Extraction procedure

The extraction procedure can be described briefly as follows. For every mineral reaction a 1/T vs. ln Kred plot was drawn. As a first step, for each 1/T vs. ln Kred plot one or two points were chosen which were thought to be very close to a plausible equilibrium line. The experimental data constraining these points had to be well determined. Both reactants and products had to be stable within close proximity.

Utilizing a least squares formalism, a set of enthalpies (ΔfH°, e.g. formation from the elements) and entropies (S°) was found for which the equilibrium lines were as close as possible to the chosen equilibrium points. In addition, the ΔfH° and S° values were constrained to a minimum deviation from their calorimetric values. Because the ln Kred equilibrium points in each plot were initially chosen manually their selection was not necessarily objective. Therefore the results after the first extraction run were not final. Successive iterations were used until an internally consistent data set was achieved.

After each run, the equilibrium lines for each mineral reaction were plotted in the 1/T vs. ln Kred plots using extracted ΔfH° and S° values from the previous run. If the selected point in the 1/T vs. ln Kred plot used for the previous iteration cycle was not identical to the current equilibrium line, but the separation of the phase assemblages with this new equilibrium line was still in agreement with the experimental results, a point from this new equilibrium line was chosen as input for the next iteration cycle. For some reactions contradictory sets of experimental results were available. If the new equilibrium line was compatible with another available set of experimental results than the one used for the previous iteration, again, a point from this new equilibrium line was chosen as input for the next iteration cycle, using the other set of experimental results as constraints.

The brackets confining the equilibrium line in each 1/T vs. ln Kred plot were of different widths. The vertical width, which is the width of the bracket with respect to ln Kred, can be used as a measure of the validity and quality of the bracket. The reciprocal width was used as a normalizing and weighting factor for this reaction in the least square algorithm. Reactions with wide brackets were less important to the least square algorithm than reactions with tight brackets. Also the reciprocals of the errors of the available calorimetric ΔfH° and S° values were used as weighting factors.

Some derived ΔfH° and S° values showed a tendency to shift away from tabulated calorimetric values. Some third law entropies especially showed a tendency drift to higher values, indicating significant disorder in these phases at elevated temperatures. When this was observed, the constraint which linked the derived So to the calorimetric value was weakened using a higher variance and therefore a lower weighting factor. For some phases calorimetric values with significant differences in ΔfH° and in some cases S° (e.g. garnets) were available. At the starting point of the extraction process it was not obvious which values were more reliable. As a consequence, if the result of an iteration cycle preferred a different calorimetric value for ΔfH° and/or S° than the one used for the previous iteration, the calorimetric value which came closest to the preferred value was used as the input parameter for the next iteration. Each iteration cycle generated a new input data set. This process was repeated several times until a stable solution was obtained.

last modified: 24.11.2008 by Matthias Gottschalk