Table 4 Several examined isotherm models for adsorptive sequestration of As(V) by Bi-ene-QD

Langmuir

Ce/qe = (1/qmax) k₁ + Ce/qmax b

RL = 1/1 + b C°

C_o and C_e the initial and equilibrium concentrations (mg L⁻¹), respectively. q_e is the magnitude of abstracted Arsenate at equilibrium (mg g⁻¹). q_max (mg g⁻¹) the maximum abstracted arsenate which is used to determine the abstraction energy and the abstraction efficiency besides b (L mg⁻¹) is Langmuir constants

C_e/q_e versus C_e

Freundlich

lnqe = ln k_f + 1/n ln Ce

q_e is the magnitude of up taken Arsenate which is corresponding to the equilibrium concentration of Arsenate in solution and C_e is the equilibrium Arsenate concentration. K_F (mg g⁻¹) is Freundlich constant, n is the intensity of the adsorbent

ln q_e versus ln C_e

Temkin

q_e = (RT/b_T) ln a_T + (RT/b_T) ln c_e

q_e = B ln a_T + B ln C_e

B = \(\frac{\mathbf{R}\mathbf{T}}{\mathbf{b}\mathbf{t}}\)

b_T (mg L⁻¹) is the Temkin isotherm constant, a_T (L g⁻¹) is the Temkin isotherm equilibrium binding constant and B is constant represents the heat of arsenate abstraction reaction (J/mol)

q_e versus ln C_e

Redlich-Peterson isotherm

\({\varvec{l}}{\varvec{n}}\left( {{\varvec{K}}}_{{\varvec{R}}{\varvec{P}} } \frac{{{\varvec{C}}}_{{\varvec{e}}}}{{{\varvec{q}}}_{{\varvec{e}}}}-1\right)={\varvec{g}}{\varvec{l}}{\varvec{n}}{{\varvec{c}}}_{{\varvec{e}}}+{\varvec{l}}{\varvec{n}}{{\varvec{a}}}_{{\varvec{R}}{\varvec{P}}}\)

C_e is the initial and equilibrium concentrations (mg.L⁻¹), respectively. q_e is the magnitude of abstracted arsenate at equilibrium (mg g⁻¹) as a_RP (Lg.mg^−g) and K_RP(L.g⁻¹) are constants

ln c_e/q_e versus ln c_e

Dubinin–Radushkevich (D–R)

ln q_e = ln q_s – (K_ad ε²)

ε = RT ln (1 + 1/Ce)

Es = \(\frac{1}{\sqrt{2 \mathbf{K}\mathbf{a}\mathbf{d}}}\)

K_ad the D–R isotherm constant which is related to the abstraction free energy per mole of the arsenate (mol²/kJ²). q_s (mg g⁻¹) is the theoretical fullness capacity ε is the Polanyi potential which is based on equilibrium, R is the universal gas constant (8.314J/mol K⁻¹) and T absolute temp. Kelvin

ln q_e versus ε²