The iterated least square algorithm, which is used here, takes advantage of the equation:
At equilibrium conditions, i.e. ΔG(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 2σ 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 ΔfH° 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 ΔfH° 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.