# Theoretical basis for the Dubinin-Radushkevitch (D-R) adsorption isotherm equation

## Abstract

The Dubinin-Radushkevitch (D-R) equation, which was originally proposed as an empirical adaptation The Polanyi adsorption potential theory, has been the fundamental equation to quantitatively describe the adsorption gases and vapors by microporous sorbents. The equation, based on the postulate that the mechanism for adsorption in micropores is that of pore-filling rather than layer-by-layer surface coverage, generally applies well to adsorption systems involving only van der Waals forces and is especially useful to describe adsorption on activated ???. The ability of the D-R equation to describe gas adsorption on porous materials has inspired many to undertake studies, both experimental and theoretical, to explain the source of the success of the D-R equation in ??? of molecular properties at the gas-solid interface. In many cases, these studies have led to extensions or modifications of the original D-R equation. Many of these attempts and the resulting extensions are reviewed and discussed here. Recently, an isotherm equation was derived for adsorption of gases and vapors on microporous ??? from statistical mechanical principles. It was shown that the D-R equation is an approximated form of this potential theory isotherm. This development is also reviewed and discussed.

### Keywords

Dubinin-Radushkevitch equation Dubinin-Astakhov equation adsorption micropore adsorption### Nomenclature

*A*adsorption potential; differential molar work of adsorption; Helmholtz free energy

*B*parameter in D-R. D-A equations, related to

*E*_{0}*B*_{0}value of the parameter

*B*for the minimum of the micropore distribution curve in the D-R-S equation*E*characteristic energy of adsorption; energy

*E*_{0}characteristics energy of adsorption for standard vapor

*g*(*E*)energy distribution function

*G*(*x*)micropore size distribution

*h*Planck's constant

*k*Boltzmann's constant

*K*constant from approximation, Eq. (41)

*m*molecular weight

*n*parameter in D-A equation

*N*number of molecules in the system

*N*_{0}Avogadro's number

*P*pressure

*P*_{m}condensation pressure

*P*^{0}saturated vapor pressure

*q*parameter of the Gamma micropore distribution

*q*_{c}parameter of the Gamma micropore distribution for a reference vapor

*q*_{r}rotational partition function

*q*_{v}vibrational partition function

*R*gas constant

*r*_{k}position coordinate of molecule “

*k*”*T*temperature

*U*(*r*_{u})potential energy between molecules

*i*and*j**V*^{s}surface area

*W*volume of micropores filled at relative pressure

*P*/*P*^{0}*W*_{0}total volume of micropore system

*x*half-width of slit pore

*X*empirical constant relating characteristic energy to pore size

*Z*_{N}^{s}partition function of the canonical ensemble

### Greek etc. Symbols

*α*(*T*)a function of temperature in Eq. (35)

*β*similarity coefficient

- Δ
dispersion (or variance)

- Δ
*G*_{ads} differential free energy of adsorption

- ε
energy

- ε
_{0} minimum adsorption energy

*η*packing fraction of the surface

*η*_{0}packing fraction of the surface at

*P*=*P*^{0}- γ
l/

*kT*- Γ
gamma function

- λ
number of dimension

- Λ
de Broglie thermal wavelength

- ϕ
potential energy function

*ρ*^{N}number density of the solid phase

*ρ*^{g}number of density of the gas phase

*θ*fractional coverage (=

*W/W*_{0})*θ*_{1}overall isotherm for microporous system

*θ*_{1}local isotherm for microporous system

- χ(ε)
energy distribution

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### References

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