. 2023 Nov 16;13:20060. doi: 10.1038/s41598-023-46967-3

Table 1.

Kinetic adsorption of different models exploiting.

Models kinetic	The equations	Parameters	References
Pseudo 1st order	$\frac{d q_{t}}{dS} = KF (q_{e} - q_{t}) (3)$ $log (q_{e} - q_{t}) = log (q_{e}) - (\frac{k_{}}{2.203}) . t (4)$	q_t : Biosorption capability over time (mg g⁻¹)	⁹
		q_e: Biosorption capability of equilibrium study (mg g⁻¹)
		k: Constant study of pseudo 1st order model (g mg⁻¹ min⁻¹)
		t: The time expressed in (min)
Pseudo 2sec order	$\frac{t}{q_{t}} = \frac{1}{k q_{e}^{2}} + \frac{1}{q_{e}} \times t (5)$	k: Constant of 2sec order model present (g mg⁻¹ min⁻¹)	¹⁰
Elovich’s Kinetic	$q_{t} = \frac{1}{β} \times ln (α β) + \frac{1}{β} ln (t) (6)$	α: The initial rate of the adsorption expressed (mg g⁻¹ min⁻¹) β: The constant of desorption in grams per milligram
Intraparticle diffusion model	$q_{t} = K_{i} \times t^{1 / 2} + χ_{i} (7)$	q_t : The amount of metal adsorbed on the solid phase is expressed in mg g⁻¹;	¹¹
		Ki: The slope and concentration value of the junction of the graph curves qt against t^0.5 were used to calculate the intraparticle diffusion coefficient (mg g⁻¹ min^-0.5)
		C: Diffusion constant (mg g⁻¹)
Bangham model	$log log (\frac{C_{0}}{C_{0} - q_{t} \times m}) = log (\frac{K_{B}}{2.303 \times V}) + α l o g (8)$	qt: The quantity of metal adsorbed at time t (mg g⁻¹)m : Mass study of adsorbent (g)	^12–14
Bangham model		ɑ are K_B the constants	^12–14