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 lowenergy perch.
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Natural systems tend toward states of minimum energy
Figure 51. Stability states. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
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)
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
z
P
T
2
2

=

G
G
VdP
SdT
T
P
T
P
2
2
1
1
P
T
1
1
ThermodynamicsFor 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)
Now consider a reaction, we can then use the equation:
dDG = DVdP  DSdT (again ignoring DX)
DG for any reaction = 0 at equilibrium
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:
0  DG298, 0.1 = DS (Teq  298) (at constant P)
0  DG298, 0.1 = DV (Peq  0.1) (at constant T)
P  T phase diagram of the equilibrium curve
How do you know which side has which phases?
Figure 271. Temperaturepressure phase diagram for the reaction: Albite = Jadeite + Quartz calculated using the program TWQ of Berman (1988, 1990, 1991). Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
dP
S
=
Thus
D
dT
V
pick any two points on the equilibrium curve
dDG = 0 = DVdP  DSdT
Figure 271. Temperaturepressure phase diagram for the reaction: Albite = Jadeite + Quartz calculated using the program TWQ of Berman (1988, 1990, 1991). Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
P
2

=
G
G
VdP
P
P
2
1
P
1
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?
Ideal Gas
= 8.3144 J mol1 K1
Figure 55. Pistonandcylinder apparatus to compress a gas. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
P x V is a constant at constant T
P
2

=
G
G
VdP
P
P
2
1
P
1
z
RT
P
2

=
G
G
dP
P
P
P
2
1
P
1
1
P
2

=
G
G
RT
dP
P
P
P
2
1
P
1
Gas PressureVolume RelationshipsSince
we can substitute RT/P vor V (for a single mole of gas), thus:
and, since R and T are certainly independent of P:
z
1
=
dx
ln
x
x
Gas PressureVolume RelationshipsAnd 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
The form of this equuation is very useful
GP, T  GT = RT ln (P/Po)
For a nonideal gas (more geologically appropriate) the same form is used, but we substitute fugacity ( f ) for P
wheref = gPg 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
GP, T  GT = DVsolids (P  0.1) + RT ln (P/0.1) (isothermal)
dP
S
=
D
dT
V
Dehydration Reactions(qualitative analysis)Figure 272. Pressuretemperature phase diagram for the reaction muscovite + quartz = Al2SiO5 + Kfeldspar + H2O, calculated using SUPCRT (Helgeson et al., 1978). Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
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
Nonideal: ai = gi Xi
where gi is the activity coefficient
n
n
Example: orthopyroxenes (Fe, Mg)SiO3
Figure 273. Activitycomposition relationships for the enstatiteferrosilite 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 RockForming Crystalline Solutions. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
=
D

o
c
d
G
G
RT
K
a
a
ln
=
c
D
K
P
,
T
P
,
T
a
b
a
a
A
B
Solutions: TX relationshipsBack 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:
Q
X
X
=
Jd
SiO
2
K
Plag
X
Ab
Compositional variationsEffect of adding Ca to albite = jadeite + quartz
plagioclase = Alrich Cpx + Q
DGT, P = DGoT, P + RTlnK
Let’s say DGoT, Pwas the value that we calculated for equilibrium in the pure Nasystem (= 0 at some P and T)
DGoT, P = DG298, 0.1 + DV (P  0.1)  DS (T298) = 0
By adding Ca we will shift the equilibrium by RTlnK
We could assume ideal solution and
All coefficients = 1
X
Jd
Plag
X
Ab
Compositional variationsSo now we have:
DGT, P = DGoT, P + RTlnsince Q is pure
DGoT, P = 0 as calculated for the pure system at P and T
DGT, P is the shifed DG due to the Ca added (no longer 0)
Thus we could calculate a DV(PPeq) that would bring DGT, P back to 0, solving for the new Peq
Effect of adding Ca to albite = jadeite + quartz
DGP, T = DGoP, T + RTlnK
numbers are values for K
Figure 274. PT 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 endmember minerals (Figure 271). Data from SUPCRT (Helgeson et al., 1978). Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
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
The Garnet  Biotite geothermometer
The Garnet  Biotite geothermometer
lnKD = 2108 · T(K) + 0.781
DGP,T = 0 = DH 0.1, 298  TDS0.1, 298 + PDV + 3 RTlnKD
Figure 275. Graph of lnK vs. 1/T (in Kelvins) for the Ferry and Spear (1978) garnetbiotite exchange equilibrium at 0.2 GPa from Table 272. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
The Garnet  Biotite geothermometer
Figure 276. 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 PressureTemperatureTime Paths. Mineral. Soc. Amer. Monograph 1.
The Garnet  Biotite geothermometer
Figure 277. Pressuretemperature diagram similar to Figure 274 showing lines of constant KD plotted using equation (2735) for the garnetbiotite exchange reaction. The Al2SiO5 phase diagram is added. From Spear (1993) Metamorphic Phase Equilibria and PressureTemperatureTime Paths. Mineral. Soc. Amer. Monograph 1.
The GASP geobarometer
Figure 278. PT phase diagram showing the experimental results of Koziol and Newton (1988), and the equilibrium curve for reaction (2737). Open triangles indicate runs in which An grew, closed triangles indicate runs in which Grs + Ky + Qtz grew, and halffilled triangles indicate no significant reaction. The univariant equilibrium curve is a bestfit 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, 216233
The GASP geobarometer
Figure 278. PT 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 PressureTemperatureTime Paths. Mineral. Soc. Amer. Monograph 1.
Figure 2710. PT diagram showing the results of garnetbiotite geothermometry (steep lines) and GASP barometry (shallow lines) for sample 90A of Mt. Moosilauke (Table 274). 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 PT conditions for the sample. The Al2SiO5 invariant point also lies within the shaded area.
Figure 2711. PT 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 garnetbiotite curve of Hodges and Spear (1982) Amer. Mineral., 67, 11181134 has been added.
PTt Paths
Figure 2712. Chemically zoned plagioclase and poikiloblastic garnet from metapelitic sample 3, Wopmay Orogen, Canada. a. Chemical profiles across a garnet (rim rim). b. Ancontent of plagioclase inclusions in garnet and corresponding zonation in neighboring plagioclase. After StOnge (1987) J. Petrol. 28, 122 .
PTt Paths
Figure 2713. The results of applying the garnetbiotite 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 StOnge (1987). The three intersection points yield PT estimates which define a PTt path for the growing minerals showing nearisothermal decompression. After Spear (1993).
Precision and Accuracy
Figure 2714. 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 bullseye. b. The shots are not precise, because of the large scatter, but they are accurate, because the average of the shots is near the bullseye. c. The shots are both precise and accurate. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
Precision and Accuracy
Figure 2715. PT diagram illustrating the calculated uncertainties from various sources in the application of the garnetbiotite geothermometer and the GASP geobarometer to a pelitic schist from southern Chile. After Kohn and Spear (1991b) Amer. Mineral., 74, 7784 and Spear (1993) From Spear (1993) Metamorphic Phase Equilibria and PressureTemperatureTime Paths. Mineral. Soc. Amer. Monograph 1.
Figure 2716. Phase diagram for the reaction: calcite + quartz = wollastonite + CO2 calculated using the program SUPCRT, assuming pCO2 = PLith. Winter (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.