## 3 Isotherm Models
On the other hand, the Langmuir sorption isotherm assumes that adsorption takes place at specific homogeneous sites within the material. The isotherm equation is derived from simple mass action kinetics, assuming chemisorption. Also, it assumes that once a template occupies a site, no further adsorption can take place at that site, all sites are energetically equivalent and there is no interaction between molecules adsorbed on neighboring sites. A saturation value should be reached beyond which no further sorption can take place. While the Langmuir model assumes that there is only a single class of binding sites, the bi-Langmuir model assumes that there are two classes of sites within the imprinted material. It is straightforward to implement the Langmuir and bi-Langmuir models using Scatchard plots to determine the binding parameters, the binding affinity (\(K\)) and the number of binding sites (\(N_{t}\)), using the general expression:  
\[\frac{B}{C}=KN_{t}-KB \tag{5}\]  
where \(B\) is the amount of adsorbed pollutant at equilibrium; \(N_{t}\) is the total number of accessible adsorption sites; \(C\) is the equilibrium concentration of template; \(K\) is the Langmuir isotherm equilibrium constant. In homogeneous systems with only one type of binding sites, by plotting \(B/C\) against \(B\), it is possible to obtain the value of \(N\) from the x-intercept and the value of \(K\) from the slope. In heterogeneous systems, Scatchard plots are not linear and the simplest model in this case is a material with two different kinds of adsorption sites. Langmuir equation is then extended to an equation with two Langmuir terms (bi-Langmuir equation):  
\[B=\frac{N_{t}K_{1}C_{1}}{1+K_{1}C_{1}}+\frac{N_{2}K_{2}C_{2}}{1+K_{2}C_{2}} \tag{6}\]  
Thus, the plot \(B/C\)_versus_\(B\) is composed of two straight lines, from which two sets of binding parameters (\(K_{1}\), \(N_{1}\) and \(K_{2}\), \(N_{2}\)) for the two classes of binding sites within the imprinted polymer can be obtained. The steeper line corresponds to the high-affinity sites while the flatter line measured the low-affinity ones. Furthermore, another isotherm model can be used which is the combination of Langmuir and Freundlich equations namely Langmuir-Freundlich (L-F) isotherm. This model, first described by [PERSON] [71, 72] was introduced for the biosorbents by the groups of [PERSON] _et al._[69, 70, 73] and [PERSON] _et al._[74, 75, 76]. The model describes an equilibrium relationship between the concentration of a bound template (\(B\)) and the equilibrium template concentration in solution (\(C\)) such that (where \(N_{t}\) is the total number of binding sites and \(K_{0}\) is the median binding affinity):  
\[B=\frac{N_{t}K_{0}^{\ \ m}C^{\ m}}{1+K_{0}^{\ \ m}C^{\ m}} \tag{7}\]  
The variable \(a\) is related to \(K_{0}\) via \(K_{0}=a^{1/m}\). The fitting parameter \"\(m\)\" is identical to the heterogeneity index of site energies from the Freundlich isotherm. The difference between the L-F model and the Freundlich one is evident at high adsorbate concentrations, for which the L-F model is able to represent the saturation behavior. At low adsorbate concentrations, the L-F equation reduces to the classical Freundlich equation. On the other hand, as m approaches unity, indicative of a completely homogeneous adsorbent surface (_i.e._, energetic equivalence of all binding sites) the L-F equation reduces to the classical Langmuir equation. Thus, the hybridised L-F isotherm is able to model adsorption of solutes at high and low concentrations onto homogeneous and heterogeneous biosorbents. Although a linear analysis is not possible for a three-parameter isotherm, the L-F isotherm can be fitted to the experimental data following the method of [PERSON] _et al._[4, 5] in which a solver function may be used to maximize the coefficient of determination (\(R^{2}\)) by iteratively varying the three fitting parameters \(N_{s}\), \(a\) and \(m\).