The formalism behind each extraction cycle is itself very similar to
the approach used by Powell &
Holland (1985). In this special case, their approach was adapted to
the 1/*T vs.* ln *Kred* plots. For each mineral reaction
*j* the reduced equilibrium constant ln *Kred *can
be expressed by

for the *i*th reaction constraint, here 1 or 2 for each reaction.

For a total of n reaction constraints this can be written as a matrix product:

The vector **A**R includes the reduced equilibrium
constant ln *Kred* for each reaction at a given temperature
*Ti.* These ln* Kred* values are the
selected points in each 1/*T vs.* ln *Kred*
plot. The elements of the rows of matrix **B** are the stoichiometric
coefficients for each phase component in each reaction. The* Δ**f**H*° and *S*° values of each phase component
are listed in vector **D**°, or:

In the next step an objective function for the optimization process must
be defined. As discussed above, the following criteria were used for the
extraction: (1) assemblages in each 1/*T vs.* *ln Kred*
plot should be perfectly separated by the equilibrium line, i.e. as close
as possible to the observed (obs) ln* Kred* values; (2)
all third law entropies and enthalpies of formation should be close to the
values determined by calorimetry. These criteria lead to the following objective
function *F*:

The superscript tab stands for a calorimetric, tabulated value. Using a least squares approach, the minimum for the objective function can be determined:

**D**° tab is a vector which includes the
tabulated, i.e. calorimetric, reference entropies and enthalpies. **E**
is the identity matrix. **D**° is the result and includes the optimized* Δ**f**H*°
and S° values of each phase component.

The following modified objective function (cf. Powell & Holland, 1985) for the weighting scheme was used for the extraction of the internally consistent data set,

where the parameters *F* are the associated variances of ln *Kred*,* Δ**f*Ho, So and therefore 1/*F* are
the weighting factors. The solution to *F* is obtained by:

The weighting matrix **WA****R**
for the manually selected points in the 1/*T vs.* ln* Kred*
plots and the tabulated calorimetric values **WD**° tab are defined as follows:

Again vector **D°** contains the optimized* Δ**f**H*° and *S*° values.

Calorimetric enthalpies and entropies of certain simple phases, especially
elements and oxides, are considered to be well known experimentally. With
this in mind,* Δ**f**H*°
and *S*° values of corundum, hematite, iron, periclase, quartz,
rutile, CO2, and O2 were used
as anchors, along with the enthalpies and entropies for andalusite, kyanite
and sillimanite from Hemingway et
al. (1991).

last modified: 24.11.2008 by Matthias Gottschalk