Thermodynamics

1
Thermodynamics

• Begin with a brief review of Chapter 5

Natural systems tend toward states of minimum
energy
2
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.
3
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)
4
Thermodynamics

• DG for a reaction of the type
• 2 A 3 B C 4 D
• 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
5
Thermodynamics
For other temperatures and pressures we can use
the equation dG VdP - SdT (ignoring DX for
now) where V volume and S entropy (both
molar)
We can use this equation to calculate G for any
phase at any T and P by integrating
If V and S are constants, our equation reduces
to GT2 P2 - GT1 P1 V(P2 - P1) - S (T2 - T1)
6
Now consider a reaction, we can then use the
equation dDG DVdP - DSdT (again ignoring DX)
DG for any reaction 0 at equilibrium
7

• 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

• 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)

8
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.
9

• pick any two points on the equilibrium curve
• dDG 0 DVdP - DSdT

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.
10
Gas Phases

• Return to dG VdP - SdT, for an isothermal
  process

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?
11
Gas Pressure-Volume Relationships

• Ideal Gas
• As P increases V decreases
• PVnRT Ideal Gas Law
• P pressure
• V volume
• T temperature
• n of moles of gas
• R gas constant
• 8.3144 J mol-1 K-1

Figure 5.5. Piston-and-cylinder apparatus to
compress a gas. Winter (2010) An Introduction
to Igneous and Metamorphic Petrology. Prentice
Hall.
P x V is a constant at constant T
12
Gas Pressure-Volume Relationships

• Since
• we can substitute RT/P for V (for a single mole
  of gas), thus
• and, since R and T are certainly independent of P

z
13
Gas Pressure-Volume Relationships

• And since
• 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 - 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

o
14
Gas Pressure-Volume Relationships

• The form of this equation is very useful
• 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

o
15
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

16
Dehydration Reactions (qualitative analysis)
Figure 27.2. Pressure-temperature phase diagram
for the reaction muscovite quartz Al2SiO5
K-feldspar H2O, calculated using SUPCRT
(Helgeson et al., 1978). Winter (2010) An
Introduction to Igneous and Metamorphic
Petrology. Prentice Hall.
17
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
• Ideal solution ai Xi n of sites in
  the phase on
• which solution takes place
• Non-ideal ai gi Xi
• where gi is the activity coefficient

n
n
18
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.
19
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
• where

20
Compositional variations

• Effect of adding Ca to albite jadeite quartz
• plagioclase Al-rich Cpx Q
• DGT, P DGoT, P RTlnK
• Lets 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

All coefficients 1
21
Compositional variations

• So now we have
• DGT, P DGoT, P RTln 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

22
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.
23
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

24
Geothermobarometry

• The Garnet - Biotite geothermometer

25
Geothermobarometry

• The Garnet - Biotite geothermometer

lnKD -2108 T(K) 0.781
DGP,T 0 DH 0.1, 298 - TDS0.1, 298 PDV 3
RTlnKD
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.
26
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.
27
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
garnet-biotite exchange reaction. The Al2SiO5
phase diagram is added. From Spear (1993)
Metamorphic Phase Equilibria and
Pressure-Temperature-Time Paths. Mineral. Soc.
Amer. Monograph 1.
28
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
Newtons 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
29
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
30
Geothermobarometry
31
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.
32
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 Bermans 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.
33
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).
34
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).
35
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 .
36
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).
37
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 H2O-saturated minimum melting
conditions. After Foster et al. (2004).
38
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.
39
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.