Micropore volume filling

V is the micropore volume filled with the adsorbed phase. 

On the other hand, it is impossible to apply the SP method to the correct description of gas adsorption in the micropores, since the adsorption in the micropores does not occur by multilayer adsorption but by micropore volume filling process. In this case, the pore fractal dimension gives a physical importance for the description of structural heterogeneity of the microporous solids. Terzyk et al.143"149 have intensively investigated the pore fractal characteristics of the microporous materials using gas adsorption isotherms theoretically simulated. 

The theory of microporous volume filling, worked out by Dubinin, is widely used for quantitative characteristic of adsorptive properties and basic varieties of porous structure. Equilibrium state equation (3) includes the saturation pressure Psal. Since the hydrogen sorption isotherms are measured within the temperature and pressure intervals comprising the regions of supercritical states of the adsorptive (Tcr — 33.24 K Pcr— 1.298 MPa), the notion of saturation pressure... 

The Dubinin-Radushkevieh (DR) equation is usually applied to describe the physical adsorption of organic vapors on microporous adsorbents. It is based on the micropore volume-filling theory and the Polanyi concept of adsorption potential. The DR equation can be expressed as... 

A similar technique is based on the theory of micropore volume filling. It states that the total microporous volume accessible to a given adsorbate can be obtained from the Dubinin-Radushkevich equation as a function of the temperature, relative pressure, and characteristic energy of adsorption. When this procedure is applied to a few linear or spherical molecules (as probes) of different but known sizes, the adsorption isotherms of these gases at the same temperature can be employed in combination with their... 

All the empirical equations dealt with in Section 3.2 are for adsorption with monolayer coverage, with the exception of the Freundlich isotherm, which does not have a finite saturation capacity and the DR equation, which is applicable for micropore volume filling. In the adsorption of sub-critical adsorbates, molecules first adsorb onto the solid surface as a layering process, and when the pressure is sufficiently high (about 0.1 of the relative pressure) multiple layers are formed. Brunauer, Emmett and Teller are the first to develop a theory to account for this multilayer adsorption, and the range of validity of this theory is approximately between 0.05 and 0.35 times the vapor pressure. In this section we will discuss this important theory and its various versions modified by a number of workers since the publication of the BET theory in 1938. Despite the many versions, the BET equation still remains the most important equation for the characterization of mesoporous solids, mainly due to its simplicity. 

Figure 5.13. Typical a, plots A, wide-pored gel with unrestricted monolayer-multilayer adsorption but no capillary adsorption 5, condensation in mesopores where curve deviates from A C, micropore adsorption micropore volume filled at Vf. a, m x/x where x, amount adsorbed at p/p - 0.40 and x - amount adkirbed at other pressures.

Stoeckli and Krachenbuehl (1981) proposed an alternative approach (more developed) to the interpretation of immersion enthalpy data, which is based on the Dubinin model of micropore volume filling. They obtained a mathematical relationship between the enthalpy of immersion of a microporous solid into a liquid whose vapor is adsorbed according to the DR equation and the parameters of the DR equation (Equation (4.12)). 

Activated carbons are adsorbents with a wide pore size distribution and consequently the precise determination of their porous structure is a rather difficult task. Since activated carbons are es.sentially microporous, most work devoted to their characterization is centred around the determination of the microporosity. Several methods have been extensively used to analyze the adsorption isotherms of nitrogen and other ad.sorptives. The micropore volume filling theory of Dubinin has been successfully used but there are well-known problems when the micropore size distribution is heterogeneous (refs. 1,2). The n-nonane preadsorption technique has also been used in the last few years but it provides information only on narrow microporosity and the results are conditioned by interconnectivity network of the porosity (refs. 3,4). The t- and a-plot methods have also been widely applied using different non-porous reference materials, the selection of which may be critical (refs. 5,6). 

According to Dubinin s ideas, the process involved is volume filling of the micropores rather than layer-by-layer adsorption on the pore walls. A second parameter is therefore the degree of filling of the micropores, defined by... 

If a Type I isotherm exhibits a nearly constant adsorption at high relative pressure, the micropore volume is given by the amount adsorbed (converted to a liquid volume) in the plateau region, since the mesopore volume and the external surface are both relatively small. In the more usual case where the Type I isotherm has a finite slope at high relative pressures, both the external area and the micropore volume can be evaluated by the a,-method provided that a standard isotherm on a suitable non-porous reference solid is available. Alternatively, the nonane pre-adsorption method may be used in appropriate cases to separate the processes of micropore filling and surface coverage. At present, however, there is no reliable procedure for the computation of micropore size distribution from a single isotherm but if the size extends down to micropores of molecular dimensions, adsorptive molecules of selected size can be employed as molecular probes. 

In very small pores the molecules never escape from the force field of the pore wall even at the center of the pore. In this situation the concepts of monolayer and multilayer sorption become blurred and it is more useful to consider adsorption simply as pore filling. The molecular volume in the adsorbed phase is similar to that of the saturated Hquid sorbate, so a rough estimate of the saturation capacity can be obtained simply from the quotient of the specific micropore volume and the molar volume of the saturated Hquid. 

From the above data, it would appear that methane densities in pores with carbon surfaces are higher than those of other materials. In the previous section it was pointed out that to maximize natural gas or methane storage, it is necessary to maximize micropore volume, not per unit mass of adsorbent, but per unit volume of storage vessel. Moreover, a porous carbon filled vessel will store and deliver more methane than a vessel filled wnth a siliea based or polymer adsorbent which has an equivalent micropore volume fraction of the storage vessel. 

The volume of the solid phase Vp is usually measured by a pycnometric technique, which measures the excluded volume of a pycnometric fluid, whose molecules cannot penetrate the solid phase of PS. A simple example of a pycnometric fluid is helium [55], The pycnometric fluid fill in all void space (pores) accessible to it, and presumably do not adsorb on the surface of PS. In the case of microporous PSs, measurement of the volume accessible for guests with various sizes allows the determination of a distribution of micropores volume vs. the characteristic size of guest molecules. This approach lays the basis of the method of molecular probes. The essence of this method is in the following we have a series of probe molecules with different mean sizes (dl>d2>d3>---). The pycnometric measurements of the excluded volume will give a series The difference A V=Vpi-Vpi(i>j) corresponds to the volume of micropores with pycnometric sizes of d in a range dt[Pg.283]

Virial Isotherm Equation. No isotherm equation based on idealized physical models provides a generally valid description of experimental isotherms in gas-zeolite systems (19). Instead (6, 20, 21, 22) one may make the assumption that in any gas-sorbent mixture the sorbed fluid exerts a surface pressure (adsorption thermodynamics), a mean hydrostatic stress intensity, Ps (volume filling of micropores), or that there is an osmotic pressure, w (solution thermodynamics) and that these pressures are related to the appropriate concentrations, C, by equations of polynomial (virial) form, illustrated by Equation 3 for osmotic pressure ... 

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