Isotherm Model

Linear equation

Plot

Significance

Langmuir

$\frac{C_e}{q_e}=\frac{1}{K_L{q}_m}+\frac{1}{q_m}{C}_e$ (4)

$\frac{C_e}{q_e}$ against C_e

1) Uniform adsorption energies

2) Monolayer formation.

3) Explains the relationship between the adsorbate concentration and the number of surface-active sites that is being adsorbed [33] [34] .

Freudlich

$\log q_e=\frac{1}{n}\log C_e+\log K_f$ (5)

$\log q_e$ against $\log C_e$

1) Surface heterogeneity and exponential distribution of the surface-active sites and their energies [35] [36] .

Temkin

$q_e=\frac{RT}{b}\ln A+\frac{RT}{b}\ln C_e$ (6)

$q_e=B\ln A+B\ln C_e$ (7)

where $B=\frac{RT}{b}$

$q_e$ against $\log C_e$

1) The heat of adsorption is a linear function (and not a logarithmic function) that decreases as a result of an increase in the surface coverage [37] .

2) There is an even distribution of binding energy in the course of adsorption [34] .

Dubnin-Radushkevich (D-R)

$\ln q_e=\ln q_m-\beta {\epsilon }^2$ (8)

where $\epsilon =RT\ln \left(1+\frac{1}{C_e}\right)$ (9)

$\ln q_e$ against ${\epsilon }^2$

1) Adsorption in micropores occurs by filling of pores and not by a layer-by-layer film formation in the pore walls (Polanyi potential theory of adsorption mechanism) [38] [39] .

2) It explains the Gaussian distribution of energy on the surface and also used to distinguish between physical and chemical adsorption of ions.