## 3 Equilibrium and Thermodynamic Aspects of Cr(VI) Sorption
### Sorption Thermodynamics  
The change of sorption isotherms (i.e., of the equilibrium conditions) with temperature provides information on the temperature-dependence of the sorption equilibrium and related thermodynamic parameters, namely the standard change in Gibbs energy (\(\Delta G^{\circ}\)), enthalpy (\(\Delta H^{\circ}\)) and entropy (\(\Delta S^{\circ}\)). The temperature-dependence of the sorption equilibrium constant (\(K\)) is generally derived from the Gibbs-Helmholtz equation [84]:  
\[\frac{\partial\Big{(}\Delta G^{\circ}/T\Big{)}}{\partial T}=-\frac{\Delta H^{ \circ}}{T^{2}} \tag{21}\]  
By using the well-known thermodynamic relationships:  
\[\Delta G^{\circ}=-RTln(K) \tag{22}\]  
\[\Delta G^{\circ}=\Delta H^{\circ}-T\Delta S^{\circ} \tag{23}\]  
and assuming that neither \(\Delta H^{\circ}\) or \(\Delta S^{\circ}\) varies appreciably with temperature, Equation (21) can be integrated to give:  
\[\ln(K)=-\frac{\Delta H^{\circ}}{R}\frac{1}{T}+\frac{\Delta S^{\circ}}{R} \tag{24}\]  
A plot of \(\ln K\) vs. 1/\(T\) (van't Hoff plot) will produce a straight line whose slope and intercept with y-axis permit the determination of \(\Delta H^{\circ}\) and \(\Delta S^{\circ}\), respectively. The value of \(K\) in Equation (24) depends on the model chosen for describing the sorption isotherm. The Langmuir equilibrium constant \(K_{L}\) is often used for this purpose in Cr(VI) sorption studies [51; 75; 35]; the sorption distribution coefficient \(K_{D}=q_{d}/C_{e}\) (determined by extrapolating to zero the plot \(\ln(q_{d}/C_{e})\) against \(C_{e}\)) is also utilized, though less frequently [64; 70]. Applying Equation (24), [PERSON] et al. [58] obtained positive values of \(\Delta H^{\circ}\), indicating that Cr(VI) sorption on zirconium oxide-immobilized alginate beads is an endothermic process. The same was reported by [PERSON] et al. [63] using anion exchanger microspheres embedded into chitosan/poly(vinyl amine), by [PERSON] et al. [68] using amine-functionalized magnetic corn stalk composites, and by many other authors (Table 1). Negative values of \(\Delta H^{\circ}\) are encountered less frequently in the literature, for example [PERSON] et al. [24] using MnO\({}_{2}\) and [PERSON] et al. [51] using dolomite. Crucially thermodynamic analyses of Cr(VI) sorption reported in the literature is often wrong [2; 7; 19; 20; 24; 26; 30; 35; 52; 53; 58; 62; 68; 73; 74; 77]. The value of \(K\) (Equation (24)) has been erroneously determined on many occasions [2; 7; 20; 24; 30; 52; 53; 58; 62; 68; 73; 77] using the following expression:  
\[K=\frac{q_{e}}{C_{e}} \tag{25}\]  
As easily inferred from any experimental isotherm curve (with the exception of linear isotherms), the \(q_{d}/C_{e}\) ratio is _not_ constant (i.e., it varies with sorbent coverage [85]), thus Equation (25) does notrepresent an equilibrium constant and cannot be used in the van't Hoff equation (Equation (24)). As already observed, a correct determination of \(K\) from the \(q_{e}\)/\(C_{e}\) ratio could be obtained by extrapolating its value at zero sorbent coverage [86; 87; 88]. For the same reasons, we consider incorrect the determination of \(K\) as:  
\[K=\frac{F_{e}}{1-F_{e}} \tag{26}\]  
where \(F_{e}\) is the fraction of Cr(VI) sorbed at equilibrium [19; 74]. Another mistake frequently observed in the literature concerns the meaning ascribed to the sign of \(\Delta G^{\circ}\). Several authors [2; 7; 19; 20; 24; 26; 32; 35; 52; 53; 58; 62; 64; 68; 73] found a \(\Delta G^{\circ}<0\) and interpreted this as an indication of spontaneous Cr(VI) sorption. Others obtained a \(\Delta G^{\circ}>0\) and concluded that the instance of Cr(VI) sorption they examined was a non-spontaneous process [4; 74; 77]. These claims have been so long held that one tends to forget that the calculated change in Gibbs energy refers to _standard conditions_.