Computation method 1: a recursive formulation
Computing the derivative of the ionic strength yields:
We obtain a recursive formulation for the derivative of the activity coefficients:
This recursive formulation can be rearranged as:
The vector contains the explicit portion of the equation. This vector differs depending on the component used to derive the activity coefficient through the stoichiometric coefficient .
The matrix contains the coefficients of the implicit portion of the equation. This matrix is the same regardless of the component used in the derivation.
A complete derivative of the activity coefficients is given by the following linear system:
or