Toth	$q_{e} = \frac{q_{m k_{L}} c_{e}}{{[1 + {(k_{L} c_{e})}^{n}]}^{1 / n}}$ $\frac{q_{e}}{q_{m}} = θ = \frac{k_{L} c_{e}}{{[1 + {(k_{L} c_{e})}^{n}]}^{1 / n}}$ $\ln \frac{q_{e}^{n}}{q_{m}^{n} - q_{e}^{n}} = n \ln k_{L} + n \ln c_{e}$		o Modification of the Langmuir equation and describes heterogeneous systems which satisfy both low and high end boundary of adsorbate concentration, both K_L & n is isotherm constant and when n = 1 reduces to Langmuir & n far from 1 shows heterogeneity, this where evaluated by nonlinear curve fitting method using sigma plot software	[56]
Koble-Carrigan	$q_{e} = \frac{A_{k} C_{e}^{p}}{1 + C_{e}^{p} B_{k}}$ $\frac{1}{q_{e}} = (\frac{1}{A_{k} C_{e}^{p}}) + \frac{B_{k}}{A_{k}}$		o A combination of both Langmuir and Freundlich isotherms and all A_k, B_k and p are isotherm constant. Solver add-in function of the Microsoft Excel and valid only when p ≥ 1	[56]
Kahn	$q_{e} = \frac{q_{\max} b_{k} C_{e}}{{(1 + b_{k} c_{e})}^{a_{k}}}$		o Used for adsorption of bi-solute sorption in dilute solution, where, 𝑎_𝑘 is isotherm exponent and b_k is isotherm constant	[57]
Radke-Prausniiz	$q_{e} = \frac{q_{M R P} K_{R P} C_{e}}{{(1 + K_{R P} C_{e})}^{M R P}}$		o Chose at low concentrations, where q_MRP = q_max, K_RP is equilibrium and constant and MRP is exponent o Reduces to linear at low concentration, at high [ ] reduces to Freundlich and When MRP = 0 becomes Langmuir isotherm model	[56]
Langmuir- Freundlich	$q_{e} = \frac{q_{M L F} {(K_{L F} C_{e})}^{n}}{1 + {(K_{L F} C_{e})}^{n}}$		o At low concentration becomes Freundlich and at high becomes the Langmuir isotherm model, where, q_MLF = q_max, K_LF is constant for heterogeneous solid; n is heterogeneity index lies b/n 0 and 1.
Jossens	$C_{e} = \frac{q_{e}}{H} \exp (F q_{e}^{p})$ $\ln (\frac{C_{e}}{q_{e}}) = - \ln (H) + F q_{e}^{p}$		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		$q_{e} = \frac{q_{m F S} K_{F S} C_{e}}{1 + q_{m} C_{e}^{M F S}}$	o Due to large number of coefficients, makes it to fit a wide range of experimental results, where, q_mFS = q_max, K_FS is equilibrium constant and MFS is exponent. And if MFS = 1 it becomes the Langmuir and high concentrations reduces to Freundlich, The constants are evaluated by nonlinear regression analysis.	[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_m_l) by measurement of tangents at different equilibrium concentrations shows that they are not constants in a broad range, where b_o is equilibrium constant and x is & Y is parameters	[17] [59]
Weber-Van Vliet		$c_{e} = p_{1} q_{e}^{(p_{2} q_{e}^{p_{3}} + p_{4})}$	o Describe wide range of adsorption systems, where p₁, p₂, p₃, & p₄ are isotherm parameters which defined by multiple nonlinear curve fitting method.	[42]
Marczewski-Jaroniec		$q_{e} = q_{M M J} {[\frac{{(k_{M J} C_{e})}^{n_{M J}}}{1 + {(k_{M J} C_{E})}^{n_{M J}}}]}^{\frac{M_{M J}}{n_{M J}}}$	o General Langmuir equation, where n_MJ and M_MJ are parameters characterize the Tells the Heterogeneity of the surface, where M_MJ describes the spreading of distribution in the path of higher adsorption energy, and 𝑛_𝑀𝐽 lesser adsorption energies
Five-Parameter Isotherms
Fritz and Schlunder	$q_{e} = \frac{q_{m} F S_{s} K_{1} C_{e}^{α_{F S}}}{1 + k_{2} C_{e}^{β_{F S}}}$		o Define more wide range of adsorption systems, where q_mFS = q_max and K₁, K₂, α_FS & β_FS are parameter constants. And this model approaches Langmuir model when the value of α_FS and β_FS equals 1; for higher concentrations it reduces to Freundlich model.	[60]