activity of each phase is reduced
Use the same form as for gases (RT ln P or ln f)
Instead of fugacity, we use activity
n
Ideal solution: ai = Xi
n = # of sites in the phase on
which solution takes place
Non-ideal: ai = gi Xi
n
where gi is the activity coefficient
Solutions: T-X relationships
Example: orthopyroxenes (Fe, Mg)SiO3
– Real vs. Ideal Solution Models
Figure 27.3. Activity-composition relationships for the enstatite-ferrosilite mixture in orthopyroxene at 600oC and 800oC. Circles are data
from Saxena and Ghose (1971); curves are model for sites as simple mixtures (from Saxena, 1973) Thermodynamics of Rock-Forming
Crystalline Solutions. Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
Solutions: T-X relationships
Back to our reaction:
Simplify for now by ignoring dP and dT
For a reaction such as:
aA + bB = cC + dD
At a constant P and T:
D G P , T = D G oP , T - RT ln K
where:
K=
c
d
a
b
ac aD
aA aB
Compositional variations
Effect of adding Ca to albite = jadeite + quartz
plagioclase = Al-rich Cpx + Q
DGT, P = DGoT, P + RTlnK
Let’s say DGoT, P was the value that we calculated for
equilibrium in the pure Na-system (= 0 at some P and T)
DGoT, P = DG298, 0.1 + DV (P - 0.1) - DS (T-298) = 0
By adding Ca we will shift the equilibrium by RTlnK
We could assume ideal solution and
K=
X
Pyx
Jd
X
Q
SiO 2
Plag
Ab
X
All coefficients = 1
Compositional variations
So now we have:
Pyx
X
Jd
o
DGT, P = DG T, P + RTln Plag
X Ab
since Q is pure
DGoT, P = 0 as calculated for the pure system at P and T
DGT, P is the shifted DG due to the Ca added (no longer 0)
Thus we could calculate a DV(P - Peq) that would bring
DGT, P back to 0, solving for the new Peq
Compositional variations
Effect of adding Ca to albite = jadeite + quartz
DGP, T = DGoP, T + RTlnK
numbers are values for K
Figure 27.4. P-T phase diagram for the reaction Jadeite + Quartz = Albite for various values of K. The equilibrium curve for K = 1.0 is
the reaction for pure end-member minerals (Figure 27.1). Data from SUPCRT (Helgeson et al., 1978). Winter (2010) An Introduction to
Igneous and Metamorphic Petrology. Prentice Hall.
Geothermobarometry
Use measured distribution of elements in coexisting
phases from experiments at known P and T to estimate P
and T of equilibrium in natural samples
Geothermobarometry
The Garnet - Biotite geothermometer
Table 27-2. Experimental results of Ferry and Spear (1978) on a Garnet-Biotite Geothermometer
T oC
Initial
Final
Final
Final
Final
X(Fe-Bt) X(Fe-Bt) X(Fe-Grt) (Mg/Fe)Grt (Mg/Fe)Bt
K
T
Kelvins
1/T
Kelvins
lnK
799
799
749
738
698
698
651
651
599
599
550
550
1.00
0.50
0.50
1.00
0.75
0.50
0.75
0.50
0.75
0.50
0.75
0.50
0.750
0.710
0.695
0.730
0.704
0.690
0.679
0.661
0.645
0.610
0.620
0.590
0.905
0.896
0.896
0.906
0.901
0.896
0.901
0.897
0.902
0.898
0.903
0.898
0.105
0.116
0.116
0.104
0.110
0.116
0.110
0.115
0.109
0.114
0.107
0.114
0.333
0.408
0.439
0.370
0.420
0.449
0.473
0.513
0.550
0.639
0.613
0.695
0.315
0.284
0.264
0.281
0.261
0.258
0.232
0.224
0.197
0.178
0.175
0.163
1072
1072
1022
1011
971
971
924
924
872
872
823
823
0.00093
0.00093
0.00098
0.00099
0.00103
