## 3 Isotherm Models  
Studies of biosorbents contain mainly factors such as: (i) the isotherm models and (ii) the affinity distribution analysis (selectivity). An adsorption isotherm is a measure of the relationship between the equilibrium concentrations of bound and free guest over a certain concentration range and is readily generated from equilibrium batch rebinding studies or chromatographic frontal-zone analysis. Binding properties can be calculated from the binding isotherm by fitting the adsorption. It is a fact that the main theories are the same compared to the commonly-used biosorbents. However, some changes in the symbols and the physical meanings differentiate the models and give other potential to the whole engineering part. Models are used everywhere for simulating any possible process [54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66]. The main  
Figure 3: Example for the synthesis of MIP for capturing/binding dye molecules. Reproduced with permission from [PERSON] [50], published by MDPI AG, 2011.  
models, which will be discussed in following, are the Freundlich, Langmuir, and Langmuir-Freundlich (L-F) equations. In general, the model chosen to explain the particular performance of a biosorbent is the simplest and most statistically significant. For example, if the data fit a model with three different binding sites better than a model with one or two kind of binding sites, the three-binding site model is often chosen, unless other independent data suggest using a different model (tetra-modal, exponential, _etc._). Before accepting a given n-site fit it is advisable to test whether the improvement is statistically significant. The Freundlich isotherm is an empirical power function for non-ideal sorption on heterogeneous surfaces as well as for multilayer sorption and is expressed by the equation:  
\[B=aC^{n} \tag{2}\]  
where \(B\) is the amount of adsorbed pollutant at equilibrium; \(a\) is the Freundlich adsorption coefficient (related with the adsorption capacity (\(N_{t}\)) and the average affinity(\(K_{0}\))); \(m\) is the Freundlich constant which represents the heterogeneity index and varies from zero to one (values approaching to zero indicate increasingly heterogeneity and one being homogeneous); \(C\) is the equilibrium concentration of template. The term \(B\) in Equation (2) was calculated from the simple mass balance equation as follows:  
\[B=\frac{\left(C_{0}-C\right)V}{M} \tag{3}\]  
where \(C_{0}\) is the initial pollutant concentration, \(C\) is the pollutant concentration at equilibrium, \(V\) is the volume of and \(M\) is the mass of biosorbent. The linearized form of Equation (2) was obtained by taking log on both sides:  
\[\log\left(B\right)=\log\left(a\right)+m\cdot\log\left(C\right) \tag{4}\]  
therefore, the plot of \(\log(B)\)_versus_\(\log(C)\) was employed to generate the intercept value of a and the slope of \(m\). The Freundlich equation has been derived by assuming an exponentially decaying adsorption site energy distribution. The data given by this model can be explained, in most cases, taking into account that the Freundlich model is a generalization of the Langmuir model applied to a heterogeneous surface with an energy distribution corresponding to an exponential decrease. Although the Freundlich model predicts that there is an indefinite increase of the adsorbed solute increasing its concentration in solution, this empirical equation is suitable for highly heterogeneous surfaces [67, 68, 69, 70] and often represents typical adsorption data over a restricted range of concentration. In spite that the Freundlich isotherm is suitable to describe the dependence between the coverage degree (\(\theta\)) and the adsorption energy, it does not follow the fundamental thermodynamic basis since it does not reduce to Henry's law at lower concentrations nor predicts the monolayer saturation at higher concentrations. Even so, several methods have been developed in order to estimate the exponential distribution of the isotherm.