## 3 Equilibrium and Thermodynamic Aspects of Cr(VI) Sorption  
### Sorption Isotherms  
The sorption equilibrium of Cr(VI), as for any other sorbate, is generally investigated using the sorption isotherms [42]. These are obtained by plotting the sorbed amount of Cr(VI) per unit mass of sorbent (\(q_{e}\)) at equilibrium against the initial Cr(VI) concentration (\(C_{e}\)). The isotherm models most frequently applied for describing Cr(VI) sorption are the Langmuir model and the Freundlich model [43]. The Langmuir model [44] (Equation (18)) is a simplistic model applicable under the assumptions that (i) the sorbate encompasses a finite number of energetically equivalent sites arranged in a superficial monolayer, and (ii) the solution behaves ideally (i.e., the solution is diluted enough for solute activity being approximated by concentration), and there are no intermolecular interactions among the free solute and/or sorbate molecules [45; 46]:  
\[q_{e}=\frac{q_{m}K_{L}C_{e}}{1+K_{L}C_{e}} \tag{18}\]  
where \(q_{m}\) and \(K_{L}\) are the maximum sorption capacity and the Langmuir equilibrium constant, respectively. The Freundlich model [47] (Equation (19)) is a power function empirical model especially useful for the practical purpose of describing sorption equilibrium data not including a sorption saturation level as the Langmuir model demands:  
\[q_{e}=K_{F}C_{e}{}^{N} \tag{19}\]  
where \(K_{F}\) and \(N\) are empirical constants. It has been suggested that the value of the \(N\) parameter gives information on the heterogeneity of sorbent sites: values of \(N\) significantly lower than 1 (concave downward isotherms) would reflect high heterogeneity, whereas values of \(N\cong 1\) (linear isotherms) would be indicative of a homogeneous energetic distribution of sorption sites. It is important to note that \(N\) is a dimensionless parameter. This notwithstanding, \(N\) is sometimes erroneously reported with units of L mg-1[48; 49]. With few exceptions [18; 50; 51; 52], if compared to, the Langmuir model fits better than the Freundlich model the experimental data for Cr(VI) sorption. [PERSON] et al. [53] obtained a higher correlation coefficient using the Langmuir model for modelling the sorption isotherms of Cr(VI) sorption by poly(N\({}^{1}\),N\({}^{1}\),N\({}^{3}\),N\({}^{3}\)-tetraallylpropane-1,3-diaminium chloride) (PTAPDAC). A better performance of the Langmuir model was confirmed by [PERSON] et al. [54] using native and chemically modified green macroalgae _Codium tometosum_ biomass as a sorbent. Similar results were obtained by [PERSON] et al. [8] using Al/Fe oxyhydroxide composite powders, [PERSON] and [PERSON] [55] using the green alga _Pseudopediastrum boryanum_ and many other authors using minerals [40; 56], synthetized materials [1; 7; 10; 20; 57; 58] or biosorbents [26; 27; 48; 59]. In some cases [6; 25; 60], a semi-empirical hybrid Langmuir-Freundlich model, also known as Sips model [61], has been used for modelling isotherm equilibrium data:  
\[q_{e}=\frac{q_{m}K_{LF}\mathcal{C}_{e}{}^{N}}{1+K_{LF}\mathcal{C}_{e}{}^{N}} \tag{20}\]  
where \(K_{LF}\) is a constant. The Sips model generally gives better fitting results than either the Langmuir or the Freundlich model. This is not surprising, because the Sips model is a more general (and therefore adaptable) equation that, for \(N=1\) or for high Cr(VI) initial concentration, reduces to the Langmuir model whereas for sufficient low Cr(VI) concentration (i.e., \(1>>K_{LF}\)\(\mathcal{C}_{e}{}^{N}\)) reduces to the Freundlich model (with \(K_{F}=q_{m}K_{LF}\)). In rare cases, the equilibrium sorption data exhibited unusual trends not amenable to modelling by any of the isotherm equations described above. For example, [PERSON] and coworkers [24] reported a bell-shaped isotherm curve for the sorption of Cr(VI) onto MnO\({}_{2}\). Normally, sorption capacity increases with increasing sorbate initial concentration and eventually reaches a plateau that corresponds to the saturation level of the sorbent.