Table 5 Different evaluated isotherm models for biosorptive removal of BF dye by Gel@GO-F-ZrSiO₄@Gel.

Langmuir

C_e/q_e = (1/q_max) k₁ + C_e/q_max b

R_L = 1/1 + b C_o

C_o and C_e is for to the initial and equilibrium concentrations (mg/L), respectively. q_e is the amount of eliminated dye at equilibrium(mg g⁻¹). q_max (mg g⁻¹) the maximum elimination efficiency which is used to investigation elimination energy and elimination capacity and b (L mg⁻¹) is Langmuir constants

Ce/qe versus Ce

Freundlich

lnq_e = ln k_f + 1/n ln C_e

q_e is the amount of eliminated dye which is connected to the

equilibrium concentration of dye in solution and C_e is the equilibrium dye concentration. K_F (mg g⁻¹) is Freundlich constant, n is the intensity of the adsorbent

log qe versus log Ce

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 = RT/b_T

b_T (mg/L) is the Temkin isotherm constant, a_T (L/g) is the Temkin isotherm equilibrium binding constant and B is constant represents the heat of elimination (J/mol)

q_e versus ln C_e

Dubinin–Radushkevich (D–R)

ln qe = lnqs − (Kad Ɛ²)

ε = RT ln (1 + 1/Ce)

Es = \(\frac{1}{\sqrt{2Kad}}\)

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

ln qe versus ε²