## 2 Materials and Methods
### Batch Experiments
#### 2.3.1 Kinetic Study for the Sorption Process of Cr(VI) in AC@SCF  
The Cr(VI) removal capacity of AC@SCF depends on the contact time as well as Cr(VI) initial concentration. Kinetic experiments were carried out at pH = 3, temperature T = 21 \({}^{\circ}\)C, and for five Cr(VI) initial concentrations (i.e., 5.23, 20.94, 26.73, 35.40, and 54.14 ppm). Samples, each of 5 mL, were withdrawn from the liquid solutions at different periods (0, 0.5, 1, 2, 3, 6, 9, and 24 h), centrifuged, and titrated for Cr(VI) concentration determination. Measurements are depicted as relative liquid Cr(VI) concentration, that is, \(C(t)\)/C\({}_{init}\) versus contact time \(t\) (h). A further transformation of these experimental measurements was carried out taking into account that the solid activated carbon concentration in the liquid was 180 mg/L. The kinetic curves are depicted as adsorbed Cr(VI) concentration in AC@SCF solid \(q(t)\) (mg/g) versus contact time \(t\) (h). Four kinetic models were used to interpret these data. The first two models were based on the mass balance equations for a continuous stirred batch reactor, assuming complete mixing (ideal) conditions. For an \(n\)th order process, this equation is as follows:  
\[\frac{dq(t)}{dt}=k_{n}\times[q_{e}-q(t)]^{n} \tag{1}\]  
where \(q(t)\) (mg\({}_{\text{CC(VI)}}\)/g\({}_{\text{AC}}\)) is the total Cr(VI) adsorbed in AC@SCF activated carbon after \(t\) (h) contact time, \(q_{e}\) (mg\({}_{\text{CC(VI)}}\)/g\({}_{\text{AC}}\)) is the total Cr(VI) adsorbed to AC@SCF activated carbon at equilibrium stage (\(t\)\(\rightarrow\)\(\infty\)), and \(k_{n}\) ((g\({}_{\text{AC}}\)/mg\({}_{\text{CC(VI)}}\))\({}^{n-1}\) h\({}^{-1}\)) is the sorption rate constant for an \(n\)th order adsorption process. The initial rate \(r_{i}\) for such an adsorption process becomes by the substitution of \(t\) = 0 and \(q(t)\) = 0 to Equation (1) and reads as:  
\[r_{i}=\left.\frac{dq(t)}{dt}\right|_{t=0}=k_{n}\times q_{e}^{n} \tag{2}\]  
By the substitution of \(n\) = 1 and \(n\) = 2 to Equations (1) and (2), we achieved the most commonly used kinetic models of pseudo-first and pseudo-second orders [29; 30]. Pseudo-First Order Kinetic Model  
For a process with a first-order kinetic (\(n\) = 1), Equation (1) is transformed to Equation (3) and assuming the boundary conditions \(q(t)\) = 0 at \(t\) = 0 and \(q(t)\) = q at \(t\) = t, the definite integration of Equation (3) leads to Equation (4). Moreover, the initial adsorption rate is derived from Equation (3) by assuming \(q_{t}\) = 0 for \(t\) = 0 and it is described by the Equation (5). \[\frac{dq_{t}}{dt}=k_{1}\times(q_{e}-q_{t}) \tag{3}\]  
\[q_{t}=q_{e}\times\left(1-e^{-k_{1}\times t}\right) \tag{4}\]  
\[r_{i}=\left.\frac{dq_{t}}{dt}\right|_{t=0}=k_{1}\times q_{e} \tag{5}\]  
Pseudo-Second Order Kinetic Model  
For a process with a second-order kinetic (\(n\) = 2), the mass balance of Equation (1) is transformed as follows [30]:  
\[\frac{dq_{t}}{dt}=k_{2}\times\left(q_{e}-q_{t}\right)^{2} \tag{6}\]  
The definite integration of the above mass balance, using \(q_{t}\) = 0 for \(t\) = 0 and \(q_{t}\) = t for \(t\) = t as boundary conditions, gives the following expression:  
\[q_{t}=\frac{q_{e}^{2}\times k_{2}\times t}{q_{e}\times k_{2}\times t+1} \tag{7}\]Furthermore, the initial rate for this second-order model is as follows:  
\[r_{i}=\left.\frac{dq_{t}}{dt}\right|_{t=0}=k_{2}\times q_{e}^{2} \tag{8}\]  
Intraparticle Diffusion Kinetic Model  
[PERSON] and [PERSON] (1962) [31] proposed Equation (9) as a semi-empirical equation for the calculation of the rate of the intraparticle diffusion process. \[q_{t}=K_{ID}\times t^{\frac{1}{2}}+c \tag{9}\]  
Similarly to the abovementioned models, the \(q_{t}\) (mgCR(VI)/gAC@SCF) variable is the adsorbate uptake on the activated carbon at time \(t\) (\(h\)). Moreover, the constant \(K_{ID}\) (mg/(g h\({}^{1/2}\))) is rate constant considering the intraparticle diffusion process and \(C\) (mg/g) is an index of the boundary layer thickness at the surface of the AC@SCF particles.