Dubinin- Radushkevich

$q_e=q_s\exp \left(-\beta {\epsilon }^2\right)$

$\ln q_e=\ln q_s-\beta {\epsilon }^2$

$\epsilon =RT\ln \left(1+\frac{1}{C_e}\right)\&E=\frac{1}{\sqrt{2\beta }}$

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 K_β = Dubinin-Radushkevich isotherm constant (mol²/kJ²), ε = Dubinin?Radushkevich isotherm constant and q_s is saturation capacity (mg/g)

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Temkin

$q_e=\frac{RT}{b}\left(k_T{c}_e\right)$

$q_e=\frac{RT}{b}\ln k_T+\frac{RT}{b}\ln c_e$

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 k_T (L/g)―Temkin isotherm equilibrium binding constant

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Flory-Huggins

$\ln \left(\frac{\theta }{C_o}\right)=n\ln \left(1-\theta \right)+\ln k_{FH}$

$\Delta G^o=-RT\ln \left(k_{FH}\right)$

$\ln \left(\frac{1000\ast q_e}{C_e}\right)=\frac{\Delta s^o}{R}-\frac{\Delta H^o}{RT}$

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 K_FH, where, n is number of adsorbates occupying adsorption sites, and K_FH is Flory-Huggins equilibrium constant (L・mol⁻¹)

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Hill-de Boer

$C_e=\frac{\theta }{\left(1-\theta \right)}\exp \left(\frac{\theta }{\left(1-\theta \right)}-\frac{k_2\theta }{RT}\right)$

$\ln \left[\frac{C_e\left(1-\theta \right)}{\left(1-\theta \right)}\right]-\frac{\theta }{\left(1-\theta \right)}=-\ln k_1-\left(\frac{k_2\theta }{RT}\right)$

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

[47]

Halsey

$q_e=\frac{1}{n_H}{I}_n{k}_H-\frac{1}{n_H}\ln c_{qe}$

o Multilayer adsorption at a relatively large distance from the surface, where K_H and n are constants

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Harkin-Jura

$\frac{1}{q_e^2}=\frac{B}{A}-\left(\frac{1}{A}\right)\log c_e$

o Multilayer adsorption having heterogeneous pore distribution, where B and A are constants

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Jovanovic

$q_e=\ln q_{\max }-KJ{C}_e$

o Assumes Langmuir plus mechanical contacts b/n adsorbate and adsorbent

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Elovich

$\frac{q_e}{q_m}=K_E{C}_e\exp \left(-\frac{q_e}{q_m}\right)$ Math_32#

o Define the kinetics of chemisorption and Multilayer adsorption, where K_E is equilibrium constant (L・mg⁻¹) and q_m is maximum adsorption capacity (mg・g⁻¹)

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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 K_n―constant for formation of complex between adsorbed molecules

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Three-Parameter Isotherms

Redlich-Peterson

$q_e=\frac{A{C}_e}{1+B{c}_e^{\beta }}$

$\ln \frac{c_e}{q_e}=\beta \ln C_e-\ln A$

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

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$q_e=\frac{A}{B}{c}_e^{1-\beta }$

$A/B=K_F$ and $\left(1-\beta \right)=1/n$

Sips

$q_e=\frac{K_s{c}_e^{{\beta }_s}}{1-a_s{c}_e^{{\beta }_s}}$

${\beta }_s\ln c_e=-\ln \left(\frac{k_s}{q_e}\right)+\ln \left(a_s\right)$

o A combination of the Langmuir and Freundlich isotherms, where a_s & K_s are isotherm constants and β_s is isotherm exponent. At low adsorbate concentration reduces to the Freundlich model and at high concentration it predicts the Langmuir model

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