for isotherm Langmuir and Freundlich models are most widely used and in case of kinetics pseudo first order and pseudo second order are the most common. I am attaching some files here, you can go through them to understand about the models. Also, be attentive about the form of model i.e. linear or non linear.
Freundlich and Langmuir adsorption isotherms are both successful models to describe the adsorption of heavy metals, HM, from water using various sorbents. Regarding the kinetics many of us agreed on the pseudo second order kinetic equation to describe the removal of HM. The following link may be of some help:
All of kinds isotherm and kinetic models can apply for considering to fit the adsorption experimental data and then decide that how the adsorption take place, according to R2
Olusola, the kinetics and equilibrium tests are integral for the design of adsorption systems. The kinetic test helps you determine how fast (rate) the metal ions can be removed by the adsorbent while the equilibrium helps elucidate how much (capacity) metal ions can be adsorbed for a given initial concentration. There are numerous kinetic and equilibrium isotherm models in the literature; many of which can assist in determining both the transport and attachment mechanisms by with the metal ions are removed. Care must be exercised when interpreting models. While they may fit your data, they may offer little insight into the mechanisms. The following links may offer some help:
Equilibrium Isotherms and kinetics are used to know the interactions between adsorbent and absorbatell. Widely Langmuir, Freundlich and Temkin models are used to estimate the maximum adsorption capacity and kinetics models : pseudo first order and pseudo secod order