Standard Isotherms

Isotherms measured on well-characterized material and are used for comparison with isotherms of unknowns are referred to as standard isotherms. Tables of a variety of standard isotherms that are described here are presented in this section. 

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Sing (see Ref. 207 and earlier papers) developed a modification of the de Boer r-plot idea. The latter rests on the observation of a characteristic isotherm (Section XVII-9), that is, on the conclusion that the adsorption isotherm is independent of the adsorbent in the multilayer region. Sing recognized that there were differences for different adsorbents, and used an appropriate standard isotherm for each system, the standard isotherm being for a nonporous adsorbent of composition similar to that of the porous one being studied. He then defined a quantity = n/nx)s where nx is the amount adsorbed by the nonporous reference material at the selected P/P. The values are used to correct pore radii for multilayer adsorption in much the same manner as with de Boer. Lecloux and Pirard [208] have discussed further the use of standard isotherms. 

Fig. 2.26 Comparison of a number of standard isotherms of nitrogen at 77 K, plotted as n/n against pip . O, Shnll x, Pierce , de Boer el , Cranston and Inkley. ...

Well as various samples of nonporous but amorphous silica. They found that the points fitted on to a common curve very closely, which may be plotted from Table 2.14. A corresponding curve, though based on fewer samples, was put forward for y-alumina. The two curves are close to one another, but the divergence between them is greater than that between different samples of the same substance. Standard isotherm data for argon (at 77 K) on silica have been obtained by various workers. ... 

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. 

The f-curve and its associated t-plot were originally devised as a means of allowing for the thickness of the adsorbed layer on the walls of the pores when calculating pore size distribution from the (Type IV) isotherm (Chapter 3). For the purpose of testing for conformity to the standard isotherm, however, a knowledge of the numerical thickness is irrelevant since the object is merely to compare the shape of the isotherm under test with that of the standard isotherm, it is not necessary to involve the number of molecular layers n/fi or even the monolayer capacity itself. 

To facilitate application of the method, Dollimore and Heal gave a standard table of the relevant parameters, based on regular intervals of P extending from 100 A down to 7 A (-values were calculated with Halsey s equation (p. 89). Table 3.2B retains the essential features of their original table, but P no longer extends below 17 A (cf. p. 160) and the /-values are now based on an experimentally determined standard isotherm.(p. 93). 

To test the Brunauer approach, it was necessary to use standard isotherms of nitrogen having the same c-constants as the experimental isotherms of Table 4.7. Since nitrogen isotherms with c > 10 have not been reported in the literature, theoretical isotherms corresponding to the c-values of Table 4.6 were calculated by Brunauer s modification of Anderson s equation, and standard a,-curves were constructed from them. The corresponding a,-plots appear in Fig. 4.15 they are no longer parallel... 

The a,-plots were based on the standard isotherms of Nj and CCI respectively, on Fransil. For calculation of. 4,(BET), the value u fCCU) = 37 was assumed. 

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 the mechanism illustrated by scheme B, significant inhibition is only realized after equilibrium is achieved. Hence the value of vs (in Equations 6.1 and 6.2) would not be expected to vary with inhibitor concentration, and should in fact be similar to the initial velocity value in the absence of inhibitor (i.e., v, = v0, where v0 is the steady state velocity in the absence of inhibitor). This invariance of v, with inhibitor concentration is a distinguishing feature of the mechanism summarized in scheme B (Morrison, 1982). The value of vs, on the other hand, should vary with inhibitor concentration according to a standard isotherm equation (Figure 6.5). Thus the IC50 (which is equivalent to Kfv) of a slow binding inhibitor that conforms to the mechanism of scheme B can be determined from a plot of vjv0 as a function of [/]. 

Figure 7.1 Concentration-response plots for a series of compounds displaying Kf9p values ranging from 100 to 0.01 nM, when studied in an enzyme assay for which the enzyme concentration is 50nM. The lines through the data sets represent the best fits to the standard isotherm equation that includes a non-unity Hill coefficient (Equation 5.4). Note that for the more potent inhibitors (where Kf" < [E]T), the data are not well fit by the isotherm equation.

The support and the catalysts were characterised by means of nitrogen adsorption, XPS, TPD and SEM. The nitrogen adsorption isotherms were determined at 77 K in a Coulter Omnisorp 1000 CX equipment, and were analysed by the BET equation (SBet), and by the t-plot for mesopore surface area (Smeso) and micropore and mesopore volume (Vmicr0, Vmeso), using the standard isotherm for carbon materials. The catalyst samples were previously outgassed at 120 °C. 

