# Chapter 27 Thermodynamics of Metamorphic Reactions

Mechanics

Oct 27, 2013 (4 years and 6 months ago)

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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 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
-

TS

Where:

G = Gibbs Free Energy

H = Enthalpy (heat content)

T = Temperature in Kelvins

S = Entropy (can think of as randomness)

Thermodynamics

D
G for a reaction

of the type:

2 A + 3 B = C + 4 D

D
G =
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

Thermodynamics

For other temperatures and pressures we can use the equation:

dG = VdP
-

SdT

(ignoring
D
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

z

z

G

G

VdP

SdT

T

P

T

P

T

T

P

P

2

1

1

1

2

1

2

2

-

=

-

If V and S are constants, our equation reduces to:

G
T2 P2

-

G
T1 P1

= V(P
2

-

P
1
)
-

S (T
2

-

T
1
)

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

d
D
G =
D
VdP
-

D
SdT

(again ignoring
D
X)

D
G for any reaction = 0 at equilibrium

Worked Problem #2 used:

d
D
G =
D
VdP
-

D
SdT

and G, S, V values for albite, jadeite and quartz to
calculate the conditions for which
D
G of the reaction:

Ab + Jd = Q

is equal to 0

from G values for each phase at 298K and 0.1 MPa calculate
D
G
298, 0.1

for the
reaction, do the same for
D
V and
D
S

D
G at equilibrium = 0
, so we can calculate an isobaric change in T that would
be required to bring
D
G
298, 0.1

to 0

0
-

D
G
298, 0.1

=
-
D
S (T
eq

-

298)

(at constant P)

Similarly we could calculate an isothermal change

0
-

D
G
298, 0.1

=
-
D
V (P
eq

-

0.1)

(at constant T)

Method:

NaAlSi
3
O
8

= NaAlSi
2
O
6

+ SiO
2

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:
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

d
D
G = 0 =
D
VdP
-

D
SdT

Thus

dP

dT

S

V

=

D

D

Figure 27.1.
Temperature
-
pressure
phase diagram for the reaction:
calculated using the program TWQ
of Berman (1988, 1990, 1991).

Winter (2010) An Introduction to
Igneous and Metamorphic
Petrology. Prentice Hall.

-

SdT, for an isothermal process:

G

G

VdP

P

P

P

P

2

1

1

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 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.

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:

G

G

VdP

P

P

P

P

2

1

1

2

-

=

z

G

G

RT

P

dP

P

P

P

P

2

1

1

2

-

=

z

z

G

G

RT

P

dP

P

P

P

P

2

1

1

2

-

=

1

Gas Pressure
-
Volume Relationships

And since

G
P2

-

G
P1

= RT
ln
P
2

-

ln
P
1

= RT

ln

(P
2
/P
1
)

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:

G
P, T

-

G
T

= RT

ln

(P/P
o
)

G of a gas at some P and T = G in the reference state (same T and 0.1 MPa)
+ a pressure term

1

x

dx

x

=

z

ln

o

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
=
g
P

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

Dehydration Reactions

Mu + Q = Kspar + Sillimanite + H
2
O

We can treat the solids and gases separately

G
P, T

-

G
T

=
D
V
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

Dehydration Reactions

(qualitative analysis)

dP

dT

S

V

=

D

D

Figure 27.2.
Pressure
-
temperature
phase diagram for the reaction
muscovite + quartz = Al
2
SiO
5

+ K
-
feldspar + H
2
O, calculated using
SUPCRT (Helgeson
et al
., 1978).
Winter (2010) 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
)

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

n

n

Solutions: T
-
X relationships

Example: orthopyroxenes (Fe, Mg)SiO
3

Real vs. Ideal Solution Models

Figure 27.3.
Activity
-
composition relationships for the enstatite
-
ferrosilite mixture in orthopyroxene at 600
o
C and 800
o
C. 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 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:

D

D

G

G

RT

K

P

T

P

T

o

,

,

=

-

ln

K

c

c

D

d

A

a

B

b

=

a

a

a

a

Compositional variations

plagioclase = Al
-
rich Cpx + Q

D
G
T, P

=
D
G
o
T, P

+ RT
l
n
K

Let’s say
D
G
o
T, P

was the value that we calculated for
equilibrium in the pure Na
-
system (= 0 at some P and T)

D
G
o
T, P

=
D
G
298, 0.1

+
D
V (P
-

0.1)
-

D
S (T
-
298) = 0

By adding Ca we will shift the equilibrium by
RT
l
n
K

We could assume ideal solution and

K

Jd

Pyx

SiO

Q

Ab

Plag

=

X

X

X

2

All coefficients = 1

Compositional variations

So now we have:

D
G
T, P

=
D
G
o
T, P

+ RT
l
n

since Q is pure

D
G
o
T, P

= 0
as calculated for the pure system at P and T

D
G
T, P

is the shifted
D
G due to the Ca added (no longer 0)

Thus we could calculate a
D
V(P
-

P
eq
) that would bring
D
G
T, P

back to 0, solving for the new P
eq

X

X

Jd

Pyx

Ab

Plag

Compositional variations

D
G
P, T

=
D
G
o
P, T

+ RT
l
n
K

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.

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

Geothermobarometry

The Garnet
-

Biotite geothermometer

Geothermobarometry

The Garnet
-

Biotite geothermometer

Figure 27.5.

Graph of
l
n
K 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.

l
n
K
D

=
-
2108

T(K) + 0.781

D
G
P,T

= 0 =
D
H
0.1, 298

-

T
D
S
0.1, 298

+ P
D
V + 3 RT
l
n
K
D

Geothermobarometry

The Garnet
-

Biotite geothermometer

Figure 27.6.
AFM projections showing the relative distribution of Fe and Mg in garnet vs. biotite at approximately 500
o
C

(a)

and 800
o
C
(b)
.
From Spear (1993)
Metamorphic Phase Equilibria and Pressure
-
Temperature
-
Time Paths
. Mineral. Soc. Amer. Monograph 1.

Geothermobarometry

The Garnet
-

Biotite geothermometer

Figure 27.7.
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 Al
2
SiO
5

phase diagram is added. From Spear (1993)
Metamorphic Phase Equilibria and Pressure
-
Temperature
-
Time Paths
. Mineral. Soc. Amer. Monograph 1.

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 650
o
C 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

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

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 THERMOBAROMET
RY,

by
Spear and Kohn (1999). The shaded area represents the bracketed estimate of the P
-
T conditions for the sample. The Al
2
SiO
5

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, Al
2
SiO
5

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
-

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.
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).

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
75
o
C, 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 (= 650
o
C 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 Holl
and
, 1994).
The shaded ellipse is the AvePT error ellipse (R. Powell, personal communication). After Tenczer et al. (2006).

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 + H
2
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
-
6 from the Pakistan Himalaya.
Monazite U
-
Pb ages of black dots are in Ma. Small
-
dashed lines are
Al
2
SiO
5

polymorph reactions and large
-
dashed curve is the H
2
O
-
saturated minimum melting conditions. After Foster et al. (2004).

Geothermobarometry

P
-
T
-
t Paths

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

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 th
e 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 (2
010) An
Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

Geothermobarometry

Precision and Accuracy

Figure 27.18.
P
-
T diagram illustrating the calculated uncertainties from various sources in the application of the garnet
-
biotite geothermomet
er
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.

Geothermobarometry

Precision and Accuracy