© A.W.Marczewski 2002

A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

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Dubinin-Radushkevich equation Isotherms of Micropore Filling Theory

Characteristic micropore-filling concentration co is smaller than cs (concentration of saturated solution). Analogously, for vapours, po < ps.

Micropore - pore with diameter below ca. 2 nm (larger pores are mesopores, 2-50nm - IUPAC - or sometimes 2-100 nm). Large micropores are often called supemicropores.
The adsorption process in micropores is much stronger than on relatively flat surfaces of meso- and macropores. The adsorbate molecule in micropore is closely surrounded by pore walls. Due to force - distance characteristics of adsorption forces (for dispersive attraction forces of a molecule near flat solid surface, F(r) ∝ 1/r3 ) in small pores the adsorbate interacts with larger number of solid wall atoms, as compared to larger pores, where closeness to one of walls means much weaker interactions with other walls.

Potential curves for DR The Micropore Volume Filling Theory was proposed by Dubinin-Radushkevich and co-workers for adsorption from gas phase. It is related to Eucken/Polany'i potential theory (characteristic adsorption potential curve a = f(A) , where a is adsorption and A = RT ln(ps/p) is adsorption potential, and A=0 at saturation point; analogously, in solutions: A = RT ln(cs/c) ).
The DR isotherm describes adsorption on a single type of uniform pores (in this respect the DR isotherm is an analogue of Langmuir-like local isotherms in adsorption on energetically heterogeneous solids). In the case of simple systems, experimental isotherms may be described by e.g. bi-DR isotherm - corresponding to bimodal micropore system. This theory was later extended by Stoeckli in a manner analogous to the global integral equation used in adsorption on energetically heterogeneous solids, allowing it to describe a continuous distribution of pore sizes (or generally pore characteristics).

F - Freundlich (DA with n=1):
Freundlich eq.
or in a more typical form:
Freundlich eq. - classic form

DR - Dubinin-Radushkevich (DA with n=2):
Dubinin-Radushkevich equation
It is very often used in log-linear form

DA - Dubinin-Astakhov:
Dubinin-Astakhov eq.

Exponential Jaroniec equation:
(A kind of virial equation for adsorption on microporous solids - F, DR and DA with integer exponent n values may be treated as special cases of this equation)
Exponential Jaroniec eq.

Stoeckli integral equation:
(Stoeckli equation describes adsorption in non-uniform micropores with distribution given by F(B) and B being a function of micropore size, B(r) )
Stoeckli integral 
equation for micropore filling
where usually local isotherm equation is Dubinin-Radushkevich eq., i.e. isotherm describing adsorption in spherical micropores of fixed pore size.

bi-DR (aka. Dubinin-Izotova) equation:
Bi-DR (aka. Dubinin-Izotova) equation may be considered a particular solution of Stoeckli equation with bimodal discrete pore distribution and adsorption on each pore type described by the Dubinin-Radushkevich equation:
bi-DR equation

DS (Dubinin-Stoeckli) equation:
Dubinin-Stoeckli equation is a solution of Stoeckli equation for a gaussian pore distribution and adsorption on each pore type described by the Dubinin-Radushkevich equation:

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