Isothermal micropore

Keywords C-constant, BET-isotherm, microporous solids, mesoporous matrix, t-plot... 

Figure 7.42 Types of gas sorption isotherm - microporous solids are characterised by a type I isotherm. Type II corresponds to macroporous materials with point B being the point at which monolayer coverage is complete. Type III is similar to type II but with adsorbate-adsorbate interactions playing an important role. Type IV corresponds to mesoporous industrial materials with the hysteresis arising from capillary condensation. The limiting adsorption at high P/P0 is a characteristic feature. Type V is uncommon. It is related to type III with weak adsorbent-adsorbate interactions. Type VI represents multilayer adsorption onto a uniform, non-porous surface with each step size representing the layer capacity (reproduced by permission of IUPAC).

Isothermal Micropore and Pore-Surface Diffusion Models. 317... 

The mass transfer coefficient can be calculated as the ratio of the estimated values of Op and Tp. 2, Isothermal Micropore and Pore-Surface Diffusion Models... 

The expressions for the FRFs for the isothermal micropore and pore-surface diffusion models were obtained for constant diffusion coefficients. If this assumption is not met, that is, if the concentration dependence of the diffusion coefficient has to be taken into account, the value estimated from the maximum of the — Imag(Fi p(diffusion coefficient corresponding to the steady-state concentration. 

For the nonisothermal case, the position of the maximum of — Imag(Fi p(diffusion mechanism. Actually, in most cases this function has two maximums, as can be seen from Figure 11.21 (from Ref. [57]). Nevertheless, the analysis of the FRFs corresponding to the nonisothermal micropore model shows that the ratio... 

DR plots for N2 adsorption Isotherms micropore filling mechanism... 

The adsorption isotherms are often Langmuirian in type (under conditions such that multilayer formation is not likely), and in the case of zeolites, both n and b vary with the cation present. At higher pressures, capillary condensation typically occurs, as discussed in the next section. Some N2 isotherms for M41S materials are shown in Fig. XVII-27 they are Langmuirian below P/P of about 0.2. In the case of a microporous carbon (prepared by carbonizing olive pits), the isotherms for He at 4.2 K and for N2 at 77 K were similar and Langmuirlike up to P/P near unity, but were fit to a modified Dubninin-Radushkevich (DR) equation (see Eq. XVII-75) to estimate micropore sizes around 40 A [186]. 

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... 

Fig. XVII-31. (a) Nitrogen adsorption isotherms expressed as /-plots for various samples of a-FeOOH dispersed on carbon fibers, (h) Micropore size distributions as obtained by the MP method. [Reprinted with permission from K. Kaneko, Langmuir, 3, 357 (1987) (Ref. 231.) Copyright 1987, American Chemical Society.]...

In Section XVII-16C there is mention of S-shaped isotherms being obtained. That is, as pressure increased, the amount adsorbed increased, then decreased, then increased again. If this is equilibrium behavior, explain whether a violation of the second law of thermodynamics is implied. A sketch of such an isotherm is shown for nitrogen adsorbed on a microporous carbon (see Ref. 226). 

The basis of the classification is that each of the size ranges corresponds to characteristic adsorption effects as manifested in the isotherm. In micropores, the interaction potential is significantly higher than in wider pores owing to the proximity of the walls, and the amount adsorbed (at a given relative pressure) is correspondingly enhanced. In mesopores, capillary condensation, with its characteristic hysteresis loop, takes place. In the macropore range the pores are so wide that it is virtually impossible to map out the isotherm in detail because the relative pressures are so close to unity. 

Type 1 isotherms, as will be demonstrated in Chapter 4, are characteristic of microporous adsorbents. The detailed interpretation of such isotherms is controversial, but the majority of workers would probably agree that the very concept of the surface area of a microporous solid is of doubtful validity, and that whilst it is possible to obtain an estimate of the total micropore volume from a Type I isotherm, only the crudest guesses can be made as to the pore size distribution. 

As will be demonstrated in Chapter 4, however, the presence of micropores distorts the Type II isotherm in a sense which is reflected in a much increased value of the constant c. In such cases the value of c is no guide at all to the course of the isotherm on the external surface. Consequently the appropriate criterion for choosing the correct f-curve for a particular system is the similarity in chemical properties and not in c-values l>etween the solid under test and the reference solid. 

It is therefore of the utmost importance to ensure that the standard isotherm is based on a solid known to be free of pores, and especially of micropores. Unfortunately, it is not easy to establish the complete absence of porosity in the solids used in adsorption isotherm measurement the unsuspected presence of such pores may well account for some, at least, of the discrepancies between different published versions of the standard isotherm for a given adsorptive. 

Deviation from the standard isotherm in the high-pressure region offers a means of detecting the occurrence of capillary condensation in the crevices l>etween the particles of a solid and in any mesopores present within the particles themselves. A convenient device for detecting deviations from the standard is the t-plot . In the next section the nature and uses of t-plots will be discussed, together with a,-plots, a later development from them. As will l>e shown, both of these plots may l>e used not only for the detection of capillary condensation in mesopores, but also for showing up the presence of micropores and evaluating their volume. 

