Table 2 Equations and variables of kinetics and isotherms models

Kinetic models

PFO

\(\ln (q_{{\text{e}}} - q_{{\text{t}}} ) = \ln q_{{\text{e}}} - k_{1} t\)

q_e (mg g⁻¹); Cd(II) ions adsorption at equilibrium, q_t (mg g⁻¹); Cd(II) retention at any time t, k₁(min⁻¹); rate constant

ln (q_e–q_t) versus t

k₁

q_e

PSO

\(\frac{t}{{q_{{\text{t}}} }} = \frac{1}{{k_{2} q_{{\text{e}}}^{2} }} + \frac{t}{{q_{{\text{e}}} }}\)

k₂ (g mg⁻¹ min⁻¹); rate constant

t/q_t versus t

q_e

k₂

Intraparticle diffusion model

\(q_{{\text{t}}} = k_{{{\text{id}}}} t^{1/2} + c\)

k_id (mg g⁻¹ min^−1/2); rate constant C; thickness of the boundary layer

q_t versus t^1/2

k_id

C

Isotherm models

Langmuir isotherm model

\(\frac{{C_{{\text{e}}} }}{{q_{{\text{e}}} }} = \frac{{C_{{\text{e}}} }}{{q_{\max } }} + \frac{1}{{bq_{\max } }}\)

C_e (mg L⁻¹); Cd(II) amount in solution at equilibrium, q_e (mg g⁻¹); amount of Cd(II) uptake by BTW, b (L mg⁻¹); Langmuir constant, q_max (mg g⁻¹); maximum adsorption capacity

C_e/q_e versus C_e

q_max

b

Freundlich isotherm model

\(\ln q_{{\text{e}}} = \ln K_{{\text{f}}} + \frac{1}{n}\ln C_{{\text{e}}}\)

K_f (mg g⁻¹) and n; Freundlich isotherm model constants

ln q_e versus ln C_e

n

K_f

D–R isotherm model

In q_e = In q_m-βε²

q_m (mg g⁻¹); monolayer adsorption capacity, β (kJ² mol⁻²); activity coefficient, ε; Polanyi potential

ln q_e versus ε²

β

q_m