Derivative of the activity coefficients versus the ionic strength : B-dot model
For the B-dot model, the activity coefficients are given in logarithmic form by:
Taking the derivative of this equation yields:

Test cases

Testing procedure
It is well-known that the initial guess of the values of the components plays a critical role in the convergence of the Newton methods [13, 21]. To test several initial guesses, we generate a large number (30 000) of activity component values according to the following procedure:
where and and are given in the description of the chemical test case to handle a representative range of concentrations.
We then obtain 30 000 realizations of the optimization procedure using the same chemical test case but different initial guesses. To analyze this large amount of data, we construct a frequency graph of the number of Newton iterations needed to reach convergence.
Chemical test cases
We propose 4 chemical test cases.

Test case with only activity correction

We first propose a test case without any chemical reactions. Any nonlinearity is only due to activity correction. The chemical system is composed of chloride ions, calcium ions, aluminum ions and thin ions. We neglect water dissociation and all chemical reactions. The details and equilibrium solutions are given in appendix 1.
Table 1: Chemical table for the test case with only activity correction.