## 3 Results and Discussion
### Practical Applications
For this reason, this mechanism could not be the controlling mechanism of the process. To observe whether the PSO and chemisorption is the only biosorption active rate limiting mechanism we fitted the experimental data using the diffusion-chemsorption model [48]. According to the R\({}^{2}\) values (Table 3) this model exhibits a fitting efficiency similar to and in many cases better to the fitting efficiency of the PSO model. According to all the above mentioned, neither the collision of the adsorbed species with the unoccupied pore surface sites nor the diffusion through pores are the only controlling mechanism [49]. For this reason, the pseudo second order model (PSO) and the diffusion-chemsorption (DC) model seems to be the most appropriate for the kinetic study of this work. Such models indicate a more complex mechanism, involving both the above-mentioned controlling stages plus a chemisorption process, which occurs on the pore surface sites by bonding the adsorbate with surface groups and may be a reaction on the surface of pores. Furthermore, the state of the adsorbate, i.e., liquid or gas, and the pore size distribution of the activated carbon are strongly interactive [50]. In the case of liquid media as an adsorbate, the existence of micropores in combination with the existence of  
Figure 12: Fitted curves of four different kinetic models (i.e., (1) pseudo first order (FFO), (2) pseudo second order (PSO), (3) intra particle diffusion (IPD), (4) diffusion–chemsorption (DC)), upon experimental kinetic datasets of five different Cr(VI) initial concentrations (i.e., (**a**) 5.5 ppm, (**b**) 10.65 ppm, (**c**) 20.7 ppm, (**d**) 40 ppm, (**e**) 53 ppm) for the APC material. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**APC-C\({}_{\text{init}}\) (ppm)** & **PFO R\({}^{2}\)** & **PSO R\({}^{2}\)** & **IPD R\({}^{2}\)** & **DC R\({}^{2}\)** \\ \hline
5.50 & 0.999 & 0.998 & 0.503 & 0.984 \\
10.65 & 0.945 & 0.986 & 0.706 & 0.998 \\
20.70 & 0.912 & 0.962 & 0.710 & 0.989 \\
40.00 & 0.914 & 0.963 & 0.727 & 0.961 \\
53.00 & 0.909 & 0.943 & 0.736 & 0.948 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: R\({}^{2}\) correlation coefficient values provided by Levenberg–Marquardt least-square fitting algorithm upon the APC experimental dataset. meso and macro pores is more effective for such adsorption processes [49]. In our case, according to the CPSM model [10; 11], APC exhibits approximately 31% macro-meso porosity, which contributes to the relatively quick intraparticle diffusion, and approximately 69% microporosity, which contributes to the relatively slow diffusion and transient adsorption, the surface bonding, the chemisorption and the surface reactions [49]. Table 4 presents the predicted equilibrium concentration for APC based on both the pseudo second order (PSO) model and the diffusion-chemisorption (DC) model. It is obvious that after a long time (theoretically infinite) of the adsorption process, the theoretical overall adsorbed amount will be around 71 mg\({}_{\text{Cr(VI)}}\)/g\({}_{\text{AC}}\), according to the PSO model, and around 90 mg\({}_{\text{Cr(VI)}}\)/g\({}_{\text{AC}}\), according to the DC model. Furthermore, it is obvious from the same table that for initial concentration equal or greater to 40 ppm, APC achieves its maximum saturation capacity. In the case of PSO mechanism and for the 5.5 ppm as the initial concentration, the predicted adsorbed amount at the equilibrium stage qe was higher than the initial concentration. This means that the total amount of the Cr(VI) was adsorbed and no equilibrium stage occurs at this experiment. Thus, no Gibbs free energy for the adsorption process could be calculated for this experiment. For an initial concentration of 10.65 ppm, 20.70 ppm, 40.00 ppm and 53.00 ppm of the calculated Gibbs free energy values are \(-26.2\) kJ/mol, \(-21.0\) kJ/mol, \(-19.0\) kJ/mol and \(-18.0\) kJ/mol, respectively. The negative sign is attributed to the spontaneous nature of the adsorption process during these experiments [20; 21; 51].