The iterated least square algorithm, which is used here, takes advantage of the equation:

At equilibrium conditions, i.e. *Δ**G*(P,T)
= 0, this equation can be rearranged to:

which has the form of a linear relationship *y = mx + b*, where 1/*T* is
the variable. In a 1/*T* *vs.* ln *Kred* plot, the equilibrium
of a chemical reaction in *P-T-x* space is reduced to a straight line with
a slope of *ΔH*°/R and an intercept
of *ΔS*°/*R*.

At any temperature only one ln Kred equilibrium
value exists. Experimental philosophy, in most cases, is to use brackets
to determine equilibrium conditions. Experimental results very rarely, if
ever, report equilibrium. If ln *Kred* values are calculated
for these non equilibrium conditions and plotted on a 1/*T* *vs.* ln* Kred* plot, these experimental constraints do not themselves
plot on the equilibrium line.

If a mineral reaction is formulated in such a way that *ΔH*°
is always positive, then experimental results for mineral reactions where
the products are stable plot above and where the reactants are stable plot
under the equilibrium line. Therefore all experimental results for a mineral
reaction can be treated as half brackets above and below this equilibrium
line. The plot for the reaction KACS-4:-1
calcite + 1 muscovite + 2 quartz <==> 1 anorthite + 1 sanidine + 1
CO2 + 1 H2O illustrates this point
(Hewitt, 1973). Here experiments
with coexisting anorthite + sanidine (empty symbols) plot below and coexisting
calcite + muscovite + quartz (filled symbols) plot above the equilibrium
line.

The calculated equilibrium line for this reaction and therefore *ΔH*°
and *ΔS*° are well constrained
by these experiments. Another advantage of a 1/*T vs.* ln Kred
plot is that it checks the consistency of all available experimental data
for the reaction in question. However, this is true only if all input data
used for the calculation of ln Kred are valid, i.e.
heat capacities, partial molar volumes, coefficients for thermal expansion
and compressibility, fugacities of fluid species, activities of components
in mixed phases, and treatment of order-disorder processes.

An additional example of a 1/*T vs.* ln *Kred* plot
for experimental results from the reaction CMS-11:
1 diopside + 3 dolomite <==> 2 forsterite + 4 calcite + 2 CO2 (Käse &
Metz, 1980; Richter, 1977, 1980).
In this particular case activities for coexisting calcite and dolomite on
their solvus are also taken into account.

For each ln *Kred* value, an error is also calculated
and plotted. The horizontal error bar designates the error in temperature,
the vertical bar incorporates all other uncertainties which contribute to
the error of ln *Kred*. These two error bars create an
error rectangle. Not all errors considered are independent of each other
(e.g. fugacities are temperature dependent). Therefore an error rectangle
is only a first, but reasonable approximation, because temperature uncertainties
for hydrothermal experiments are usually less than 10°C and less than
15°C for piston cylinder runs. In this temperature range fugacities
and other parameters such as molar volumes of solids and activities of the
phase components vary little.

For an experimental result to be in accordance with the equilibrium line,
at least one corner of the error rectangle associated with an experimental
result, which is considered to represent the 2*σ *region of the calculated ln Kred value, must be on the correct side
of the equilibrium line. However, plotting the real shape of the error region
onto a 1/*T vs.* ln *Kred* plot would reduce the readability of the plot dramatically.
In the plot for the reaction CMS-11, the
symbol for the experimental half bracket at 1000 MPa and 858°C, which
shows stable coexistence of diopside + dolomite, plots on the wrong side
of the equilibrium line. The actual upper right corner of the associated
error rectangle, however, lies on the correct side. So within the error
range of the experimental results an equilibrium line can be drawn which
separates the stable coexisting phases diopside+dolomite from forsterite+calcite.

An equilibrium line can not be chosen arbitrarily, but is defined by
*ΔH*° and *ΔS*°
of the reaction. The constant standard values of* Δ**f**H*° and *S*° for each phase component
contribute to its slope and intercept. If a set of such 1/*T vs.* ln *Kred* plots is evaluated simultaneously, one for each mineral
reaction considered, optimized values for *Δ**f**H*° and *S*° can be derived. It is,
however, important to note that the available experimental results used
in a 1/*T vs.* ln *Kred* plot only bracket the equilibrium
line and are not used as data points through which the equilibrium line
is fited. Again, the experiments are used to constrain the calculated equilibrium
line. It is of no importance if some of these experimental constraints are
far from the equilibrium line, as long as they are on the correct side.

last modified: 24.11.2008 by Matthias Gottschalk