Dubinin- Radushkevich

q e = q s exp ( β ε 2 )

ln q e = ln q s β ε 2

ε = R T ln ( 1 + 1 C e ) & E = 1 2 β

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)



q e = R T b ( k T c e )

q e = R T b ln k T + R T 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 kT (L/g)―Temkin isotherm equilibrium binding constant



ln ( θ C o ) = n ln ( 1 θ ) + ln k F H

Δ G o = R T ln ( k F H )

ln ( 1000 q e C e ) = Δ s o R Δ H o R T

o Account for the characteristic surface coverage of the adsorbed adsorbate on the adsorbent and the spontaneity of the process using Δ 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 = θ ( 1 θ ) exp ( θ ( 1 θ ) k 2 θ R T )

ln [ C e ( 1 θ ) ( 1 θ ) ] θ ( 1 θ ) = ln k 1 ( k 2 θ R T )

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



q e = 1 n H I n k H 1 n H ln c q e

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



1 q e 2 = B A ( 1 A ) log c e

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



q e = ln q max K J C e

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



q e q m = K E C e exp ( q e q m ) 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)



K 1 C e = θ ( 1 θ ) ( 1 + k n θ )

1 C e ( 1 θ ) = k 1 θ + 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


q e = A C e 1 + B c e β

ln c e q e = β 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


q e = A B c e 1 β

A / B = K F and ( 1 β ) = 1 / n


q e = K s c e β s 1 a s c e β s

β s ln c e = ln ( k s q e ) + ln ( a s )

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