Frenkel-Halsey-Hill isotherm

Tpjjjj = an empirical constant in the Frenkel-Halsey-Hill isotherm = 2 to 3 T = position of particle i... 

The intercept on the adsorption axis, and also the value of c, diminishes as the amount of retained nonane increases (Table 4.7). The very high value of c (>10 ) for the starting material could in principle be explained by adsorption either in micropores or on active sites such as exposed Ti cations produced by dehydration but, as shown in earlier work, the latter kind of adsorption would result in isotherms of quite different shape, and can be ruled out. The negative intercept obtained with the 25°C-outgassed sample (Fig. 4.14 curve (D)) is a mathematical consequence of the reduced adsorption at low relative pressure which in expressed in the low c-value (c = 13). It is most probably accounted for by the presence of adsorbed nonane on the external surface which was not removed at 25°C but only at I50°C. (The Frenkel-Halsey-Hill exponent (p. 90) for the multilayer region of the 25°C-outgassed sample was only 1 -9 as compared with 2-61 for the standard rutile, and 2-38 for the 150°C-outgassed sample). 

To overcome the problems which arise from a limited size range and the different adsorption properties of different adsorbates, one may use the entire adsorption isotherm obtained with just one probe molecule instead. There are two approaches both leading to a Frenkel-Halsey-Hill type equation ... 

Since z oc V, this may be written In (p0/p) = const. V where V is the volume of gas adsorbed. A more general form of this isotherm, called the Frenkel-Halsey-Hill (FHH) isotherm, treats the power dependence of V as an unknown n and writes... 

Fractal geometry has been used to describe the structure of porous solid and adsorption on heterogeneous solid surface [6-8]. The surface fractal dimension D was calculated from their nitrogen isotherms using both the fractal isotherm equations derived from the FHH theory. The Frenkel-Halsey-Hill (FHH) adsorption isotherm applies the Polanyi adsorption potential theory and is expressed as ... 

Important trends in N2 isotherm when the PS beads are used as a physical template are shown in Table 1 and Fig. 2. In Table 1, PI is the alumina prepared without any templates, P2 is prepared without ]4iysical template (PS bead), P3 is prepared without chemical template (stearic acid), and P4 is prepared with all templates. For above 10 nm of pore size and spherical pore system, the Barrett-Joyner-Halenda (BJH) method underestimates the characteristics for spherical pores, while the Broekhoff-de Boer-Frenkel-Halsey-Hill (BdB-FHH) model is more accurate than the BJH model at the range 10-100 nm [13]. Therefore, the pore size distribution between 1 and 10 nm and between 10 and 100 nm obtained from the BJH model and BdB-FHH model on the desorption branch of nitrogen isotherm, respectively. N2 isotherm of P2 has typical type IV and hysteresis loop, while that of P3 shows reduced hysteresis loop at P/Po ca. 0.5 and sharp lifting-up hysteresis loop at P/Po > 0.8. This sharp inflection implies a change in the texture, namely, textural macro-porosity [4,14]. It should be noted that P3 shows only macropore due to the PS bead-free from alumina framework. 

Unlike the first two methods, the third method requires only a single adsorption isotherm. The analysis of the single isotherm to obtain Ds is performed by using a modified Frenkel-Halsey-Hill (FHH) theory. The original FHH theory was developed by Frenkel, Halsey,and HilP° and was later extended to fractal surfaces by Pfeifer et al. ... 

Tang, P. Chew, N.Y.K. Chan, H.-K. Raper, J. Limitation of determination of surface fractal dimension using N2 adsorption isotherms and modified Frenkel-Halsey-Hill theory. Langmuir 2003, 7, 2632-2638. 

Many different equations have been used to interpret monolayer—multilayer isotherms [7, 11, 18, 21, 22] (e.g., the equations associated with the names Langmuir, Vohner, HiU-de Boer, Fowler-Guggenheim, Brunauer-Emmett-Teller, and Frenkel-Halsey-Hill). Although these relations were originally based on adsorption models, they are generally applied to the experimental data in an empirical manner and they all have Hmitations of one sort or another [7, 10, 11]. 

It has been pointed out above that the Wheeler-Ono approach (see Sec. III.l) to the idealized mathematically plane surface problem is the rigorous approach, though actual numerical calculations based on the general equations are not practical. On the other hand, the Frenkel-Halsey-Hill method (see Sec. III.4) is essentially a very approximate solution of this same problem resulting in a simple and surprisingly successful isotherm equation, Eq. (38), for 0 not too small. This method can be applied to capillary condensation (see Sec. III.5) and is capable of accounting for isotherm types II to V (1,55,75). 

The Frenkel-Halsey-Hill (FHH) isotherm was originally developed to describe the growth of thick films and wetting phenomena on a flat surface and was later extended to studying adsorption on fractal surfaces [3, 55]. In contrast to BET theory, FHH theory applies to long-range adsorbate-absorbent interactions and its approach is closely related to the so-called potential theory of adsorption of Eucken and Polanyi (see Ref. [35]). 

The Frenkel-Halsey-Hill (FHH) isotherm has found much utilization due to the range specified for its application. It seems especially handy for porosity determinations. It seems to work well between relative pressures in the range 0.4-0.9. The equation is... 

The form of an adsorption isotherm on a geometrically flat substrate in the thick film regime can be described by the so called Frenkel-Halsey-Hill (FHH) theory, under the eussumption of complete wetting of the surface by the film (ref. 4). The FHH vapor pressure is of the form ... 

The Frenkel-Halsey-Hill (FHH) equation has been used to analyse the behaviour of the isotherms at higher relative pressures (7). It has been applied in the form ... 

Frenkel, Halsey and Hill theory of adsorption and isotherm equation... 

The other molecular probe method is the single-probe method (SP method), which is separately proposed by Avnir and Jaroniec,93 and Pfeifer et al.108-112 In the SP method, a single adsorption isotherm is analyzed using a modified FHH theory. The FHH model was developed independently by Frenkel,113 Halsey,114 and Hill,115 and describes the multilayer adsorption coverage. Since the SP method uses only one probe molecule, this method is more convenient than the MP method. However, there are many theoretical limitations in applying the SP method to determination of the surface fractal dimension. Therefore, it is really necessary to discuss whether the SP method is an adequate tool to investigate the surface fractal dimension or not before applying the SP method to certain system. 

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