Table 1.

Equilibrium modelling equations.

Langmuir Isotherm	Plot
Non-linear form	$Q_e$ = $\frac{Q_m K_L{C}_e}{1+K_L{C}_e}$
Type-I	$\frac{C_e}{Q_e}=\left(\frac{1}{Q_m}\right) C_e$ + $\left(\frac{1}{b_L{Q}_m}\right)$   $\frac{C_e}{Q_e} Vs C_e$
Type-II	$\frac{1}{Qe}=\left(\frac{1}{b_{L Q_m}}\right) \frac{1}{C_e}$ + $\frac{1}{Q_m}$   $\frac{1}{Q_e} Vs \frac{1}{C_e}$
Type-III	$Q_e=Q_m-\left(\frac{1}{b_L}\right) \frac{Q_e}{C_e}$    $Q_e Vs \frac{Q_e}{C_e}$
Freundlich Isotherm
Non-linear form	$Q_e$ = $K_F$ $C_e$1/n
Linear form	ln ($Q_e)$ = ln ($K_F$) + $\left(\frac{1}{n}\right)$ ln ($C_e$)    ln ($Q_e) Vs\ln \left(C_e\right)$
Adsorption kinetics
(i) Pseudo-first order	dqt/dt = K1 (qe − qt)
(ii) Pseudo-second order	dqt/dt = K (qe − qt) 2
Thermodynamics
	KC = CA/Ce
	ΔG° = ΔH° − TΔS°
	ΔG° = −RTlinKc

Qm (mg/g) is the saturated monolayer adsorption capacity; bL (L/mg) is the constant related to the energy of sorption; Ce (mg/L) and Qe (mg/g) are the equilibrium liquid phase concentrations and amount of solute adsorbed at equilibrium, respectively; K_F is the constant related to adsorption capacity; n is the constant related to the adsorption intensity or degree of favourability of adsorption. qe and qt are the sorption capacity at equilibrium and at time t, K₁ and K is the rate constant of pseudo-first order and -second-order constants. Kc is the constant of equilibrium, CA is the concentration of solid phase, Ce is the equilibrium concentration. ΔG° is the Gibbs free energy, ΔH° is the enthalpy change, ΔS° is the entropy change. T(K) is the absolute temperature, R is the gas constant (8.314 J/mol K).