## 2 Materials and Methods
### Adsorption Kinetics, Equilibrium Models, and Thermodynamics  
Adsorption Kinetics: The biosorbent adsorption kinetics were investigated using three different kinetic models: pseudo-first order, pseudo-second order, and intra-particle diffusion. These models were applied to the data set produced from the MB concentration experiments at 30 \({}^{\circ}\)C for 50, 100, and 200 ppm. These models can identify the adsorption kinetics of the dye removal processes as they include the mass transfer steps during the adsorption process. Furthermore, pseudo-first- and second-order models are often used to predict the adsorption rates for batch adsorbers [62; 63; 64]. Pseudo-first-order and second-order adsorption rate models have limitations due to the adsorption rate constant being dependent on the adsorbate concentration and the amount of adsorbent charged to the adsorber [63; 65]. The pseudo-first-order (Equation (3)) and second-order (Equation (4)) kinetic models equations are presented below [62; 66; 67], along with the intra-particle diffusion model (Equation (5)) [6; 66]. \[\text{Pseudo-First Order}\ \ \ \ \ \log(\text{q}_{\text{eq}}-\text{q}_{\text{t}})= \log(\text{q}_{\text{eq}})-(\text{k}_{1,\text{ad}}/2.303)\times\text{t} \tag{3}\]  
\[\text{Pseudo-Second Order}\ \ \ \ \text{t}/\text{q}_{\text{t}}=1/(\text{k}_{2,\text{ad}} \times\text{q}_{\text{eq}}^{2})+(1/\text{q}_{\text{eq}})\times\text{t} \tag{4}\]  
\[\text{Intra-particle Diffusion}\ \ \ \ \text{q}_{\text{t}}=\text{k}_{3,\text{ad}} \times\sqrt{\text{t}+\text{C}} \tag{5}\]  
where t (min) is the adsorption time, q\({}_{\text{t}}\) (mg/g) is the adsorption capacity at time t, q\({}_{\text{eq}}\) (mg/g) is the adsorption capacity of adsorbent at the equilibrium point, K\({}_{1,\text{ad}}\) (1/min), and k\({}_{2,\text{ad}}\) (g/mg\(\cdot\)min) and k\({}_{3,\text{ad}}\) (mg/g\(\cdot\)min\({}^{1/2}\)) are the constant rates of pseudo-first-order, pseudo-second-order, and intra-particle diffusion models, respectively. Adsorption equilibrium models: Langmuir and Freundlich adsorption isotherms were used to assess adsorption equilibrium and the maximum adsorption capacity of biomass feed-stocks and chars. The models were applied to experimental data obtained from the varying MB concentration experiments as described earlier in the method section. It is known that the effluent quantity at the solid-liquid interface increases non-linearly with the concentration at the equilibrium conditions. Langmuir (Equation (6)) and Freundlich (Equation (7)) adsorption isotherms are the most frequently used models in the literature [6; 7; 29; 66; 68; 69; 70] to describe the non-linear equilibrium between \(\rm q_{eq}\) and \(\rm C_{eq}\) at a constant temperature and are described as follows [62,64]:  
\[\rm Langmuir\ Model\qquad C_{eq}/\rm q_{eq}=C_{eq}/Q^{0}+1/(Q^{0}\times b) \tag{6}\]  
\[\rm [PERSON]\ Model\qquad\ln(q_{eq})=ln(K_{F})=(1/n)\times ln(C_{eq}) \tag{7}\]  
where, \(\rm q_{eq}\) (mg/g) is the adsorption capacity of adsorbent at the equilibrium, \(\rm C_{eq}\) (mg/L) is the dye concentration at equilibrium, \(\rm Q^{0}\) is the maximum adsorption capacity, b is the adsorption bonding energy, \(\rm K_{F}\) ((mg/g\(\cdot\)mg/L)\({}^{n}\) is the Freundlich adsorption constant, and n is the Freundlich adsorption constant (0 \(<\) n \(<\) 1). Adsorption thermodynamics: The adsorption process thermodynamic parameters indicate whether a process is favourable [7]. Gibbs free energy change \(\rm\Delta G^{\circ}\), standard enthalpy \(\rm\Delta H^{\circ}\), and standard entropy \(\rm\Delta S^{\circ}\) of the kinetic data were thus analysed to gain a greater insight into the effect of temperature on the adsorption.