Table 4 Sorption isotherm models for MB uptake by AC_(H).

Langmuir

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

\(R_{L} = 1/\left( {1 + bC_{0} } \right)\)

\(R_{L}\) > 1 (unfavorable adsorption)

\(R_{L}\) = 1 (linear adsorption)

\(0 < R_{L}\) < 1 (favorable adsorption)

\(R_{L}\) = 0 (irreversible adsorption)

\({\text{q}}_{{\text{e}}} = \frac{{{\text{q}}_{\max } {\text{b C}}_{{\text{e}}} }}{{1 + {\text{b C}}_{{\text{e}}} }}\)

C_e (mg/L): equilibrium concentration of the residual MB in the solution

q_e (mg/g): removed amount of MB at equilibrium

q_max (mg/g): maximum adsorption capacity

b (L/mg): Langmuir constant

C₀: Initial MB concentration

R_L: Equilibrium parameter of Langmuir equation

Langmuir¹⁰⁷

Weber and Chakravorti³⁹

Freundlich

\(\log q_{e\,} = \,\,\log k_{F} + \,\frac{1}{n}\,\log \,C_{e}\)

\({\text{q}}_{{\text{e}}} { } = {\text{KF}}\left( {{\text{C}}_{{\text{e}}} 1/{\text{n}}} \right)\)

C_e (mg/L): equilibrium concentration of the residual MB in the solution

q_e (mg/g): removed amount of MB at equilibrium

k_F (mg/g): MB adsorption capacity

n: heterogeneity factor

Freundlich⁴⁰

Temkin

\({\text{q}}_{{\text{e}}} = {\text{B}}\ln {\text{A}} + {\text{B}}\ln {\text{C}}_{{\text{e}}}\)

\(B = \,\,RT/b\)

\({\text{q}}_{{\text{e}}} { } = {\text{B ln}}\left( {{\text{AC}}_{{\text{e}}} } \right)\)

\(B = \,\,RT/b\)

A (L/g): Temkin isotherm constant (the equilibrium binding constant corresponding to the maximum binding energy)

B (J/mol): Temkin constant related to heat of sorption

b: Temkin isotherm constant (slope)

R: The gas constant (8.314 J/mol K)

T : the absolute temperature at 298 K

Tempkin and Pyzhev⁴¹

Dubinin–Radushkevich (D–R)

Ln \({\text{q}}_{{\text{e}}}\) = Ln q_max − Kε²

ε = RTLn(1 + 1/C_e)

\({\text{q}}_{{\text{e}}} = {\text{q}}_{{{\text{max}}}} {\text{exp}}\left( { - {\text{K}}\upvarepsilon ^{2} } \right)\)

\(\upvarepsilon = {\text{RTln}}\left( {1 + \frac{1}{{{\text{C}}_{{\text{e}}} }}} \right)\)

\({\text{E}}_{{{\text{DR}}}} = \sqrt {\frac{1}{{2{\text{K}}_{{{\text{DR}}}} }}}\)

Dubinin and Radushkevich¹⁰⁶