  Column chromatography is a laboratory and industrial technique used to
separate different substances mixed in a solution. Mathematically, it can be
modelled using non-linear partial differential equations whose main ingredients
are the adsorption isotherms, which are non-linear functions modelling the
affinity between the different substances in the solution and the solid
stationary phase filling the column. The goal of this work is twofold. Firstly,
we aim to extend the techniques of Donat, Guerrero and Mulet (Appl. Numer.
Math. 123 (2018) 22-42) to other adsorption isotherms. In particular, we
propose a family of generalized Langmuir-type isotherms and prove that the
correspondence between the concentrations of solutes in the liquid phase (the
primitive variables) and the conserved variables is well defined and admits a
global smooth inverse that can be computed numerically. Secondly, to establish
the well-posedness of the mathematical model, we study the eigenstructure of
the Jacobian of the mentioned correspondence and use this characteristic
information to get oscillation-free sharp interfaces on the numerical
approximate solutions. To do so, we determine the structure of the Jacobian
matrix of the system and use it to deduce its eigenstructure. We combine the
use of characteristic-based numerical fluxes with a second-order
implicit-explicit scheme proposed in the cited reference and perform some
numerical experiments with T\'oth's adsorption isotherms to demonstrate that
the characteristic-based schemes produce accurate numerical solutions with no
oscillations, even when steep gradients appear in the solutions.
