## 2 Materials and Methods
### Experimental Design of Adsorption Kinetics
At a certain time after the beginning of the experiment, i.e., 6, 9, 12, 24, 48, 96, 192, and 288 h, one bottle of each adsorbent was taken and the phosphorus concentration (PO\({}_{4}\)-P) in the solution was measured. The control bottles with the ammonium solution and the orthophosphate solution remained in the heated chamber for 144 and 288 h, respectively. The amount of ammonium and orthophosphate adsorbed at equilibrium was calculated with the following equation:  
\[\mathrm{q_{e}}=\frac{(\mathrm{C_{i}}-\mathrm{C_{e}})}{\mathrm{M}}\mathrm{V} \tag{1}\]  
where \(\mathrm{q_{e}}\) (\(\mathrm{\mu g/g}\)) is the adsorption capacity at equilibrium; \(\mathrm{C_{i}}\) and \(\mathrm{C_{e}}\) (mg/L) are the initial and equilibrium concentrations of NH\({}_{4}\)-N or PO\({}_{4}\)-P in the solution, respectively; V (L) is the volume of solution; and M (kg) is the mass of the adsorbent (i.e., CG-Z, FG-Z, and CG-G added to each glass bottle). The amount of NH\({}_{4}\)-N and PO\({}_{4}\)-P adsorbed, q\({}_{\rm f}\) (\(\rm\upmu g/g\)), at time t (h) was calculated with the following equation:  
\[\rm q_{t}=\frac{(C_{i}-C_{t})}{M}V \tag{2}\]  
where \(C_{i}\) and \(C_{t}\) (mg/L) are the initial concentration and the concentration at time t of NH\({}_{4}\)-N or PO\({}_{4}\)-P in the solution, respectively; V (L) is the volume of solution (i.e., 500 mL); and M (kg) is the mass of the adsorbent (i.e., CG-Z, FG-Z, and CG-G added to each glass bottle). The data collected from the above adsorption experiments were used to find which adsorption kinetic model best describes the adsorption of ammonium and orthophosphate on the three adsorbent materials. In the present work, three kinetic models were tested, the pseudo-first-order and pseudo-second-order models [37; 38; 39] and the intra-particle diffusion model (or the Weber and Morris model) [40; 41]. The pseudo-first-order model and its linear form and the pseudo-second-order model are described by Equations (3), (4), and (5), respectively:  
\[\rm q_{t}=q_{e}[1-exp(-k_{1}t)] \tag{3}\]  
\[\rm log\big{(}q_{e}-q_{t}\big{)}=\rm log\,q_{e}-\frac{k_{1}}{2.303}t \tag{4}\]  
\[\rm\frac{1}{q_{t}}=\frac{1}{q_{e}}+\frac{1}{k_{2}q_{e}^{2}t} \tag{5}\]  
where q\({}_{t}\) (mg/kg) is the amount of adsorbed substance (i.e., NH\({}_{4}\)-N or PO\({}_{4}\)-P) at time t (h) and k\({}_{1}\) (1/h) and k\({}_{2}\) (kg/(mg h)) are the constant rates of the pseudo-first- and pseudo-second-order equations, respectively. The following criteria were used to evaluate the fit of the kinetic models to the experimental data: (a) the correlation coefficient (R\({}^{2}\)), (b) the root mean square error (RMSE) and the normalized objective function (NOF), and (c) the mean absolute error (MAE) [42]. The RMSE, NOF, and MAE are expressed by the following equations:  
\[\rm RMSE=\sqrt{\frac{\sum\limits_{i=1}^{N}(P_{i}-O_{i})^{2}}{N}} \tag{6}\]  
\[\rm NOF=\frac{RMSE}{O_{mean}} \tag{7}\]  
\[\rm MAE=\frac{\sum\limits_{i=1}^{N}|O_{i}-P_{i}|}{N} \tag{8}\]  
where O\({}_{i}\) represents the measured (or observed) values; P\({}_{i}\) represents the predicted values; O\({}_{mean}\) is the mean of the measured values; and N is the total number of measurements. The acceptable value of NOF is between 0.0 and 1.0, and the optimum value of both NOF and MAE is 0.0. The intra-particle diffusion model is described by the following equation [40; 41]:  
\[\rm q_{t}=k_{id}t^{0.5}+C \tag{9}\]  
where k\({}_{id}\) (mg/(kg\(\cdot\)h\({}^{0.5}\))) is the intra-particle diffusion rate constant and C (mg/kg) is the intercept, which indicates the thickness of the boundary layer.