They may be proportional or not, it always depends on the activating agent used and experimental activating conditions. High surface areas are associated with high micorporous volumes but the evaluation of the pore widths requests the use of other equations like Dubinin-Raduskehvich equation, alpha-s or t method (for micropores). In order to obtain pore size distributions you can use DFT method to N2 adsorption data (micro and mesopores) or DRS equation to CO2 adsorption data (micropores).
No they are not linked. You can measure a high surface area on a non porous oxide: fumed silicas can have surface area as high as ca. 400 m2.g-1 without significant porosity, whereas an oxide can contains a few pores of small diameter and still show a very low surface area. The values that are roughly linked are the surface, the pore diameter and the pore volume, but there is not a general relation between them, it will depend, among other parameters, on the geometry of the pores : for exemple for a sample containing cylindral pores without an homogeneous pore diameter, the relation, that can be easily demonstrated, between pore diameter (d) and surface area (s) is d=4V/S where V is the pore volume) . But if the pore size distribution is broad and complex, this relation is not strictly true.
For an accurate determination of the pores size of a solid, you will need to record the full N2 adsorption and desorption isotherms (i.e. between P/P0=0.1 or less and P/P0=1 and back to at least P/P0=0.3) and apply models such as the BJH model to estimate the pore diameter from the pressure at which capillary condensation occurs (the models are usually implemented in the software of the measurement device).
You can also, using the relation given above for cylindrical pores, measure the surface area + the total pore volume (a "BET" measurement + one extra point at high P/P0, for example at 0.9, but not too close to P/P0=1, where other phenomena occurs that would lead to an overestimation of the pore volume). This will give you a rough estimate of the pore diameter.
@Juliette. I respect your detailed post on the subject. I post here my Letter to the Editor on the subject, where Capillary Pressure in Porous Media is derived from first principles of surface phenomena, using concept of Specific Surface. And I can do the same, using expression for mean radius in porous media, which is r = (K/fi)^(1/2), where K is permeability of porous media, fi is the porosity. So, not so many degrees of freedom remain in relating Sv and r.
Porosity and pore size distribution is measured by mercury porosimetry. Here mercury is intruded in to the pore by the application of the pressure. The basic principle is based on Wash burn equation, where pore diameter is inversely proportional to the applied pressure.
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