Adsorption potential distribution

The adsorption potential A is defined as the change in the Gibbs free energy taken with the minus sign [93]  

Here p denotes the saturation vapor pressure, T is the absolute temperature, and R is the universal gas constant. 

Characteristic adsorption curves ftn- nitrc en on active carbons at 77 K 

The amount adsorbed a (after amvosion to the adsorbate volume V) measures the pore volume accessible to adsorption. If Vt is the maximum volume adsorbed, the difference Vt-V represents the unoccupied pore volume associated with the adsorption potentials smaller than A [94, 95], and denotes the tKjn-normalizEd int ral distribution function of the adsorption 

In terms of the c iidensatiDn approximation [10] the adsorption potential distribution (APD), X(A), gives essraitially the same information as the distribution function of tiie adsorption enei because in this case A is equal to the adsorption energy f expressed with respect to the energy tliat characterize the standard state [97], Le., 

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Gil and Grange (1996) also attempted to apply the DR and DA equations (i.e. Equations (4.39) and (4.45)) - the latter is given later in this chapter as Equation (11.1). The pillared clays appeared to give bimodal adsorption potential distributions, but the significance of these findings is not entirely clear. 

The energetic heterogeneity associated with micropores of microporous materials can be characterised by means of the adsorption potential distribution, and the distributions (X(A)), related to the Dubinin equations ... 

Jaroniec et al. [15] also proposed a simple thermodynamic approach to characterize microporous solids (JGC model). The MPSD, J(x), is related to the adsorption potential distribution through the following equation ... 

The DA equation was applied in the nitrogen relative pressure range between 10 and 0.2 to estimate both E and Yppo - These parameters are necessary to calculate the adsorption potential distribution (see Eq. (2)). VppHK values were evaluated from the MPSDs as the adsorbed volume corresponding to a pore size of 20 A [22], These results have been included in Table 1. 

BAsap-Al)773 and (BAsap-Al/K y,-2)773 can be considered as representative samples of the solids synthesised in this work, and their MPSDs are compared in Figures 2, 3 and 4. It has been pointed out that in some cases the adsorption potential distributions (Fig. 2) can be related with the MPSD of the samples [11]. Differences between the maxima of the distributions and their width can be observed, suggesting that (BAsap-Al, /,-2)773 shows smaller micropores than that of (BAsap-Al)773. The MPSDs derived from the HK slitlike model (Fig. 3) present two maxima in both cases, 5.1 and 7.4 A for (BAsap-Al)773 and 5.3... 

Figure 3. (a) Isotherm calculated by density functional theory for a 4.1 run wide cylindrical pore with uniform surface potential (solid points). The solid line is the reconstructed isotherm for a flat surface having the adsorptive potential distribution of MCM-41. (b) Normalized isotherm for a 4.1 run MCM-41 (solid points) compared to the composite model for the same pore size. Note that the height of the pore-filling step is accurately accounted for. 

The scatter seen in the calculated points is a result of the discrete nature of the adsorptive potential distribution. 

Figure 3. Adsorption potential distributions for the MCM-41 samples studied obtained by a simple differentiation of nitrogen adsorption isotherms.

The A(w)-curves shown in Figure 2 were used to calculate the dA/dw derivative, which is necessary for conversion of APD to PSD via equation (2). Adsorption potential distributions for the MCM-41 samples studied are shown in Figure 3. As can be seen in this figure there is a correlation between the position of the distribution maximum and the pore width. 

In terms of this adsorption, properties of various adsorbents, among them the inorganic sorbents can be determined. It must be emphasized that inorganic sorbents such as silica, alumina, titania, complex carbon - mineral sorbents, apatites, e.t.c., are both structurally and energetically heterogeneous. Their total heterogeneity may be described by the kinds of adsorption potential distribution function which is one of the most significant characteristics of the aforementioned solids. 

While DFT allows us to calculate values for q(p, e), it of course provides no analytic form for the function, and in general the form of f(e) is also unknown. However, by using carefully designed numerical methods, model isotherms calculated by MNLDFT can be used in carrying out the inversion of the discrete form of the integral equation of adsorption. In this way one can determine the effective adsorptive potential distribution of the adsorbent from the experimental adsorption isotherm. The method used can be expressed by... 

Figure 7.4 (a) The expeiimental data (points) of Fig. 7.3 fitted by Eqn (7.12) using the deconvolution method (solid line), (b) The adsorptive potential distribution for the Sterling graphite. 

Figure 7.6 (a) The adsorptive potential distribution of a natural, low surface area graphite, (b) The same material after a fine-grinding procedure, showing a slightly broadened distribution and increased surface area. 

Kruk, M., Jaroniec, M., and Gadkaree, K. (1999). Determination of the specific surface area and the pore size of microporous carbons from adsorption potential distributions. Langmuir, 15 1442-8. 

Figure 18.7 Adsorption potential distribution (APD) of various carbon blacks (CB) the curves were vertically shifted for better presentation.

Figure 18.8 Adsorption potential distribution (APD) of ordered mesoporous carbon (OMC). (Reprinted with permission from Ref. [13].)...

Differentiation of the integral Eqs. (44) and (46) with respect to A gives equations for the adsorption potential distribution[104,120] ... 

Energetic heterogeneity of a microporous solid generated by the overlapping of adsorption forces from the opposite micropore walls can be descaibed by the adsorption potential distribution in micropores This distribution associated with Eq. (62) is given by ... 

Since the relationships between A and x are known for difierent pore ranges and different pore geometries [13, 143, 153-157], they can be utilized to convert the adsorption potential distribution to the pore volume distribution via the following equation ... 

The HK micropore volume distribution for a slit-like microporous structure can be obtained by multiplying the adsorption potential distribution [see Eq. (24) and Fig. 10] by Eq. (77). For cylindrical and spherical micropore geometries another expressions for the derivative dAldx should be used [160]. An illustration of the HK pore volume distributions is shown in Fig. 12 for the WV-A900, BAX 1500 and NP5 active carbons. Similarly, the mesopore volume distribution can be calculated from the multilayer and capillary condensation range of the adsorption isotherm. In this case, the corrected Kelvin equation should be used to calculate the derivative dAldx. 

While the adsorption potential distribution is a model-independent thermod3noamic function, the pore volume distributions are obtained by assuming the relationship between the adsorption potential and the pore width. Thus, the adsorption potential distribution can be considered an imique and primary characteristics of a given adsorption system, whereas the... 

Another important conclusion concerns the geometrical heterogeneity of nanoporous carbons, which is characterized by the micropore and mesopore volume distributions. The current work demonstrates that in terms of the condensation approximation both these dishibutions are directly related to the adsorption potential distribution. As shown the pore volume distribution can be obtained by multiplication of the adsorption potential distribution... 

Knowing the parameters n and q, the differential adsorption potential distribution X(A) is obtained as ... 

Today, there is probably universal agreement that almost all real surfaces are significantly heterogeneous. The importance of physical adsorption in chemical engineering processes involving separations and catalysis has led to increased interest in this subject and to the need to explore ways to characterize the surfaces of engineering adsorbents in terms of their adsorptive potential distributions. 

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