© A.W.Marczewski 2002

A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

Potential Curves for DR isotherm

**NOTE**

*Characteristic micropore-filling concentration c _{o} is smaller than c_{s} (concentration of saturated solution). Analogously, for vapours, p_{o} < p_{s}.*

For vapour adsorption, the adsorption potential

Analogously, for liquid adsorption, the adsorption potential is given by:

In a so-called "potential adsorption theory" of Eucken and Polany'i, it is assumed that in the space around each solid one may find some isopotential surfaces (for flat homogeneous surfaces they are planes) delimiting adsorbate which is adsorbed at pressures (or concentrations) lower than those corresponding to the potential value and that such isopotential surfaces are specific for a given solid surface. If a plot is made of adsorbed volume (amount of condensed adsorbate) *vs.* adsorption potential, then such "potential curve" is characteristic for a given system. At the same time, it was proved experimentarily, that for a given solid surface and a series of adsorbates and adsorption temperatures, their plots may be scaled in such a way, that all experimental points fit a single curve (in fact also for many different solids this also proved true). In such a case, a so called adsorption-limiting volume V_{s} (i.e. maximum adsorbed volume, or a volume adsorbed at A=0 - may be different for each solid/adsorbate system) is used to scale adsorbed volume.

**Legend.**
The course of Dubinin-Radushkevich (DR) isotherm corresponds to the typical shape of characteristic potential curve found for microporous solids. In the following pictures, the characteristic potential curves obtained for T = 293K and 3 different B_{2} values are presented ( B_{2} = 3e-9; 1e-8; 3e-8 - it corresponds to B_{2}(RT)^{2} = 0.0178; 0.0593; 0.178 or - if one prefers to use log_{10} scaling - B_{2}[ln(10) RT]^{2} = 0.0944; 0.315; 0.944 ).

Rotated and scaled model potential curves, adsorbate volume (V/V_{s}) *vs.* A, adsorption potential [J], calculated for Dubinin-Radushkevich isotherm, T = 293K and 3 values of B_{2} parameter (see legend).

Scaled model potential curves, A, adsorption potential [J] *vs.* adsorbate volume (V/V_{s}), calculated for Dubinin-Radushkevich isotherm, T = 293K and 3 values of B_{2} parameter (see legend).

Scaled model potential curves, A, adsorption potential [J] *vs.* adsorbate volume (V/V_{s}), calculated for Dubinin-Radushkevich isotherm, T = 293K and 3 values of B_{2} parameter (the same B_{2} as above; in the picture the values of B_{2}(RT)^{2} are shown) (see legend).

Linear Dubinin-Radushkevich plot: log_{10}(V) *vs.* [log_{10}(p_{s}/p)]^{2} , calculated for Dubinin-Radushkevich isotherm, T = 293K and 3 values of B_{2} parameter (in the picture the values of B_{2}(ln10 RT)^{2} are shown) (see legend). One has to be always aware which form of DR equation is used. In evaluation of experimental data this decimal logarithms are often used, however such obtained values of (slope is = B_{2}(ln10 RT)^{2} ) parameters are scaled when compared to natural logarithm plots (slope is = B_{2}(RT)^{2} ). Sometimes DR equation is defined in such a way, that the "convoluted" parameter B = B_{2}(RT)^{2} is used, however, it hides the fact that the structural parameter B_{2} is temperature independent.

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