Table 1 Isotherm and kinetic adsorption models

 Langmuir; Eq. (3)

\( {q}_e=\frac{q_m\ {K}_L\ {C}_e}{1+{K}_L\ {C}_e} \)

q_e: the equilibrium loading of Ni (II) (mg g^− 1), q_m: the maximum loading of Ni (II) per g of adsorbent (mg g^− 1), K_L: constant of Langmuir equilibrium for the affinity between the adsorbent and Ni (II) (L mg^− 1).

 Freundlich; Eq. (4)

\( {q}_e={K}_f\ {C}_e^n \)

K_f: constant of Freundluch for the adsorption strength ((mg g^− 1) (mg L^− 1)ⁿ), n: the adsorption intensity

 Temkin; Eq. (5)

q_e = B ln(K_T C_e)

B: the adsorption heat constant (J moL^− 1); B= \( \frac{R_T\ }{b} \), R: gas constant (8.314 J mol^− 1 K^− 1), T: temperature (K), and K_T: constant of Temkin binding (L mg^− 1).

 Separation factor; Eq. (6)

\( SF=\frac{1}{1+{K}_L{C}_0} \)

SF: Separation factor (dimenstionless), C_0: initial adsorbate concentration (mg L^− 1).

 pseudo-first-order; Eq. (7)

q_t = q_e (1 − exp(−k₁ t)

k₁: PFO constant rate (min^− 1), t: contact time (min).

 pseudo-second-order; Eq. (8)

\( {q}_t=\frac{q_e^2\ {k}_2\ t}{1+{k}_2{q}_e\ t} \)

k₂: PSO constant rate (g mg^− 1 min^− 1).