One should note that this test case is not chemically realistic. Moreover, its numerical value comes only from activity correction if the unknowns of the nonlinear system are the logarithms of the activity components . Otherwise, if the unknowns are component concentrations , the problem becomes trivial and linear, and its solution is the total concentration .

Phosphoric acid test case

This test presents reactions between phosphoric acid and salt water. It includes 4 components and 8 chemical species. We handle only acid-base reactions: water dissociation and the 3 phosphoric acid reactions. A table including the stoichiometric coefficients, equilibrium constants, total concentrations and equilibrium solutions is given in appendix 2.

Gallic acid test case

This test case was proposed by Brassard and Bodurtha [20]. It includes 3 components and 17 chemical species. It is a classical test case, and many difficulties in convergence have been reported while solving it by using Newton or Newton-like algorithms [13, 14, 21]. A table including the stoichiometric coefficients, equilibrium constants, total concentrations and equilibrium solutions is given in appendix 3.

Iron-chromium test case

This test case concerns the rehabilitation of chromium-contaminated industrial soil using an iron-chromium reduction [2, 22]. Chromium (VI), which is the most toxic and mobile form of chromium, is reduced by iron (II) to yield chromium (III), which has a much lower solubility and is less toxic [23]. This test is reported to be a very difficult one [14, 21], so here we use some favorable testing conditions to increase the convergence of the Newton algorithm. A table including the stoichiometric coefficients, equilibrium constants, total concentrations and equilibrium solutions is given in appendix 4.

Results

Study of the test case with only activity correction during one resolution.
We first present two scenarios for the test case with only activity correction : one with a low ionic strength and one with a high ionic strength. The objective is to determine the influence of the activity correction on the Newton procedure depending on the algorithm used. For the situation with a low ionic strength, this influence is expected to be negligible, whereas we expect a greater impact in the situation with a high ionic strength.
For the low ionic strength situation, the initial component activities are 5.0 10-7 mol. L-1 for all components. The ionic strength is 7.80 10-6 mol. L-1, and we obtain the species concentrations and activity values, which are given in Table 2 . Also in Table 2 , we show the first Newton steps proposed by the fixed-point algorithms (inner and outer) and by the full Newton algorithm.
Table 2: Initial values for the situation with a low ionic strength.