Toth | ${q}_{e}=\frac{{q}_{m{k}_{L}}{c}_{e}}{{\left[1+{\left({k}_{L}{c}_{e}\right)}^{n}\right]}^{1/n}}$ $\frac{{q}_{e}}{{q}_{m}}=\theta =\frac{{k}_{L}{c}_{e}}{{\left[1+{\left({k}_{L}{c}_{e}\right)}^{n}\right]}^{1/n}}$ $\mathrm{ln}\frac{{q}_{e}^{n}}{{q}_{m}^{n}-{q}_{e}^{n}}=n\mathrm{ln}{k}_{L}+n\mathrm{ln}{c}_{e}$ | o Modification of the Langmuir equation and describes heterogeneous systems which satisfy both low and high end boundary of adsorbate concentration, both K | [56] | |

Koble-Carrigan | ${q}_{e}=\frac{{A}_{k}{C}_{e}^{p}}{1+{C}_{e}^{p}{B}_{k}}$ $\frac{1}{{q}_{e}}=\left(\frac{1}{{A}_{k}{C}_{e}^{p}}\right)+\frac{{B}_{k}}{{A}_{k}}$ | o A combination of both Langmuir and Freundlich isotherms and all A | [56] | |

Kahn | ${q}_{e}=\frac{{q}_{\mathrm{max}}{b}_{k}{C}_{e}}{{\left(1+{b}_{k}{c}_{e}\right)}^{{a}_{k}}}$ | o Used for adsorption of bi-solute sorption in dilute solution, where, 𝑎 | [57] | |

Radke-Prausniiz | ${q}_{e}=\frac{{q}_{MRP}{K}_{RP}{C}_{e}}{{\left(1+{K}_{RP}{C}_{e}\right)}^{MRP}}$ | o Chose at low concentrations, where q o Reduces to linear at low concentration, at high [ ] reduces to Freundlich and When MRP = 0 becomes Langmuir isotherm model | [56] | |

Langmuir- Freundlich | ${q}_{e}=\frac{{q}_{MLF}{\left({K}_{LF}{C}_{e}\right)}^{n}}{1+{\left({K}_{LF}{C}_{e}\right)}^{n}}$ | o At low concentration becomes Freundlich and at high becomes the Langmuir isotherm model, where, q |
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Jossens | ${C}_{e}=\frac{{q}_{e}}{H}\mathrm{exp}\left(F{q}_{e}^{p}\right)$ $\mathrm{ln}\left(\frac{{C}_{e}}{{q}_{e}}\right)=-\mathrm{ln}\left(H\right)+F{q}_{e}^{p}$ | o Work based on energy distribution of adsorbate-adsorbent interactions at heterogeneous adsorption sites, where H (Henry’s), p & F are constants o p is characteristic of the adsorbent regardless of temperature and the nature. | [57] | |

Four-Parameter Isotherms | ||||

Fritz-Schlunder | ${q}_{e}=\frac{{q}_{mFS}{K}_{FS}{C}_{e}}{1+{q}_{m}{C}_{e}^{MFS}}$ | o Due to large number of coefficients, makes it to fit a wide range of experimental results, where, q | [58] | |

Bauder | ${q}_{e}=\frac{{q}_{m}{b}_{o}{C}_{e}^{1+x+y}}{1+{b}_{o}{C}_{e}^{1+x}}$ | o Used to estimate the Langmuir coefficients (b and q | [17] [59] | |

Weber-Van Vliet | ${c}_{e}={p}_{1}{q}_{e}^{\left({p}_{2}{q}_{e}^{{p}_{3}}+{p}_{4}\right)}$ | o Describe wide range of adsorption systems, where p | [42] | |

Marczewski-Jaroniec | ${q}_{e}={q}_{MMJ}{\left[\frac{{\left({k}_{MJ}{C}_{e}\right)}^{{n}_{MJ}}}{1+{\left({k}_{MJ}{C}_{E}\right)}^{{n}_{MJ}}}\right]}^{\frac{{M}_{MJ}}{{n}_{MJ}}}$ | o General Langmuir equation, where n | ||

Five-Parameter Isotherms | ||||

Fritz and Schlunder | ${q}_{e}=\frac{{q}_{m}F{S}_{s}{K}_{1}{C}_{e}^{{\alpha}_{FS}}}{1+{k}_{2}{C}_{e}^{{\beta}_{FS}}}$ | o Define more wide range of adsorption systems, where q | [60] | |