. 2024 Feb 4;10(3):e25303. doi: 10.1016/j.heliyon.2024.e25303

Table 2.

Adsorption isotherm, kinetic and thermodynamic equations and their parameters.

Model	Equations	Description	References
Isotherms adsorption models
Langmuir	$\frac{1}{q_{e}} = \frac{1}{{K_{L} C}_{e}} + \frac{a_{L}}{K_{L}}$ (2) $q_{e} = \frac{K_{L} C_{e}}{1 + a_{L} C_{e}}$ (3)	q_e is metal concentration on the zeolite at equilibrium (mg g⁻¹); q_max is the maximum adsorption capacity (mg g⁻¹) and k_L is the Langmuir constant (L mg⁻¹); a_L (L mg⁻¹) is linked to the energy of adsorption; and K_L/a_L is the monolayer adsorbent capacity	[71]
Freundlich	$\log q_{e} = b_{F} \log C_{e} + \log K_{f}$ (4) $q_{e} = k_{f} {C_{e}}^{(1 / n)}$ (5)	k_f is related to adsorption capacity and 1/n is the adsorption intensity.	[71,72]
Dubinin Radushkevich	${\ln q}_{e} = {\ln q}_{\max} ‐ β ɛ^{2}$ (6) $ɛ = R T \ln (1 + \frac{1}{C_{e}})$ (7) $E_{L} = \frac{1}{\sqrt{2 β}}$ (8)	β is the Dubinin Radushkevich constant (mol²/kJ²), T is the absolute temperature (K); R is the gas constant (8.314 J mol K⁻¹), E_L is the mean free energy (kJ mol⁻¹)	[73]
Kinetic models
Pseudo first order (PFO)	${\ln (q}_{e} ‐ q_{t}) = \ln q_{e} - k_{1} t$ (9) $q_{t} = q_{e} (1 - e^{- k_{1} t})$ (10)	q_e and q_t are the amount of metal ion adsorbed (mg g⁻¹) at equilibrium and at time t (min), respectively, and k₁ (min⁻¹) is the PFO rate constant	[71,72]
Pseudo second order (PSO)	$\frac{t}{q_{t}} = \frac{1}{k_{2} q_{e}^{2}} + \frac{1}{q_{e}}$ t (11) $q_{t} = \frac{q_{e}^{2} k_{2} t}{1 + q_{e} k_{2} t}$ (12)	q_e and q_t represent the quantity of metal ion adsorbed (mg g⁻¹) at equilibrium and at time t (min), respectively, and k₂ (g mg⁻¹ min⁻¹) represent the PSO rate constant	[71,72]
Mixed order model	$q_{t} = q_{e} \frac{1 - e^{- k t}}{1 - f_{2} e^{- k t}}$ (13)	q_t is the adsorption capacity at time t (mg g⁻¹) f₂ is the mixed order coefficient	[74]
Intraparticle diffusion	$q_{t} = k_{i p} \sqrt{t} + c_{i p}$ (14)	k_ip is the diffusion coefficient (mg g⁻¹min ^−1(1/2)) c_ip is the intraparticle diffusion constant (mg g⁻¹)	[74]
Thermodynamic parameters
Adsorption thermodynamic parameters	$Δ G = - R T l n K_{c}$ (15) $\ln K_{c} = \frac{Δ S}{R} - \frac{Δ H}{R T}$ (16)	ΔG is Gibbs free energy change; ΔS is standard entropy change; ΔH is enthalpy change, R is the gas constant, T is the absolute temperature and K_c is the distribution coefficient	[74]