Pore-Filling Isotherm Equations

Adsorption isotherms in the micropore region may start off looking like one of the high BET c-value curves of Fig. XVII-10, but will then level off much like a Langmuir isotherm (Fig. XVII-3) as the pores fill and the surface area available for further adsorption greatly diminishes. The BET-type equation for adsorption limited to n layers (Eq. XVII-65) will sometimes fit this type of behavior. Currently, however, more use is made of the Dubinin-Raduschkevich or DR equation. Tliis is Eq. XVII-75, but now put in the form... 

Flat Surface Isotherm Equations The classification of isotherm equations into two broad categories for flat surfaces and pore filling reflec ts their origin. It does not restrict equations developed for flat surfaces from being apphed successfully to describe data for porous adsorbents. 

The surface area S resulting from analyses of these isotherms by BET equation in the p/po range 0.05 to 0.2 and the mesoporous volume Vm, measured at the top of the pore-filling step are reported in Table 1. 

An example of the adsorption to one such material is shown in Fig. 9.16. The siliceous material, called MCM-41, contains cylindrical pores [397], With increasing pressure first a layer is adsorbed to the surface. Up until a pressure of P/Po 0.45 is reached, this could be described by a BET adsorption isotherm equation. Then capillary condensation sets in. At a pressure of P/Po 0.75, all pores are filled. This leads to a very much reduced accessible surface and practically to saturation. When reducing the pressure the pores remain filled until the pressure is reduced to P/Pq rs 0.6. The hysteresis between adsorption and desorption is obvious. At P/Po 0.45 all pores are empty and are only coated with roughly a monolayer. Adsorption and desorption isotherms are indistinguishable again below P/Po 0.45. 

In all examined cases, equation 2 provided a very good fit of the first part of the isotherm. As this equation represents a layer-by-layer adsorption rather than a pore filling mechanism, this strongly suggests a different adsorption behaviour for the very first part of the isotherms. The fact of Um being proportional to Wo shows this type of adsorption to be exclusively related to the micropore system. 

A novel approach is reported for the accurate evaluation of pore size distributions for mesoporous and microporous silicas from nitrogen adsorption data. The model used is a hybrid combination of statistical mechanical calculations and experimental observations for macroporous silicas and for MCM-41 ordered mesoporous silicas, which are regarded as the best model mesoporous solids currently available. Thus, an accurate reference isotherm has been developed from extensive experimental observations and surface heterogeneity analysis by density functional theory the critical pore filling pressures have been determined as a function of the pore size from adsorption isotherms on MCM-41 materials well characterized by independent X-ray techniques and finally, the important variation of the pore fluid density with pressure and pore size has been accounted for by density functional theory calculations. The pore size distribution for an unknown sample is extracted from its experimental nitrogen isotherm by inversion of the integral equation of adsorption using the hybrid models as the kernel matrix. The approach reported in the current study opens new opportunities in characterization of mesoporous and microporous-mesoporous materials. 

Specific surface areas (Sbet for investigated samples, were calculated from the linear form of the BET equation over the range of relative pressure between 0.05 and 0.4, taking the cross-sectional area of the nitrogen molecule to be 16.2 A. The pore size distributions were derived from the desorption isotherm using the BJH method [14]. The total pore volumes, V, were calculated from adsorption isotherms at p/Po 0-98 by assuming that complete pore filling by the condensate had occurred. 

The benzene adsorption/desorption data were used to analyze the porous structure of activated carbons. The BET specific surface area, Sbet, was estimated from the linear BET plot. The adsorption process in microporous materials is well described by the pore filling model. Taking into account the heterogeneity of micropore structure, a special form of Dubinin-Radushkevich equation, the two-term DR isotherm was applied [6,7] allowing for determination of micropore volumes and adsorption energies ... 

Classical thermodynamic models of adsorption based upon the Kelvin equation [21] and its modihed forms These models are constructed from a balance of mechanical forces at the interface between the liquid and the vapor phases in a pore filled with condensate and, again, presume a specihc pore shape. Tlie Kelvin-derived analysis methods generate model isotherms from a continuum-level interpretation of the adsorbate surface tension, rather than from the atomistic-level calculations of molecular interaction energies that are predominantly utihzed in the other categories. 

