Table 1.

Equations of the models commonly used for the theoretical analysis of the sorption data.

Equation Number	Equation 1	Definition
Kinetic models
(3)	$q_t=q_e\left(1-ex{p}^{-k_{1}t}\right)$	Pseudo-first order (PFO) model
(4)	$q_t=\frac{q_e^2{k}_{2}t}{1+q_e{k}_{2}t}$	Pseudo-second order (PSO) model
(5)	$q_t=k_{id}{t}^{0.5}+C$	Intraparticle diffusion (IPD) model
Isotherm models
(6)	$q_e=\frac{q_m{K}_L{C}_e}{1+K_L{C}_e}$	Langmuir model
(7)	$q_e=K_F{C}_e^N$	Freundlich model
(8)	$q_e=q_{DR}{e}^{-K_{DR}{\epsilon }^2}$	Dubinin–Radushkevich (D–R) model
(9)	$q_e=\frac{q_m{a}_S{C}_e^N}{1+a_S{C}_e^N}$	Sips model
(10)	$ln{K}^o=\frac{\Delta S^o}{R}-\frac{\Delta H^o}{RT}$	Van’t Hoff equation
(11)	$K^o=K_D\times M_{adsorbate}\times 55.5$	Standard equilibrium constant
(12)	$\Delta G^o=-RTln{K}^o$	Standard Gibbs free energy of sorption, kJ mol−1

¹q_e and q_t—amount of metal ion sorbed at equilibrium and at time t, min, respectively; k₁—rate constant of the PFO kinetic model, min⁻¹; k₂—rate constant of the PSO kinetic model, g/mg × min; k_id—intraparticle diffusion rate constant, mg/g × min^1/2; C—constant describing the effect of boundary layer thickness, mg/g; q_m—maximum theoretical sorption capacity, mg/g; K_L—Langmuir constant, L/mg; K_F—Freundlich constant, mg/g × mg^N × L^N; N—measure of both the nature and strength of adsorption process, and of active sites distribution, related to the surface heterogeneity; the larger is its value, the more heterogeneous the sorbent system is; q_DR—maximum sorption capacity of the metal ion, mg/g; K_DR—D–R isotherm constant, mol²/kJ²; ε—Polanyi potential; a_S—Sips constant; ΔS°—entropy, kJ/mol × K; ΔH°—enthalpy, kJ/mol; R—the gas constant, J/mol × K; T—the temperature in Kelvin; M_adsorbate—is the abbreviation of atomic mass of each metal ion.