Table 1. Different kinetic, isotherm, and thermodynamic models.

Kinetic models
	Equation	Nomenclature	Reference
Pseudo-first-order	$\log \left(q_{e1}-q_t\right)=\log q_{e1}-\frac{k_1}{2.303}t$	qe1 and qt: biosorption capacity [mg g-1] at equilibrium and at any time t [h], respectively; k1: rate constant of the model [h-1]	[33]
Pseudo-second-order	$\frac{t}{q_t}=\frac{1}{k_2{q}_{e2}^2}+\frac{1}{q_{e2}}t$	qe2 and qt: biosorption capacity [mg g-1] at equilibrium and at any time t [h], respectively; k2: rate constant of the model [h-1]	[34]
Intraparticle diffusion	qt = kidt0.5 + c	qt: biosorption capacity [mg g-1] at any time t [h]; kid: intraparticle diffusion rate constant [mg g-1 h-0.5]; c: model intercept	[35]
Elovich	$q_t={\frac{1}{B_E}\mathrm{l}\mathrm{n}\left(A_E{B}_E\right)+\frac{1}{B}}_{E}lnt$	AE: initial biosorption rate [mg g-1 h -1]; BE: desorption constant of the model [g mg-1]	[16]
Fractional power	qt = kfptv	v: fractional power rate constant [h-1]; kfp: fractional power model constant [mg g-1]	[16]
Isotherm models
Two-parameter models
Langmuir	$q_e=\frac{q_m{K}_L{C}_e}{1+K_L{C}_e}$	qe and qm: biosorption capacity at equilibrium and maximum biosorption capacity [mg g-1], respectively; Ce: liquid phase concentration of dye at equilibrium [mg L-1]; KL: Langmuir constant [L mg-1]	[30]
Freundlich	$q_e=K_F{C_e}^{\frac{1}{n_F}}$	KF: constant of the Freundlich model [(mg g-1)(mg L-1)-1/nF]; nF: heterogeneity factor.	[16]
Temkin	$q_e=\frac{RT}{B_T}\mathrm{l}\mathrm{n}\left(A_T{C}_E\right)$	T: absolute temperature [K]; R: ideal gas constant [8.314 J mol-1 K-1]; BT: constant related to heat of biosorption [J mol-1]; AT: Temkin isotherm constant [L mg-1]	[35]
Halsey	$q_e={(K_H/C_e)}^{1/n_H}$	KH: Halsey isotherm model constant [L g-1]-1/nH; nH: Halsey model exponent	[16]
Dubinin-Radushkevich	$q_e=q_m\mathrm{e}\mathrm{x}\mathrm{p}(-B_{DR}{E}_{DR}^2)$	BDR: biosorption energy constant [mol2 J-2]; EDR: Polanyi potential [kJ mol-1]	[36]
Three-parameter models
Sips	$q_e=q_m\frac{{\left(K_S{C}_e\right)}^{1/n_s}}{1+{\left(K_S{C}_e\right)}^{1/n_s}}$	Ks: Sips constant; 1/ns: Sips model exponent	[30]
Toth	$q_e=\frac{q_m{b}_{T}Ce}{{(1+{(b}_T{Ce)}^{{n_T}^{-1}})}^{n_T}}$	bT: Toth model constant [L mg-1]-1/nT; nT: Toth model exponent	[36]
Redlich-Peterson	$q_e=\frac{K_{RP}{C}_e}{1+A_{RP}{C}_e^{B_{RP}}}$	KRP: Redlich-Peterson model isotherm constant [L g-1]; ARP: Redlich-Peterson model constant [L mg-1]BRP; BRP: Redlich-Peterson model exponent	[36]
Radke-Prausnitz	$q_e=\frac{A_R{R}_R{C}_e^{B_R}}{A_R{+R}_R{C}_e^{B_{R-1}}}$	AR: [L g-1] and RR [L mg-1]: Radke-Prausnitz model constants; BR: Radke-Prausnitz model exponent	[16]
Thermodynamic models
Gibbs free energy change	Δ G° = −R T ln Kc $K_C=\frac{q_e}{C_e}$	T: absolute temperature [K]; R: ideal gas constant [8.314 J mol-1 K-1]; Kc: equilibrium constant [L g-1]; Δ G°: Gibbs free energy change [J mol-1]	[37]
Entropy change	∂ΔG° = ∂T ∂ΔS°	T: absolute temperature [K]; Δ S°: entropy change [J mol-1 K-1]	[37]
Enthalpy change	Δ H° = ΔG° − T ΔS°	Δ H°: enthalpy change [J mol-1]	[16]