Metamorphic Thermodynamics

Metamorphic Thermodynamics

(Chapter 27)

last update:07/19/04

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:

2 A + 3 B = C + 4 D

DG = S (n G)_products - S(n G)_reactants

= G_C + 4G_D - 2G_A - 3G_B

The side of the reaction with lower G will be more stable

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:

G_{T2 P2} - G_{T1 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

Gas Pressure-Volume Relationships

The form of this equation is very useful

G_{P, T} - G_T = RT ln (P/P^o)

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

e.g. Mu + Q = Kspar + Sillimanite + H₂O

We can treat the solids and gases separately

G_{P, T} - G_T = DV_solids (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

Pressure-temperature phase diagram for the reaction

muscovite + quartz = Al₂SiO₅ + K-feldspar + H₂O

calculated using SUPCRT (Helgeson et al., 1978).

Winter (2001) 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

Ideal solution: a_i = X_i

n = # of sites in the phase on which solution takes place

Non-ideal: a_i = g_i X_i where g_i is the activity coefficient

Example: orthopyroxenes (Fe, Mg)SiO₃ - Real vs. Ideal Solution Models

Activity-composition relationships for the enstatite-ferrosilite mixture in orthopyroxene at 600^oC and 800^oC. 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 (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

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:

Effect of adding Ca to albite = jadeite + quartz

plagioclase = Al-rich Cpx + Q

DG_{T,
P} = DG^o_{T,
P} + RTlnK

Let’s say DG^o_{T,
P} was the value that we calculated for equilibrium in the pure Na-system (= 0 at some P and T)

DG^o_{T,
P} = DG_{298,
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

So now we have:

DG_{T,
P} = DG^o_{T,
P} + RTln since Q is pure

DG^o_{T,
P} = 0 as calculated for the pure system at P and T

DG_{T,
P} is the shifted DG due to the Ca added (no longer 0)

Thus we could calculate a DV(P-P_eq) that would bring DG_{T,
P} back to 0, solving for the new P_eq

Effect of adding Ca to albite = jadeite + quartz

DG_{P,
T} = DG^o_{P,
T} + RTlnK

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 (2001) 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

The Garnet - Biotite geothermometer

lnK_D = -2108 · T(K) + 0.781

DG_P,T = 0 = DH _{0.1,
298} - TDS_{0.1,
298} + PDV + 3 RTlnK_D

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

Pressure-temperature diagram similar to Figure 27-4 showing lines of constant K_D 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.

The GASP geobarometer
	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
	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 1.

Example

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.

P-T phase diagram calculated by TWQ 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.

P-T-t Paths

	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 .
	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).

Precision and Accuracy

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 (2001) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

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.