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Thermodynamics

Begin with a brief review of Chapter 5

Natural systems tend toward states of minimum energy

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Energy States

Unstable:falling or rolling

Stable:at rest in lowestenergy state

Metastable:in low-energyperch

Figure 5.1.Stability states. Winter (2010) An Introduction to Igneous

and Metamorphic Petrology. Prentice Hall.

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Gibbs Free Energy

Gibbs free energy is a measure of chemicalenergy

Gibbs free energy for aphase:

G = H - TS

Where:

G = Gibbs Free Energy

H = Enthalpy (heat content)

T = Temperature in Kelvins

S = Entropy (can think of as randomness)

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Thermodynamics

DG for a reactionof 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

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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

zzG G VdP SdTT P T P

T

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)

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Now consider a reaction, we can then use the equation:

dDG = DVdP - DSdT (again ignoring DX)

G for any reaction = 0 at equilibrium

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Worked Problem #2 used:

dDG = DVdP - DSdT

and G, S, V values for albite, jadeite and quartz tocalculate the conditions for which DG of the reaction:

Ab + Jd = Q

is equal to 0

from G values for each phase at 298K and 0.1 MPa calculate DG298, 0.1for 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, 0.1to 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)

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 22.688

From Helgeson et al. (1978).

Table 27-1.Thermodynamic Data at 298K and

0.1 MPa from the SUPCRT Database

Method:

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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 toIgneous and Metamorphic

Petrology. Prentice Hall.

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pick any two points on the equilibrium curve

dDG = 0 = DVdP - DSdT

ThusdP

dT

S

V=

D

D

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 toIgneous and Metamorphic

Petrology. Prentice Hall.

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

G G VdPP PP

P

2 11

2

- =z

Gas 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 increasesHow can we define the relationship between V and P for a gas?

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Gas Pressure-Volume Relationships

Ideal Gas

As P increases V decreases

PV=nRTIdeal Gas Law P = pressure

V = volume

T = temperature

n = # of moles of gas R = gas constant

= 8.3144 J mol-1K-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.

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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

- =z

zG G RT P dPP P PP

2 11

2

- = 1

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And since

GP2- GP1= RT ln P2- ln P1= RTln(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= RTln(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 x=

zln

o

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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

wheref = gP gis 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

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Dehydration Reactions

Mu + Q = Kspar + Sillimanite + H2O

We can treat the solids and gases separately

GP, T- GT= DVsolids(P- 0.1) + RTln(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

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Dehydration Reactions

(qualitative analysis)

dP

dT

S

V=

D

D

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.

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Solutions: T-X relationships

Ab = Jd + Q was calculated forpurephases

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 placeNon-ideal: ai= giXi

where giis theactivity coefficient

n

n

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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.

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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:

D DG G RT KP T P T

o

, ,= - ln

K cc

D

d

A

a

B

b=

a a

a a

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Compositional variations

Effect of adding Ca to albite = jadeite + quartz

plagioclase = Al-rich Cpx + Q

DGT, P= DGo

T, P+ RTlnK

Lets say DGo

T, Pwas the value that we calculated forequilibrium 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 JdPyx

SiO

Q

Ab

Plag=X X

X

2 All coefficients = 1

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So now we have:

DGT, P= DGo

T, P+ RTln since Q is pure

DGo

T, P= 0 as calculated for the pure system at P and TDGT, Pis the shifted DG due to the Ca added (no longer 0)

Thus we could calculate a DV(P - Peq

) that would bring

DGT, Pback to 0, solving for the new Peq

X

X

JdPyx

Ab

Plag

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Effect of adding Ca to albite = jadeite + quartz

DGP, T= DGo

P, T+ RTlnKnumbers 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.

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Geothermobarometry

Use measured distribution of elements in coexistingphases from experiments at known P and T to estimate P

and T of equilibrium in natural samples

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Geothermobarometry

The Garnet - Biotite geothermometer

ToC 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 -1.155

799 0.50 0.710 0.896 0.116 0.408 0.284 1072 0.00093 -1.258

749 0.50 0.695 0.896 0.116 0.439 0.264 1022 0.00098 -1.330

738 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 -1.342

698 0.50 0.690 0.896 0.116 0.449 0.258 971 0.00103 -1.353

651 0.75 0.679 0.901 0.110 0.473 0.232 924 0.00108 -1.459

651 0.50 0.661 0.897 0.115 0.513 0.224 924 0.00108 -1.497

599 0.75 0.645 0.902 0.109 0.550 0.197 872 0.00115 -1.623

599 0.50 0.610 0.898 0.114 0.639 0.178 872 0.00115 -1.728

550 0.75 0.620 0.903 0.107 0.613 0.175 823 0.00122 -1.741

550 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 -1.386

601 0.25 0.392 0.797 0.255 1.551 0.164 874 0.00114 -1.807

697 0.75 0.574 0.804 0.244 0.742 0.329 970 0.00103 -1.111

697 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

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Geothermobarometry


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

DGP,T= 0 = DH 0.1, 298- TDS0.1, 298+ PDV + 3 RTlnKD

D

H P V 1 SK

3R T 3R ln

-D - D D =

o

D

52 090 2 494P MPaT C 273

19 506 12 943 K

, .

. . ln

= -

-

h b

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Geothermobarometry


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.

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Geothermobarometry


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 Al2SiO5phase diagram is added. From Spear (1993)Metamorphic Phase Equilibria and Pressure-Temperature-

Time Paths. Mineral. Soc. Amer. Monograph 1.

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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

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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 End-

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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 0.46

Prp 10 An 14

Alm 73 Ab 83

Sps 14 Or 3

Grs 3From 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

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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 Al2SiO5invariant point

also lies within the shaded area.

Geothermobarometry

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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, Al2SiO5and 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 mineralcomposition 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 shouldproduce a tighter bracket ofP-T-Xconditions.

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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

thermobarometryusing 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 and75oC, respectively.

A third independent reaction involving the phases present was found (Figure 27.12b).

Notice how the uncertainty increaseswhen the third reaction is included, due to the

effect of the larger uncertainty for this reaction on the correlatedoverall uncertainty.

The average P-T value is higher due to the third reaction, and maybe consideredmore reliable when based on all three.

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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 Thni (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 maybest be practiced

using thepseudosectionapproach of

THERMOCALC (or Perple_X), in which aparticular 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 (andconsiderably 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 Thni(2001) and can further constrain the

equilibrium P and T.

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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

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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

Al2SiO5polymorph 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 placingthe 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 dcollement 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 andthermal models of orogeny.

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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

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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)

F S (1993) M t hi Ph E ilib i d P T t Ti P th Mi l S A M h 1

Geothermobarometry

Precision and Accuracy

Documents

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