If micropores are introduced into a solid which originally gave a standard Type II isotherm, the uptake is enhanced in the low-pressure region and the isotherm is correspondingly distorted. The effect on the t-plot is indicated in... 

Fig. 2.28 Effect of microporosity of the isotherm and t- (or a,-) plot, (a) (A) is the isotherm on a nonporous sample of the adsorbent (B) is the isotherm of the same solid when micropores have been introduced into it. [b) t- (or a,-) plots corresponding to the isotherms of (a). (Schematic only.)...

As will be demonstrated in Chapter 4, an isotherm which is reversible and of Type II is quite compatible with the presence of micropores. If such pores are present, the isotherm will be distorted in the low-pressure region, the value of c will be greatly enhanced, and the specific surface derived by the BET procedure will be erroneously high. In particular, a BET specific surface in excess of - 500m g" should be taken as a warning that... 

If a solid contains micropores—pores which are no more than a few molecular diameters in width—the potential fields from neighbouring walls will overlap and the interaction energy of the solid with a gas molecule will be correspondingly enhanced. This will result in a distortion of the isotherm, especially at low relative pressures, in the direction of increased adsorption there is indeed considerable evidence that the interaction may be strong enough to bring about a complete filling of the pores at a quite low relative pressure. 

In the simplest case, adsorption in a microporous solid leads to an isotherm of Type I consequently it is convenient to approach the subject by a discussion, from a classical standpoint, of Type I isotherms. 

If the isotherm is of Type I with a sharp knee and a plateau which is horizontal (cf. Fig. 4.10) the uptake n, at a point close to saturation, say p/p = 0-95, is then a measure of the micropore volume when converted to a liquid volume (by use of the density of the liquid adsorptive), it may be taken as actually equal to the micropore volume. 

More often, however, microporosity is associated with an appreciable external surface, or with mesoporosity, or with both. The effect of microporosity on the isotherm will be seen from Fig. 4.11(a) and Fig. 4.12(a). In Fig. 4.11(a) curve (i) refers to a powder made up of nonporous particles and curve (ii) to a solid which is wholly microporous. However, if the particles of the powder are microporous (the total micropore volume being given by the plateau of curve (ii)), the isotherm will assume the form of curve (iii), obtained by summing curves (i) and (ii). Like isotherm (i), the composite isotherm is of Type II, but because of the contribution from the Type 1 isotherm, it has a steep initial portion the relative enhancement of adsorption in the low-pressure region will be reflected in a significantly increased value of the BET c-constant and a shortened linear branch of the BET plot. 

Figure 4.12(a) refers to the case where micropores are present along with mesopores. The composite isotherm (iii), like the isotherm (ii) of the mesoporous substance itself, is of Type IV, and again has a steep initial branch with an increased value of c. 

Fig. 4.11 (o) Adsorption isotherm for (i) a powder made up of nonporous particles (ii) a solid which is wholly microporous (iii) a powder with the same external surface as in (i) but made up of microporous particles having a total micropore volume given by the plateau of isotherm (ii). The adsorption is expressed in arbitrary units, (b) t-Plots corresponding to isotherms (i) and (iii). The o,-plots are similar, except for the scale of... 

Thus, whilst a powder composed of nonporous particles gives rise to an isotherm of Type II, the converse is not necessarily true if a solid yields a Type II isotherm, it is not necessarily free of micropores. Similarly, though a Type IV isotherm signifies the presence of mesoporosity, it does not prove the absence of microporosity. - ... 

A high value of the BET constant c is a useful preliminary indication of the presence of microporosity, but it does not enable one to estimate the micropore volume itself, that is in effect to break down the composite isotherm (iii) into its components (i) and (ii). 

The table convincingly demonstrates how the unsuspected presence of micropores can lead to an erroneous value of the specific surface calculated from a Type II isotherm by application of the standard BET procedure. According to the foregoing analysis, the external specific surface of the solid is 114m g" the micropore volume (from the vertical separation of isotherms A and E) is 105 mm g but since the average pore width is not precisely known, the area of the micropore walls cannot be calculated. Thus the BET figure of 360m g calculated from isotherm E represents merely an apparent and not a true surface area. 

The t and a.-methods, the nature of which was explained in Chapter 2, may be used to arrive at a value of the micropore volume. If the surface of the solid has standard properties, the t-plot (or a,-plot) corresponding to the isotherm of the nonporous powder in Fig. 4.11(a) will be a straight line passing through the origin (cf. curve (i) of Fig. 4.11(6)) and having a slope proportional to the specific surface of the powder. For the microporous powder which yields the isotherm (iii).of Fig. 4.11(a), the t-plot (or Oj-plot) will have the form of curve (iii) of Fig. 4.11(6) the linear branch of this curve will be parallel to curve (i), since it corresponds to the area of the outside of the particles which is identical with that of the nonporous parent particles. 

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). 

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