Thermodynamics

ThermodynamicsBegin 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 - TSWhere:

G = Gibbs Free Energy

H = Enthalpy (heat content)

T = Temperature in Kelvins

S = Entropy (can think of as randomness)

Thermodynamics

G for a reaction of the type:

2 A + 3 B = C + 4 D

G = (n G)products - (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:

dG = VdP - SdT (ignoring X 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

zzG G VdP SdTT P T PT

T

P

P

2 1 11

2

1

2

2

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:

dG = VdP - SdT (again ignoring X)

G for any reaction = 0 at equilibrium

Worked Problem #2 used:

dG = VdP - SdT and G, S, V values for albite, jadeite and quartz to calculate the conditions for which G of the reaction:

Ab + Jd = Q is equal to 0

from G values for each phase at 298K and 0.1 MPa calculate G298, 0.1 for the reaction, do the same for V and S

G at equilibrium = 0, so we can calculate an isobaric change in T that would be required to bring G298, 0.1 to 0

0 - G298, 0.1 = -S (Teq - 298) (at constant P) Similarly we could calculate an isothermal change

0 - G298, 0.1 = -V (Peq - 0.1) (at constant T)

Mineral S(J) G (J) V

(cm3/mol)

Low Albite 207.25 -3,710,085 100.07 Jadeite 133.53 -2,844,157 60.04 Quartz 41.36 -856,648 22.688From Helgeson et al. (1978).

Table 27-1. Thermodynamic Data at 298K and0.1 MPa from the SUPCRT Database

Method:

NaAlSi3O8 = NaAlSi2O6 + SiO2

P - T phase diagram of the equilibrium curveHow 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

dG = 0 = VdP - SdT

ThusdP

dT

S

V

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.

Return to dG = VdP - SdT, for an isothermal process:

G G VdPP PP

P

2 11

2

z

Gas PhasesGas Phases

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.


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:

G G VdPP PP

P

2 11

2

z

G GRT

PdPP P

P

P

2 11

2

zzG G RT

PdPP P

P

P

2 11

2

1


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

1

xdx xz ln

o


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 = P 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

Dehydration Reactions

• Mu + Q = Kspar + Sillimanite + H2O

• We can treat the solids and gases separately

GP, T - GT = Vsolids (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)

dPdT

SV

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.

Solutions: T-X relationshipsAb = 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 activityIdeal solution: ai = Xi n = # of sites in the phase on

which solution takes place

Non-ideal: ai = i Xi

where i is the activity coefficient

n

n

Solutions: T-X relationshipsExample: 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 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:

GG GG RTRT KKPP TT PP TToo

,, ,, lnln

KK cccc

DDdd

AAaa

BBbb

aa aa

aa aa

Compositional variationsEffect of adding Ca to albite = jadeite + quartz

plagioclase = Al-rich Cpx + Q

GT, P = GoT, P + RTlnK

Let’s say GoT, P was the value that we calculated for

equilibrium in the pure Na-system (= 0 at some P and T)

GoT, P = G298, 0.1 + V (P - 0.1) - S (T-298) = 0

By adding Ca we will shift the equilibrium by RTlnK

We could assume ideal solution and

K JdPyx

SiOQ

AbPlag

X X

X2 All coefficients = 1

Compositional variationsSo now we have:

GT, P = GoT, P + RTln since Q is pure

GoT, Pas calculated for the pure system at P and T

GT, P is the shifted G due to the Ca added (no longer 0)

Thus we could calculate a V(P - Peq) that would bring GT, P back to 0, solving for the new Peq

X

XJdPyx

AbPlag

Compositional variations

Effect of adding Ca to albite = jadeite + quartz

GP, T = GoP, 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 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 reaction for pure end-member minerals (Figure 27.1). Data from SUPCRT (Helgeson et alet al., 1978). ., 1978). Winter (2010) An Introduction to Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall. 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

GeothermobarometryThe Garnet - Biotite geothermometer

T oC Initial X(Fe-Bt)

Final X(Fe-Bt)

Final X(Fe-Grt)

