q e = q m k L c e [ 1 + ( k L c e ) n ] 1 / n

q e q m = θ = k L c e [ 1 + ( k L c e ) n ] 1 / n

ln 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 KL & 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



q e = A k C e p 1 + C e p B k

1 q e = ( 1 A k C e p ) + B k A k

o A combination of both Langmuir and Freundlich isotherms and all Ak, Bk and p are isotherm constant. Solver add-in function of the Microsoft Excel and valid only when p ≥ 1



q e = 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, 𝑎𝑘 is isotherm exponent and bk is isotherm constant



q e = q M R P K R P C e ( 1 + K R P C e ) M R P

o Chose at low concentrations, where qMRP = qmax, KRP 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


Langmuir- Freundlich

q e = 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, qMLF = qmax, KLF is constant for heterogeneous solid; n is heterogeneity index lies b/n 0 and 1.


C e = q e H exp ( F q e p )

ln ( 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.


Four-Parameter Isotherms


q e = 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, qmFS = qmax, KFS 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.



q e = 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 qml) by measurement of tangents at different equilibrium concentrations shows that they are not constants in a broad range, where bo 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.



q e = q M M J [ ( k M J C e ) n M J 1 + ( k M J C E ) n M J ] M M J n M J

o General Langmuir equation, where nMJ and MMJ are parameters characterize the Tells the Heterogeneity of the surface, where MMJ describes the spreading of distribution in the path of higher adsorption energy, and 𝑛𝑀𝐽 lesser adsorption energies

Five-Parameter Isotherms

Fritz and Schlunder

q e = 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 qmFS = qmax and K1, K2, αFS & βFS are parameter constants. And this model approaches Langmuir model when the value of αFS and βFS equals 1; for higher concentrations it reduces to Freundlich model.