Dubinin- Radushkevich ${q}_{e}={q}_{s}\mathrm{exp}\left(-\beta {\epsilon }^{2}\right)$ $\mathrm{ln}{q}_{e}=\mathrm{ln}{q}_{s}-\beta {\epsilon }^{2}$ $\epsilon =RT\mathrm{ln}\left(1+\frac{1}{{C}_{e}}\right)&E=\frac{1}{\sqrt{2\beta }}$ o Fit for intermediate range of adsorbate concentrations. Used to calculate energy (E) and physical, if the value is less than 8 and chemical if the E values are in between 8 and 16 kJ・mol−1 Kβ = Dubinin-Radushkevich isotherm constant (mol2/kJ2), ε = Dubinin?Radushkevich isotherm constant and qs is saturation capacity (mg/g)  Temkin ${q}_{e}=\frac{RT}{b}\left({k}_{T}{c}_{e}\right)$ ${q}_{e}=\frac{RT}{b}\mathrm{ln}{k}_{T}+\frac{RT}{b}\mathrm{ln}{c}_{e}$ o Effective only for an intermediate range of adsorbate concentrations and gives information for adsorbate/adsorbate interactions, where b (J/mol) is Temkin isotherm constant kT (L/g)―Temkin isotherm equilibrium binding constant  Flory-Huggins $\mathrm{ln}\left(\frac{\theta }{{C}_{o}}\right)=n\mathrm{ln}\left(1-\theta \right)+\mathrm{ln}{k}_{FH}$ $\Delta {G}^{o}=-RT\mathrm{ln}\left({k}_{FH}\right)$ $\mathrm{ln}\left(\frac{1000\ast {q}_{e}}{{C}_{e}}\right)=\frac{\Delta {s}^{o}}{R}-\frac{\Delta {H}^{o}}{RT}$ o Account for the characteristic surface coverage of the adsorbed adsorbate on the adsorbent and the spontaneity of the process using $\Delta {G}^{o}$ value obtained from KFH, where, n is number of adsorbates occupying adsorption sites, and KFH is Flory-Huggins equilibrium constant (L・mol−1)  Hill-de Boer ${C}_{e}=\frac{\theta }{\left(1-\theta \right)}\mathrm{exp}\left(\frac{\theta }{\left(1-\theta \right)}-\frac{{k}_{2}\theta }{RT}\right)$ $\mathrm{ln}\left[\frac{{C}_{e}\left(1-\theta \right)}{\left(1-\theta \right)}\right]-\frac{\theta }{\left(1-\theta \right)}=-\mathrm{ln}{k}_{1}-\left(\frac{{k}_{2}\theta }{RT}\right)$ o Defines the case a mobile adsorption and later interaction among adsorbed molecules, where K1 is constant (L・mg−1) and K2 is the energetic constant of the interaction between adsorbed molecules (kJ・mol−1), A positive K2 means attraction and negative value means repulsion between adsorbed species  Halsey ${q}_{e}=\frac{1}{{n}_{H}}{I}_{n}{k}_{H}-\frac{1}{{n}_{H}}\mathrm{ln}{c}_{qe}$ o Multilayer adsorption at a relatively large distance from the surface, where KH and n are constants  Harkin-Jura $\frac{1}{{q}_{e}^{2}}=\frac{B}{A}-\left(\frac{1}{A}\right)\mathrm{log}{c}_{e}$ o Multilayer adsorption having heterogeneous pore distribution, where B and A are constants  Jovanovic ${q}_{e}=\mathrm{ln}{q}_{\mathrm{max}}-KJ{C}_{e}$ o Assumes Langmuir plus mechanical contacts b/n adsorbate and adsorbent  Elovich $\frac{{q}_{e}}{{q}_{m}}={K}_{E}{C}_{e}\mathrm{exp}\left(-\frac{{q}_{e}}{{q}_{m}}\right)$ Math_32# o Define the kinetics of chemisorption and Multilayer adsorption, where KE is equilibrium constant (L・mg−1) and qm is maximum adsorption capacity (mg・g−1)  Kiselev ${K}_{1}{C}_{e}=\frac{\theta }{\left(1-\theta \right)\left(1+{k}_{n}\theta \right)}$ $\frac{1}{{C}_{e}\left(1-\theta \right)}=\frac{{k}_{1}}{\theta }+{k}_{i}{k}_{n}$ o Localized monomolecular layer, valid only when θ > 0.68 and Kn―constant for formation of complex between adsorbed molecules  Three-Parameter Isotherms Redlich-Peterson ${q}_{e}=\frac{A{C}_{e}}{1+B{c}_{e}^{\beta }}$ $\mathrm{ln}\frac{{c}_{e}}{{q}_{e}}=\beta \mathrm{ln}{C}_{e}-\mathrm{ln}A$ o A mixture of the Langmuir and Freundlich isotherms (the mechanism of adsorption is a blend of the two) and applicable both for homogeneous or heterogeneous systems. A & B are RP isotherm constants (L・g−1) and β is exponent lies b/n 0 and 1 for heterogeneous adsorption system  ${q}_{e}=\frac{A}{B}{c}_{e}^{1-\beta }$ $A/B={K}_{F}$ and $\left(1-\beta \right)=1/n$ Sips ${q}_{e}=\frac{{K}_{s}{c}_{e}^{{\beta }_{s}}}{1-{a}_{s}{c}_{e}^{{\beta }_{s}}}$ ${\beta }_{s}\mathrm{ln}{c}_{e}=-\mathrm{ln}\left(\frac{{k}_{s}}{{q}_{e}}\right)+\mathrm{ln}\left({a}_{s}\right)$ o A combination of the Langmuir and Freundlich isotherms, where as & Ks are isotherm constants and βs is isotherm exponent. At low adsorbate concentration reduces to the Freundlich model and at high concentration it predicts the Langmuir model