Final (Mg/Fe)Grt

Final (Mg/Fe)Bt

K T Kelvins

1/T Kelvins

lnK

799 1.00 0.750 0.905 0.105 0.333 0.315 1072 0.00093 -1.155799 0.50 0.710 0.896 0.116 0.408 0.284 1072 0.00093 -1.258749 0.50 0.695 0.896 0.116 0.439 0.264 1022 0.00098 -1.330738 1.00 0.730 0.906 0.104 0.370 0.281 1011 0.00099 -1.271698 0.75 0.704 0.901 0.110 0.420 0.261 971 0.00103 -1.342698 0.50 0.690 0.896 0.116 0.449 0.258 971 0.00103 -1.353651 0.75 0.679 0.901 0.110 0.473 0.232 924 0.00108 -1.459651 0.50 0.661 0.897 0.115 0.513 0.224 924 0.00108 -1.497599 0.75 0.645 0.902 0.109 0.550 0.197 872 0.00115 -1.623599 0.50 0.610 0.898 0.114 0.639 0.178 872 0.00115 -1.728550 0.75 0.620 0.903 0.107 0.613 0.175 823 0.00122 -1.741550 0.50 0.590 0.898 0.114 0.695 0.163 823 0.00122 -1.811

601 0.50 0.500 0.800 0.250 1.000 0.250 874 0.00114 -1.386601 0.25 0.392 0.797 0.255 1.551 0.164 874 0.00114 -1.807697 0.75 0.574 0.804 0.244 0.742 0.329 970 0.00103 -1.111697 0.25 0.468 0.796 0.257 1.137 0.226 970 0.00103 -1.487

Table 27-2. Experimental results of Ferry and Spear (1978) on a Garnet-Biotite Geothermometer

y = -2108x + 0.7811

-1.900

-1.800

-1.700

-1.600

-1.500

-1.400

-1.300

-1.200

-1.100

-1.000

0.00090 0.00095 0.00100 0.00105 0.00110 0.00115 0.00120 0.00125

1/T (K)

lnK


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.

lnKD = -2108 · T(K) + 0.781

GP,T = 0 = H 0.1, 298 - TS0.1, 298 + PV + 3 RTlnKD

D

H P V 1 SK

3R T 3Rln

o

D

52 090 2 494P MPaT C 273

19 506 12 943 K

, .

. . ln


Figure 27.6. AFM projections showing the relative distribution of Fe and Mg in garnet vs. biotite at approximately 500 oC (a) and 800oC (b). From Spear (1993) Metamorphic Phase Equilibria and Pressure-Temperature-Time Paths. Mineral. Soc. Amer. Monograph 1.


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.

GeothermobarometryThe 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

GeothermobarometryThe 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

Wt. % Oxides Garnet Biotite Muscovite Plagioclase

SiO2 37.26 34.22 44.50 64.93

Al2O3 21.03 18.97 34.50 22.59

TiO2 1.23 0.40

FeO 32.45 17.50 0.70 MgO 2.46 9.98 0.46 MnO 6.08 0.12 0.02 CaO 1.03 0.01 0.03 2.90

Na2O 0.27 1.64 9.36

K2O 7.79 8.05 0.45

Total 100.31 90.09 90.30 100.23

Si 3.00 5.43 6.17 2.84

AlIV 2.00 2.57 1.83 1.17

AlVI 0.98 3.81 Ti 0.15 0.04 Fe 2.19 2.32 0.08 Mg 0.30 2.36 0.10 Mn 0.42 0.02 0.00 Ca 0.09 0.00 0.14 Na 0.08 0.44 0.83 K 1.58 1.42 0.03

Fe/(Fe+Mg) 0.88 0.50 0.46Prp 10 An 14Alm 73 Ab 83Sps 14 Or 3Grs 3

From Hodges and Spear (1982) and Spear (1993).

Table 27-3. Mineral Compositions, Formulas, and End-Members for Sample 90A from Mt. Moosilauke, New

Hampshire

Cations

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 Al 2SiO5 invariant point

also lies within the shaded area.

Geothermobarometry

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

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 27.12. 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).

GeothermobarometryTHERMOCALC (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 27.13. 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).

GeothermobarometryThermobarometry 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 27.14. 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

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).

GeothermobarometryP-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 27.17. 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.

GeothermobarometryPrecision 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.

GeothermobarometryPrecision and Accuracy

Thermodynamics

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