Nitrogen adsorption was performed at -196 °C in a Micromeritics ASAP 2010 volumetric instrument. The samples were outgassed at 80 °C prior to the adsorption measurement until a 3.10 3 Torr static vacuum was reached. The surface area was calculated by the Brunauer-Emmett-Teller (BET) method. Micropore volume and external surface area were evaluated by the alpha-S method using a standard isotherm measured on Aerosil 200 fumed silica [8]. Powder X-ray diffraction (XRD) patterns of samples dried at 80 °C were collected at room temperature on a Broker AXS D-8 diffractometer with Cu Ka radiation. Thermogravimetric analysis was carried out in air flow with heating rate 10 °C min"1 up to 900 °C in a Netzsch TG 209 C thermal balance. SEM micrographs were recorded on a Hitachi S4500 microscope. 

Fmpirical methods can be applied in order to determine the validity of the BFT surface area. The derived standard isotherms can be obtained by normalization of the y-axis (volume adsorbed) of adsorption isotherms. It is strongly recommended that data should always be derived from standard isotherms related to a nonpor-ous sample of the same type of material. Various methods have been established like the as-method where the quantity of gas adsorbed V], is related to the value at a relative pressure of 0.4. In the t-plot, the vertical axis is normalized in relation to the average thickness of the adsorbed layer. The shape of the constructed reduced isotherms reveal the presence or absence of micropores and allows the determination of their volume [79, 80]. 

Typical methods for testing microporosity consist in the comparison of a standard isotherm (mostly an isotherm of a non-(micro)porous sample of identical chemical composition), with the isotherm under study. 

A convenient method is provided by the t-plot of Lippens and De Boer.37 It consists of plotting the volume of gas adsorbed vs t, the statistical thickness of the adsorbed film, t is a function of p/p0, as measured in the standard isotherm, according to the Halsey equation or one of its modifications.38,39,40,41... 

However, for the purpose of testing for conformity to the standard isotherm, a knowledge of the numerical thickness is irrelevant. 

Concept and Use of Standard Isotherms (the f-Plot Method and the Os-Method)... 

Considering the nature of the forces involved in the physical adsorption process (see Section 4.2.1), it is evident that the adsorption isotherm of a given adsorptive on a particular solid at a given temperature depends on the nature of both the gas and the solid, and therefore, each adsorbate-adsorbent system has a unique isotherm. In spite of this, a number of attempts have been made to express the adsorption isotherm data in a normalized form. It was seen that, for a large number of nonporous solids (type II isotherms), the plot of n/nm versus P/P° can be represented by a single curve, called the standard isotherm. Among these related attempts, the t- and as-methods are the most widely used. 

The use of standard isotherms as a tool to characterize the porosity of solids (even for micropo-rous solids as it was shown) by means of one of the two methods described above may be appropriate. However, it is necessary to keep in mind that all of them are subject to the same limitation, that is, the difficulty of an appropriate choice of a nonporous reference material. The as-method chooses the standard isotherm according to the chemical nature of the sample to be studied, whereas the /-method chooses it independent of the nature of the nonporous adsorbent. 

The Broekhoff-de Boer t method for the determination of surface areas is not based on a new theory. It is an empirical method in which the adsorption isotherm of a material of unknown surface area is compared with a standard isotherm, the common t curve, valid for f materials with a surface area of 1 m2. 

The most straightforward form of as-plot is Type 11(a) in Figure 6.1, which is for a typical Type II isotherm with a moderate value of C ( 100). The extensive range of linearity and the zero intercept are the result of unrestricted monolayer-multilayer adsorption on a non-porous solid of very similar surface structure to that of the reference material. In this case the shapes of the experimental and standard isotherms are virtually identical and therefore the slope of the as-plot is directly proportional to the ratio of the surface areas, a(S)/aref. Thus, if the value of aKl is already known, it is a simple matter to calculate atest, which we denote a(S) to indicate it is calculated by the as-method. 

In principle, a /-plot can be used to assess the micropore capacity provided that the standard multilayer thickness curve has been determined on a non-porous reference material with a similar surface structure to that of the microporous sample. In our view, it is not safe to select a standard isotherm with the same BET C value (i.e. the procedure recommended by Brunauer (1970) and Lecloux and Pirard (1979)) since this does not allow for the fact that the sub-monolayer isotherm shape is dependent on both the surface chemistry and the micropore structure. 

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