The Brunauer-Emmett-Teller (B.E.T.) and B.D.D.T. isotherms [d and e in Table 14.3] account for pore filling via multiple layers instead of just a monolayer, and they use C/Q, that tends toward unity as the pores are completely filled. The B.D.D.T. isotherm includes the number of layers explicitly (m), as well as a heat of adsorption term (q). The B.E.T. isotherm is mostly used to estimate surface areas, not for process calculations [see Equation (14.10)]. 

A pore size distribution (PSD) of a sample is a measure of the cumulative or differential pore volume as a function of pore diameter. PSDs can be calculated from adsorption isotherms based on an analysis which accounts for capillary condensation into pores. This analysis (14.16) uses a model of the pore structure combined with the Kelvin equation (12) to relate the pore size to the value of p/Po at which pore "filling" occurs. Due to limitations in this technique, only pores with diameters from about 3 to 50 nm, called mesopores (14), can be characterized. This pore size range, however, is typical of many porous samples of interest. For samples with pores smaller or larger than this range, alternative techniques, such as mercury intrusion for large pores (14.16), are typically more suitable. 

Sometimes the adsorption isotherm has been experimentally determined only for a certain temperature, for instance for the room temperature. The equation of Dubinin-Astakov can be used as a basis to extrapolate loadings to other temperatures when the relative pore filling v/v x is plotted against the adsorption potential e = RT %) since the characteristic energy e is constant for a certain adsorptive-adsorbent combination, see Fig. 2.4-5. The Henry coefficient of a certain component / depends on the temperature according to the equation of van t Hoff ... 

The method of Neimark [7, 8, 33, 34] for the determination of the surface fractal dimension of a microporous solid is based on the adsorption isotherm equation that was developed by Kiselev [35]. This equation relates the surface area of pores filled by the adsorbate S(x) to the amount of adsorbed molecules N(x) at a given relative pressure x = pf po. ... 

Both Equations (6.10) and (6.13) require some comments. If Equation (6.10) is used to analyze experimental data, then a problem often encountered is related to adsorption hysteresis in the mesopores. The results of calculations may depend on which branch of the isotherm (adsorption or desorption) is used as the basis of calculations. In such cases, one should preferentially employ the desorption branch in order to conform to the Kelvin theory of pore filling [8, 27, 46]. Nonetheless, values of the fractal dimension that are calculated from the adsorption and desorption isotherms should be comparable for those samples showing fractal properties over a sufficiently wide range of pore sizes [7, 8]. 

For pores having characteristic pressure less than P, the pores are filled with adsorbate at maximum density, while pores having characteristic energy P greater than P will have partial filling of which the density is given in eq. (6.6-15). Thus, the adsorption isotherm equation is ... 

An alternative view was suggested by Cohan who suggested that capillary condensation occurs along both adsorption and desorption branches of the isotherm, the difference being due to a difference in the shape of the meniscus. During adsorption the pore fills radially and a cylindrical meniscus is formed as sketched in Figure 2.12a. Under these conditions dv/ds = r-(rather than jl as assumed in the Kelvin equation) and with 0 = 0 ... 

Cranston and Inkley Method. Cranston and Inkley (39) used the known thick-. ness, /, of the Him of nitrogen on the inner walls of the pores, along with the diameter of pores filled by nitrogen according to the Kelvin equation, to develop a procedure for calculating the volume and size of pores from the desorption or adsorption isotherm. Use is made of the portion of the isotherm for p/p above 0.3 where at least a monomolecular layer of nitrogen is adsorbed. 

The gas adsorption-desorption technique relates to the adsorption of nitrogen (or, less commonly, carbon dioxide, argon, xenon, and krypton), at cryogenic temperatures, via adsorption and capillary condensation from the gas phase, with subsequent desorption occurring after complete pore filling. An adsorption-desorption isotherm is constructed based upon the relationship between the pressure of the adsorbate gas and the volume of gas adsorbed/desorbed. Computational analysis of the isotherms based on the BET (Brunauer-Emmett-Teller) (Brunauer et al. 1938) and/or BJH (Barrett-Joyner-Halenda) (Barrett et al. 1951) methods, underpinned by the classical Kelvin equation, facilitates the calculation of surface area, pore volume, average pore size, and pore size distribution. 

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