0.00103
0.00108
0.00108
0.00115
0.00115
0.00122
0.00122
-1.155
-1.258
-1.330
-1.271
-1.342
-1.353
-1.459
-1.497
-1.623
-1.728
-1.741
-1.811
601
601
697
697
0.50
0.25
0.75
0.25
0.500
0.392
0.574
0.468
0.800
0.797
0.804
0.796
0.250
0.255
0.244
0.257
1.000
1.551
0.742
1.137
0.250
0.164
0.329
0.226
874
874
970
970
0.00114
0.00114
0.00103
0.00103
-1.386
-1.807
-1.111
-1.487
-1.900
Geothermobarometry
The Garnet - Biotite geothermometer
lnKD = -2108 · T(K) + 0.781
DGP,T = 0 = DH 0.1, 298 - TDS0.1, 298 + PDV + 3 RTlnKD
-DH - PDV  1  DS
ln K D =
 
3R
 T  3R
T
o
52 ,090  2.494P  MPa 
C =
- 273
19.506 - 12.943 ln K D
Figure 27.5. Graph of lnK vs. 1/T (in Kelvins) for the Ferry and Spear (1978) garnet-biotite exchange equilibrium at 0.2 GPa from Table
27.2. Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
Geothermobarometry
The Garnet - Biotite geothermometer
Figure 27.6. AFM projections showing the relative distribution of Fe and Mg in garnet vs. biotite at approximately 500oC (a) and 800oC (b).
From Spear (1993) Metamorphic Phase Equilibria and Pressure-Temperature-Time Paths. Mineral. Soc. Amer. Monograph 1.
Geothermobarometry
The Garnet - Biotite geothermometer
Figure 27.7. Pressure-temperature diagram similar to Figure 27.4 showing lines of constant KD plotted using equation (27.35) for the garnetbiotite exchange reaction. The Al2SiO5 phase diagram is added. From Spear (1993) Metamorphic Phase Equilibria and Pressure-TemperatureTime Paths. Mineral. Soc. Amer. Monograph 1.
Geothermobarometry
The GASP geobarometer
Figure 27.8. P-T phase diagram showing the
experimental results of Koziol and Newton (1988),
and the equilibrium curve for reaction (27.37).
Open triangles indicate runs in which An grew,
closed triangles indicate runs in which Grs + Ky +
Qtz grew, and half-filled triangles indicate no
significant reaction. The univariant equilibrium
curve is a best-fit regression of the data brackets.
The line at 650oC is Koziol and Newton’s estimate
of the reaction location based on reactions
involving zoisite. The shaded area is the
uncertainty envelope. After Koziol and Newton
(1988) Amer. Mineral., 73, 216-233
Geothermobarometry
The GASP geobarometer
Figure 27.98. P-T diagram contoured for equilibrium curves of various values of K for the GASP geobarometer reaction: 3 An = Grs + 2 Ky +
Qtz. From Spear (1993) Metamorphic Phase Equilibria and Pressure-Temperature-Time Paths. Mineral. Soc. Amer. Monograph
Table 27-3. Mineral Compositions, Formulas, and EndMembers for Sample 90A from Mt. Moosilauke, New
Hampshire
Geothermobarometry
Wt. % Oxides
SiO2
Al2O3
TiO2
FeO
MgO
MnO
CaO
Na2O
K2O
Total
Si
IV
Al
AlVI
Ti
Fe
Mg
Mn
Ca
Na
K
Fe/(Fe+Mg)
Garnet
37.26
Biotite
34.22
21.03
18.97
34.50
22.59
32.45
2.46
6.08
1.03
1.23
17.50
9.98
0.12
0.01
0.27
0.40
0.70
0.46
0.02
0.03
1.64
2.90
9.36
7.79
90.09
8.05
90.30
0.45
100.23
3.00
Cations
5.43
6.17
2.84
2.00
2.57
1.83
1.17
0.98
0.15
2.32
2.36
0.02
0.08
1.58
3.81
0.04
0.08
0.10
0.00
0.00
0.44
1.42
0.14
0.83
0.03
0.50
0.46
100.31
2.19
0.30
0.42
0.09
0.88
Prp 10
Alm 73
Sps 14
Grs 3
From Hodges and Spear (1982) and Spear (1993).
