## 3 Results and Discussion
### Chromium (VI) Adsorption Reactions and Stability Constants  
The acid-based properties of calcium phosphate were described by the protonation and deprotonation reactions of the phosphate surface functional groups, as shown in Equations (1) and (2), \(\overline{>\text{SOH}}\)[53,54]:  
\[\overline{>\text{SOH}}+\text{H}^{+}\leftrightarrow\overline{>\text{S}(\text{ OH}_{2})^{+}}\quad\text{K}^{+} \tag{1}\]  
\[\overline{>\text{SOH}}\quad\overline{<\text{S}\text{O}^{-}}+\text{H}^{+}\quad \text{K}^{-} \tag{2}\]  
\(\text{K}^{+}\) and \(\text{K}^{-}\) were the surface stability constants, and the on lined species referred to the solid phase. The adsorption reaction of chromium (VI) on calcium phosphate can be expressed as follows:  
\[(>\overline{\text{SOH})_{1}(\text{H}_{i}\text{CrO}_{4})\text{H}_{-n}}+\text{ nH}^{+};\quad 1=1;2,\quad\text{i}=1;2 \tag{3}\]  
where n is the number of protons, which ranges from \(-1\) to \(+1\). The initial ionization of \(\text{H}_{2}\text{CrO}_{4}\) is relatively strong, so \(\text{HCrO}^{-}4\) was the main species found at pH \(>5\), and i = 1 at equilibrium (Equation (3)), under these conditions. The symbol \(\text{H}_{-n}\), stands for both hydrogen atoms (n \(<0\)) and for OH groups (n \(>0\)). Taking into account that \(\text{H}_{2}\text{O}\equiv\text{H}_{-1}+\text{H}_{\text{I}}\), the surface complexes \(\overline{>(\text{SOH})_{1}(\text{H}_{i}\text{CrO}_{4})\text{H}_{-n}}\) noted thereafter \(\text{C}_{\text{In}}\), represented a general formulation of species, differentiated by water composition. So \(\overline{>(\text{SOH})_{1}(\text{H}_{2}\text{CrO}_{4})\text{H}_{-n}}\) could be \(\overline{>(\text{SOH})_{1}(\text{H}\text{CrO}_{4})\text{H}_{-n+1}}\) + H\({}_{2}\)O or \(\overline{>(\text{SOH})_{1}(\text{CrO}_{4})\text{H}_{2-n}}\) + 2H\({}_{2}\)O, and even \(\overline{>(\text{S})_{1}(\text{H}_{2}\text{CrO}_{4})\text{H}_{-n-1}}\) or \(\overline{>(\text{S})_{1}(\text{H}\text{CrO}_{4})\text{H}_{-n-1+1}}\) + H\({}_{2}\)O or \(>(\text{S})_{1}(\text{H}_{2}\text{CrO}_{4})\text{H}_{-n-1+2}}\) + 2H\({}_{2}\)O. The surface complexation constant for the relationship (3) is:  
\[\text{K}_{\text{In}}=\frac{\left[(\overline{>\text{SOH})_{1}(\text{H}_{i} \text{CrO}_{4})\text{H}_{-n}}\right]\left[\text{H}^{+}\right]^{n}}{\left[ \overline{>\text{SOH}}\right]^{1}\left[\text{H}_{i}\text{CrO}_{4}\text{(}^{ \text{i}-2)+}\right]} \tag{4}\]  
The distribution coefficient being:  
\[\text{D}=\frac{\left[(\overline{>\text{SOH})_{1}(\text{H}_{i}\text{CrO}_{4}) \text{H}_{-n}}\right]}{\left[\text{H}_{i}\text{CrO}_{4}\text{(}^{\text{i}-2)+ }\right]} \tag{5}\]  
where [\(\text{H}_{i}\text{CrO}_{4}\text{(}^{\text{i}-2)+}\)] represents the equilibrium concentration of Cr (VI) in solution, it was obtained that:  
\[\text{log}\,\text{D}=\text{log}\,\text{K}_{\text{In}}+\text{log}\,\text{m}+ \text{npH} \tag{6}\]  
where \(\overline{>\text{SOH}}=\text{m}\), was the concentration of sorbent used in \(\text{g}/\text{L}\). Assuming that the first approximation is the essential formation of mononuclear (l = 1), the surface complexes of Equation (6) become:  
\[\text{log}\,\text{D}=\text{log}\,\text{K}_{\text{In}}+\text{log}\,\text{m}+ \text{npH} \tag{7}\]  
The variation of the distribution coefficient with pH allowed us to define the nature of the \(\text{C}_{\text{In}}\)-adsorbed species. The values of (l,n) were obtained according to:  
\[\left(\frac{\text{$\delta$logD}}{\text{$\delta$pH}}\right)_{\text{m}}=\text{n} \tag{8}\]So, the surface complexes of Cr (VI) with calcium phosphate could be well described from the experimental data logD = f(pH), shown in Figure 1. The analysis of the obtained results showed that the plots of logD = f(pH) were linear at various pH ranges. The straight lines of the slope correspond to the mean values of n, and varied between \(-1\) and \(1\). The value of the surface complexes (1,n), involved in this case were (l,0), (l,1), (l,2), and (l,\(-1\)).