# Practical Aspects of Mineral Thermobarometry

## Thermobarometer Calibration

The aim of this section will be to cover briefly such topics such as

- Conditions of heterogeneous equilibrium
- The equilibrium constant and its P-T dependence.
- A simple geothermometer: garnet-biotite Fe-Mg exchange
- A geobarometer based on a net-transfer reaction with large volume change
- How thermometers and barometers are calibrated
- Self-consistent thermodynamic data sets and their application to thermobarometry

You can read about these subjects in Spear 1993, chapter 8, pp. 241-244, and chapter 15, pp. 511-537.

Here's some provisional content, from lecture course handouts:

#### Condition for heterogeneous equilibrium

[Spear, p. 511]. A balanced chemical reaction such as Grs + 2 Ky + Qtz = 3 An could also be written in the form 3 An - Grs - 2 Ky - Qtz = 0, reminding us that the mass of each system component is conserved, and that reaction coefficients can be positive ("products") or negative ("reactants"). At equilibrium, a similar relationship holds among the chemical potentials of phase components (n are the reaction coefficients and there are m phase components):

i.e. there is no free energy difference between "reactants" and "products".

#### Expand the expression for heterogeneous equilibrium

[Spear, p. 512]. Adding up, for all the phase components, the
expressions µ_{i} = µ°_{i} +
*RT*ln(*a*_{i}), gives us

*G*_{R} = 0 =
*G*^{0} + *RT*ln*K*

where *K* is the **equilibrium constant **or activity
product, which depends on measured phase compositions.
*G*^{0}
is the standard state free energy change - something we can calculate
if we have thermochemical data. Expanding
*G*^{0} to see what thermochemical data are
needed, we get:

The subscripts and limits reflect the fact that thermochemical data are generally tabulated at 298K and 1 bar.

#### Linear approximation for solid-solid reactions

[Spear, p. 523]. ** C_{P} = 0.
**The

*C*

_{P}terms adjust

*H*and

*S*for different

*P*and

*T*. The heat capacities of solid phases vary sympathetically, so

*C*

_{P}is small, and can often be ignored, particularly over small ranges of temperature.

** V constant**. The volume integral brings the
enthalpies from their reference pressure

*P*

^{0}(1 bar) up to the pressure of interest (usually several thousand bars).

*V*is not a function of

*P*or

*T*if all solids are roughly equally compressible, so the volume integral becomes

*V*(

*P*-

*P*

^{0}). Also,

*P*

^{0}is negligible compared to

*P*.

This gives us the greatly simplified expression

_{}

which, for a given value of ln*K*, is the equation of a
straight line on a *P*-*T* diagram. The curve for
ln*K* = 0 (i.e. *K* = 1) is the curve for pure phases, such
as might be determined in the experimental laboratory. Other values
of ln*K* displace the curve across the *P*-*T*
diagram.

#### Dependence of ln*K* on *T* and *P*

[Spear, pp. 515-6]

_{}

_{}

where *S* and
*V* are calculated at the
*P* and *T* of interest. Also remember the
Clausius-Clapeyron equation:

_{}; or, more strictly,

_{}

Therefore, a good geobarometer needs a large
*V*, in
order to have a large sensitivity to *P* and for curves of
constant *K* to have a low slope on the *P*-*T*
diagram.

### Geothermometry and geobarometry using multivariant equilibria

At equilibrium

DG

_{P,T}= DH_{1,T}- T.DS_{T}+ (P-1).DV + RT.lnK = 0

K is the equilibrium constant, the product of the activities of end members in the phases of variable composition.

#### The Grt-Pl-Ky-Qtz geobarometer

Ca_{3}Al_{2}Si_{3}O_{12}
+ 2Al_{2}SiO_{5} + SiO_{2} =
3CaAl_{2}Si_{2}O_{8}

grossular(Grt) + kyanite + quartz = anorthite(Pl)

Substituting thermochemical data, dividing through by the gas constant R, and expanding K gives the calibration:

7635 - 19.66T + 0.7963(P-1) + 3Tln(X

_{Ca,Pl }/X_{Ca,Grt}) = 0

DV is large, because of the density difference between open feldspar and close-packed garnet and kyanite structures. The P-T slope is relatively low, making this equilibrium useful as a barometer. Uncertainty on calculated P is about ± 1.3 kbar.

#### The Grt-Bt Fe-Mg exchange thermometer

Fe_{3}Al_{2}Si_{3}O_{12}
+ KMg_{3}AlSi_{3}O_{10}(OH)_{2} =
Mg_{3}Al_{2}Si_{3}O_{12} +
KFe_{3}AlSi_{3}O_{10}(OH)_{2}

Almandine (Grt) + Phlogopite (Bt) = Pyrope(Grt) + Annite (Bt)

DG for cation exchange is small, and the calibration here is fitted to exchange equilibrium experiments

6266 - 2.35T + 0.029(P-1) + 3TlnK

_{D}= 0

K_{D} is most simply expressed as (Mg/Fe)_{Grt}
/(Mg/Fe)_{Bt}

DV is very small, and cation exchange equilibria are generally insensitive to pressure. The usually-quoted uncertainty on calculated T is ± 50°.

#### Application

An amphibolite-facies metapelitic schist from the Eastern Alps
contains garnet of composition (% end members)
Alm_{80}Sps_{3}Prp_{13.5}Grs_{3.5} ,
biotite with Mg/Fe = 0.91, and plagioclase An15 , together in
apparent equilibrium with kyanite and quartz. From these data we
calculate the two P-T curves. Their intersection (with a surrounding
box of uncertainty) gives us an estimate of the conditions of
equilibration.

## Self-consistent data-sets

[ Material to incorporate from Petrology course lecture slides ]

This page last modified 12 October 2004