## 3 Results and Discussion
### Chromium (VI) Adsorption Reactions and Stability Constants
#### 3.3.2 Equilibrium Constants  
In the Log(D)= f(pH),Figure 1 plots the various contact times and sorbent amounts, and the straight lines show slopes ranging from \(-0.4\) to \(0.6\). In the cases when the surface complexes are mononuclear (l = 1), apparent equilibrium constants, \(\mathrm{K_{ap}}\) = \(\mathrm{K_{ln}}\) or \(\mathrm{K^{\prime}}\)\({}_{1n}\), were obtained from the origin ordinates A = LogK\({}_{1n}\) + Log(m) or logK\({}^{\prime}\)\({}_{1n}\) + Log(m) \(-4\). From the obtained results, it can be that the variations of logK\({}_{1n}\) = f(n) (Figure 3) and logK\({}^{\prime}\)\({}_{1n}\) = f(n) (not shown), were linear under all experimental conditions. Taking into account that \(\mid\)n\(\mid\) also represented the fraction (x) of the predominant 1H\({}^{+}\) or 1 OH\({}^{-}\) exchange reactions contributing to the overall adsorption equilibrium, the apparent constants (K\({}_{\rm ap}\)) of the overall equilibrium were given by the expressions shown in Equations (14) and (15):  
\[\rm{K_{ap}=(K_{10})^{(1-\mid n\mid)}(K_{1\pm 1})^{\mid n\mid}} \tag{14}\]  
\[\rm{logK_{ap}=(1-\mid n\mid)logK_{10}+\mid n\mid logK_{1\pm 1}} \tag{15}\]  
with \(K_{1\pm 1}\) = K\({}_{11}\) or K\({}_{1-1}\). As an example, the overall partition equilibrium prevailing for pH > 5.5, and involving successive exchange reactions \(n\) = 0 and \(-1\), can be summarized in the following reactions:  
\[(1-x)(\overline{>\rm{SOH}}+\rm{HCrO_{4}}^{-}\leftrightarrow\overline{>S(OH)( HCrO_{4})H_{0}}=\overline{>S(CrO_{4})}+H_{2}O):\quad(K_{10})^{(1-x)},\ n=0 \tag{16}\]  
Overall adsorption reactions are:  
\[(x)(\overline{>\rm{SOH}}+\rm{HCrO_{4}}^{-}+H_{2}O\leftrightarrow(\overline{> \rm{SOH}(HCrO_{4})H_{1}}=\overline{>S(HCrO_{4})}+H_{2}O)+OH^{-}:\quad(K_{1-1}) ^{x},\ n=-1 \tag{17}\]  
\[\overline{>\rm{SOH}}+\rm{HCrO_{4}}^{-}+nH_{2}O\leftrightarrow(\overline{>\rm{ SOH}(HCrO_{4})H_{n}}=\overline{>S(HCrO_{4})}+H_{2}O)+nOH^{-}:\quad(K_{1n}),\ -1\leq n\leq 0 \tag{18}\]  
Where x is the molar fraction of the sorption mechanism with 1 OH\({}^{-}\) exchange. The apparent constant, K\({}_{\rm ap}\) = K\({}_{\rm{In}}\), is then given by:  
\[\rm{logK_{In}}\rm{=(1-x)logK_{10}+xlogK_{11}=(1-\mid n\mid)\ logK_{10}+\mid n \mid\ logK_{10}} \tag{19}\]  
As shown above, the K\({}_{\rm ap}\) value could be determined experimentally, and was variable, depending on the surface charge and pH. Therefore, the constants K\({}_{10}\) and K\({}_{1\pm 1}\) were the intersect and the slope of logK\({}_{\rm ap}\) = f(n), respectively, whereas K\({}_{1-1}\) was the opposite of the slope. From this information, the intrinsic constants K\({}_{10}\), for H\({}^{+}\) (formation of \(\overline{>S(OH)(HCrO_{4})H_{0}}\equiv\overline{>S(HCrO_{4})}\)) and OH\({}^{-}\) (formation of \(\overline{>S(OH)(HCrO_{4})H_{0}}\equiv\overline{>S(CrO_{4})}\)) adsorption reaction exchange, were determined, with corresponding logarithmic values of logK\({}_{10}\) = 0.2 and 0.1, respectively. The intrinsic constants (K\({}_{\rm int}\) = K\({}_{10}\)) were obtained in general, by extrapolating the apparent constants to a zero surface charge [57; 58]. Figure 3: Variations of log Kap = f(n). As discussed previously, the variations of logK\({}_{\rm ap}\)= f(n) could be due to the contribution of the electrostatic effect (K\({}_{\rm col}\)), related to K\({}_{\rm ap}\)[60; 61] according to:  
\[{\rm K}_{\rm int}={\rm K}_{\rm ap}{\rm K}_{\rm col}={\rm K}_{\rm ap}\exp(-{\rm nF} {\rm F}{\rm V}_{0}/{\rm RT}) \tag{20}\]  
\[{\rm K}_{\rm ap}={\rm K}_{\rm int}\,({\rm K}_{\rm col})^{-1}=\,({\rm K}_{10})^{(1 -|n|)}\,({\rm K}_{1\pm 1})^{|n|} \tag{21}\]  
In the particular case of \(\,|n\,|=1\), Equation (22) was obtained:  
\[{\rm K}_{\rm ap}={\rm K}_{\rm int}({\rm K}_{\rm col})^{-1}={\rm K}_{1\pm 1} \tag{22}\]  
where \(\psi_{0}\) was the surface potential, and R, T, and F were the gas constant, absolute temperature, and Faraday constant, respectively.