Muscovite Plagioclase
44.50
64.93
An 14
Ab 83
Or 3
Geothermobarometry
Figure 27.10. P-T diagram showing the results of garnet-biotite geothermometry (steep lines) and GASP barometry (shallow lines) for sample
90A of Mt. Moosilauke (Table 27.4). Each curve represents a different calibration, calculated using the program THERMOBAROMETRY, by
Spear and Kohn (1999). The shaded area represents the bracketed estimate of the P-T conditions for the sample. The Al2SiO5 invariant point
also lies within the shaded area.
Geothermobarometry
TWQ and THERMOCALC accept mineral
composition data and calculate equilibrium
curves based on an internally consistent set of
calibrations and activity-composition mineral
solution models.
Rob Berman’s TWQ 2.32 program calculated
relevant equilibria relating the phases in sample
90A from Mt. Moosilauke.
TWQ also searches for and computes all
possible reactions involving the input phases, a
process called multi-equilibrium calculations
by Berman (1991).
Output from these programs yields a single
equilibrium curve for each reaction and should
produce a tighter bracket of P-T-X conditions.
Figure 27.11. P-T phase diagram calculated by TQW 2.02 (Berman, 1988, 1990, 1991) showing the internally consistent reactions between
garnet, muscovite, biotite, Al2SiO5 and plagioclase, when applied to the mineral compositions for sample 90A, Mt. Moosilauke, NH. The
garnet-biotite curve of Hodges and Spear (1982) Amer. Mineral., 67, 1118-1134 has been added.
Geothermobarometry
THERMOCALC (Holland and Powell) also based on an internally-consistent dataset
and produces similar results, which Powell and Holland (1994) call optimal
thermobarometry using the AvePT module.
THERMOCALC also considers activities of each end-member of the phases to be
variable within the uncertainty of each activity model, defining bands for each
reaction within that uncertainty (shaded blue).
Calculates an optimal P-T point within the correlated uncertainty of all relevant
reactions via least squares and estimates the overall activity model uncertainty.
The P and T uncertainties for the Grt-Bt and GASP equilibria are about  0.1 GPa and
75oC, respectively.
A third independent reaction involving the phases present was found (Figure 27.12b).
Notice how the uncertainty increases when the third reaction is included, due to the
effect of the larger uncertainty for this reaction on the correlated overall uncertainty.
The average P-T value is higher due to the third reaction, and may be considered
more reliable when based on all three.
Figure 27.12. Reactions for the garnet-biotite geothermometer and GASP geobarometer
calculated using THERMOCALC with the mineral compositions from sample PR13 of Powell
(1985). A P-T uncertainty ellipse, and the “optimal” AvePT ( ) calculated from correlated
uncertainties using the approach of Powell and Holland (1994). b. Addition of a third
independent reaction generates three intersections (A, B, and C). The calculated AvePT lies
within the consistent band of overlap of individual reaction uncertainties (yet outside the ABC
triangle).
Geothermobarometry
Thermobarometry may best be practiced
using the pseudosection approach of
THERMOCALC (or Perple_X), in which a
particular whole-rock bulk composition is
defined and the mineral reactions delimit a
certain P-T range of equilibration for the
mineral assemblage present.
The peak metamorphic mineral assemblage:
garnet + muscovite + biotite + sillimanite +
quartz + plagioclase + H2O, is shaded (and
considerably smaller than the uncertainty
ellipse determined by the AvePT approach).
