Table 1.

Common models for fitting the experimental sorption data (isotherms, kinetic and column).

Fitting Models	Nonlinear	Linear	Parameters
Isotherms
Langmuir	$q_e=\frac{q_m{K}_L{C}_e}{1+K_L{C}_e}$	$\frac{C_e}{q_e}=\frac{C_e}{q_m}+\frac{1}{K_L{q}_m}$	qe (mg/g) = sorbed amount at equilibrium; Ce (mg/L) = equilibrium concentration; qm (mg/g) = maximum sorbed amount. KL (L/mg) = Langmuir adsorption constant; KF = Freundlich constant; 1/n = parameter related to surface heterogeneity; aS (L/mg) = Sips adsorption constant.
Freundlich	$q_e=K_F{C}_e^{1/n}$	$ln{q}_e=ln{K}_F+\frac{1}{n}ln{C}_e$
Sips	$q_e=\frac{a_S{C}_e^{1/n}}{1+a_S{C}_e^{1/n}}$	$\frac{1}{n}ln{C}_e=-ln\left(\frac{a_S}{q_e}\right)+ln{a}_S$
Kinetics
Pseudo-first-order (PFO)	$q_t=q_e\left(1-e^{-k_{1}t}\right)$	$ln\left(q_e-q_t\right)=ln{q}_e-k_{1}t$	qe (mg/g) = sorbed amount at equilibrium; qt (mg/g) = sorbed amount at time t; k1 (min−1) = PFO model rate constant; k2 (g/mg∙min) = PSO model rate constant; t (min) = time.
Pseudo-second-order (PSO)	$q_t=\frac{k_2{q}_e^{2}t}{1+k_2{q}_{e}t}$	$\frac{1}{q_t}=\frac{1}{k_2{q}_e^2}+\frac{1}{q_e}$
Column experiments
Thomas	$\frac{C_t}{C_0}=\frac{1}{1+exp\left[\left(\frac{k_{TH}{q}_{0}m}{Q}\right)-k_{TH}{C}_{0}t\right]}$	ln(C0/Ct–1) = (kTH∙q0∙m)/Q-kTH∙C0∙t	C0 (mg/L) = initial concentration; Ct (mg/L) = concentration at time t; Ce (mg/L) = equilibrium concentration; q0 (mg/g) = maximum sorption capacity; Q (mL/min) = flow rate. kTH (L/mg∙min) = Thomas rate constant; kYN (mL/mg∙min) = Yoon–Nelson rate constant; τ (min) = time required to reach 50% breakthrough; t (min) = time.
Yoon–Nelson	$\frac{C_t}{C_0}=\frac{1}{1+exp\left[\left(k_{YN}\left(\mathsf{\tau }-\mathrm{t}\right)\right)\right]}$	ln[Ce/(C0–Ce)] = kYN∙t-τ∙kYN