Table 2.

Non-linear form of Kinetic, isotherm, and thermodynamic equations and their parameters.

Eq.	Kinetic equations
5	PFO	$Q_{\mathrm{t}}=Q_{\mathrm{e},\mathrm{cal}.}\cdot (1-e^{-k_1\cdot t})$	$Q_{\mathrm{e},\mathrm{cal}.}$: the calculated adsorption capacity at any time t (mg g−1) k1: the PFO rate constant of adsorption (min−1)
6	PSO	$Q_{\mathrm{t}}=\frac{Q_{\mathrm{e},\mathrm{cal}.}^2\cdot k_2\cdot t}{1+Q_{\mathrm{e},\mathrm{cal}.}\cdot k_2\cdot t}$	k2: the PSO rate constant of adsorption (g mg−1 min−1)$h(h=Q_{\mathrm{e},\mathrm{cal}.}^2\cdot k_2)$: the initial adsorption rate (mg g−1 min−1)
7	Elovich	$Q_t=\frac{1}{\beta }\ln (\alpha \cdot \beta )\cdot t$	α: the initial adsorption rate (mg g−1 min−1)β: kinetic parameter related to the activation energy for chemisorption and extent of surface coverage (g mg−1)
8	IPD	$Q_{\mathrm{t}}=k_{\mathrm{IPD}}\cdot t^{0.5}+C$	kIPD: intra-particle diffusion rate constant (mg g−1 min−0.5)C: intra-particle diffusion constant related to the thickness of the boundary layer (mg g−1)
	Isotherm equations
9	Langmuir	$Q_{\mathrm{e}}=\frac{Q_{\mathrm{m},\mathrm{cal}.}{K}_{\mathrm{L}}{C}_{\mathrm{e}}}{1+K_{\mathrm{L}}{C}_{\mathrm{e}}}$	$Q_{\mathrm{m},\mathrm{cal}.}$: the theoretical maximum adsorption capacity (mg g−1) KL: the Langmuir equilibrium constant (L mg−1) RL ($R_{\mathrm{L}}=[1/(1+K_{\mathrm{L}}\cdot C_{\mathrm{e}}\left)\right]$): the separation factor (−)
10	Freundlich	$Q_{\mathrm{e}}=K_{\mathrm{F}}{C}_{\mathrm{e}}^{1/n}$	KF: Freundlich constant ((mg g−1) (L mg−1)1/n) n: adsorption intensity (−)
11	R-P	$Q_{\mathrm{e}}=\frac{K_{\mathrm{R}-\mathrm{P}}{C}_{\mathrm{e}}}{1+{\alpha }_{\mathrm{R}-\mathrm{P}}{C}_{\mathrm{e}}^{\mathrm{g}}}$	KR−P: Redlich-Peterson isotherm constant (L g−1) αR−P: Redlich-Peterson isotherm constant (mg L−1)−g g: Redlich-Peterson isotherm binding constant (0 < g < 1)
	Thermodynamic equations
12	$K_{\mathrm{e}}^{\mathrm{o}}=\frac{1000\cdot K_L\cdot MW\cdot {\left[adsorbate\right]}^{\mathrm{o}}}{\gamma }$		$K_{\mathrm{e}}^{\mathrm{o}}$: thermodynamic equilibrium constant (−) MW: the molecular weight of the adsorbate (g mol−1) [absorbate]°: the unitary standard concentration of the adsorbate (1 mol L−1) γ: the coefficient of activity (−)
13	$\ln K_{\mathrm{e}}^{\mathrm{o}}=-\frac{\mathrm{\Delta }{H}_{\mathrm{ads}.}^{\mathrm{o}}}{R}\cdot \frac{1}{T}+\frac{\mathrm{\Delta }{S}_{\mathrm{ads}.}^{\mathrm{o}}}{T}$		R: the universal gas constant (J mol−1 k−1) $\mathrm{\Delta }{H}_{\mathrm{ads}.}^{\mathrm{o}}$: standard enthalpy changes of adsorption (kJ mol−1) $\mathrm{\Delta }{S}_{\mathrm{ads}.}^{\mathrm{o}}$: standard entropy changes of adsorption (J mol−1 K−1)
14	$\mathrm{\Delta }{G}_{\mathrm{ads}.}^{\mathrm{o}}=\mathrm{\Delta }{H}_{\mathrm{ads}.}^{\mathrm{o}}-T\cdot \mathrm{\Delta }{S}_{\mathrm{ads}.}^{\mathrm{o}}$		$\mathrm{\Delta }{G}_{\mathrm{ads}.}^{\mathrm{o}}$: standard Gibbs free changes of adsorption (kJ mol−1)