The calculated compositions of garnet,
biotite, and plagioclase within the shaded
area are also contoured (inset). They
compare favorably with the reported
mineral compositions of Habler and Thöni
(2001) and can further constrain the
equilibrium P and T.
Figure 27.13. P-T pseudosection calculated by THERMOCALC for a computed average composition in NCKFMASH for a pelitic
Plattengneiss from the Austrian Eastern Alps. The large + is the calculated average PT (= 650oC and 0.65 GPa) using the mineral data of
Habler and Thöni (2001). Heavy curve through AvePT is the average P calculated from a series of temperatures (Powell and Holland, 1994).
The shaded ellipse is the AvePT error ellipse (R. Powell, personal communication). After Tenczer et al. (2006).
Geothermobarometry
P-T-t Paths
Figure 27.14. Chemically zoned plagioclase and poikiloblastic garnet from meta-pelitic sample 3, Wopmay Orogen, Canada. a. Chemical
profiles across a garnet (rim  rim). b. An-content of plagioclase inclusions in garnet and corresponding zonation in neighboring plagioclase.
After St-Onge (1987) J. Petrol. 28, 1-22 .
Geothermobarometry
P-T-t Paths
Figure 27.15. The results of applying the garnet-biotite geothermometer of Hodges and Spear (1982) and the GASP geobarometer of Koziol
(1988, in Spear 1993) to the core, interior, and rim composition data of St-Onge (1987). The three intersection points yield P-T estimates which
define a P-T-t path for the growing minerals showing near-isothermal decompression. After Spear (1993).
Geothermobarometry
P-T-t Paths
Recent advances in textural geochronology have allowed age
estimates for some points along a P-T-t path, finally placing
the “t” term in “P-T-t” on a similar quantitative basis as P and
T.
Foster et al. (2004) modeled temperature and pressure
evolution of two amphibolite facies metapelites from the
Canadian Cordillera and one from the Pakistan Himalaya.
Three to four stages of monazite growth were recognized
texturally in the samples, and dated on the basis of U-Pb
isotopes in Monazite analyzed by LA-ICPMS.
Used the P-T-t paths to constrain the timing of thrusting
(pressure increase) along the Monashee décollement in
Canada (it ceased about 58 Ma b.p.), followed by
exhumation beginning about 54 Ma.
Himalayan sample records periods of monazite formation
during garnet growth at 82 Ma, followed by later monazite
growth during uplift and garnet breakdown at 56 Ma, and a
melting event during subsequent decompression.
Such data combined with field recognition of structural
features can elucidate the metamorphic and tectonic history
of an area and also place constraints on kinematic and
thermal models of orogeny.
Figure 27.16. Clockwise P-T-t paths for samples D136 and D167 from
the Canadian Cordillera and K98-6 from the Pakistan Himalaya.
Monazite U-Pb ages of black dots are in Ma. Small-dashed lines are
Al2SiO5 polymorph reactions and large-dashed curve is the H2Osaturated minimum melting conditions. After Foster et al. (2004).
Geothermobarometry
Precision and Accuracy
Figure 27.17. An illustration of precision vs. accuracy. a. The shots are precise because successive shots hit near the same place
(reproducibility). Yet they are not accurate, because they do not hit the bulls-eye. b. The shots are not precise, because of the large scatter, but
they are accurate, because the average of the shots is near the bulls-eye. c. The shots are both precise and accurate. Winter (2010) An
Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
Geothermobarometry
Precision and Accuracy
Figure 27.18. P-T diagram illustrating the calculated uncertainties from various sources in the application of the garnet-biotite geothermometer
and the GASP geobarometer to a pelitic schist from southern Chile. After Kohn and Spear (1991b) Amer. Mineral., 74, 77-84 and Spear (1993)
From Spear (1993) Metamorphic Phase Equilibria and Pressure-Temperature-Time Paths. Mineral. Soc. Amer. Monograph 1.