Experimental isotherm

The characteristic isotherm concept was elaborated by de Boer and coworkers [90]. By accepting a reference from a BET fit to a standard system and assuming a density for the adsorbed film, one may convert n/rim to film thickness t. The characteristic isotherm for a given adsorbate may then be plotted as t versus P/P. For any new system, one reads t from the standard r-curve and n from the new isotherm, for various P/P values. De Boer and co-work-ers t values are given in Table XVII-4. A plot of t versus n should be linear if the experimental isotherm has the same shape as the reference characteristic isotherm, and the slope gives E ... 

Equation (2.6) is the familiar Langmuir equation for the case when adsorption is confined to a monolayer. In practice B is an empirical constant and cannot be evaluated from the relationship in Equation (2.7). The question as to how well the Langmuir equation reproduces experimental isotherms will be dealt with in Chapter 4. 

A number of attempts have been made to modify the BET equation so as to obtain better agreement with the experimental isotherm data in the multilayer region. One of the most recent is that of Brunauer and his co-workers ... 

The steps may be so chosen as to correspond to consecutive points on the experimental isotherm. In practice it is more convenient to divide the desorption process into a number of standard steps, either of relative pressure, or of pore radius, which is of course a function of relative pressure. The amount given up during each step i must be converted into a liquid volume i , (by use of the normal liquid density) in some procedures the conversion is deferred to a late stage in the calculation, but conceptually it is preferable to undertake the conversion at the outset. As indicated earlier, the task then becomes (i) to calculate the contribution dv due to thinning of the adsorbed film, and thus obtain the core volume associated with the mean core radius r by the subtraction = t ... 

The uptake, in column 10, has been converted into a liquid volume at the outset rather than at a later stage as in the original papers. Columns 1-8 are based on Table 3.2. In the original papers the values of p/p° corresponded to actual points on the experimental isotherm, but the work sheet is much simplified by the choice of standard intervals of pjp° (or of r " cf. p. 135). 

W2 is of course obtained from the experimental isotherm as the difference between the uptakes at the starting point and at P2/p° respectively, expressed as a liquid volume. 

In using the table for pore size calculations, it is necessary to read off the values of the uptake from the experimental isotherm for the values of p/p° corresponding to the different r values given in the table. Unfortunately, these values of relative pressure do not correspond to division marks on the scale of abscissae, so that care is needed if inaccuracy is to be avoided. This difficulty can be circumvented by basing the standard table on even intervals of relative pressure rather than of r but this then leads to uneven spacings of r . Table 3.6 illustrates the application of the standard table to a specific example—the desorption branch of the silica isotherm already referred to. The resultant distribution curve appears as Curve C in Fig. 3.18. 

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... 

When the value of c exceeds unity, the value of n can be derived from the slope and intercept of the BET plot in the usual way but because of deviations at low relative pressures, it is sometimes more convenient to locate the BET monolayer point , the relative pressure (p/p°) at which n/n = 1. First, the value of c is found by matching the experimental isotherm against a set of ideal BET isotherms, calculated by insertion of a succession of values of c (1, 2, 3, etc., including nonintegral values if necessary) into the BET equation in the form ... 

The adsorption free energy and other parameters may be determined, provided that a proper adsorption isotherm is identified and is fitted to experimental data. However, it is usually difficult to unequivocally choose an appropriate isotherm an experimental isotherm may well be fitted to a multitude of theoretical isotherms having several adjustable parameters. If the adsorption isotherm at a very small surface coverage is accessible experimentally, the adsorption free energy can be determined from the limiting slope of the isotherm, as all isotherms reduce to Henry s law when 6 0 ... 

A major advantage of the simple model described in this paper lies in its potential applicability to the direct evaluation of experimental data. Unfortunately, it is clear from the form of the typical isotherms, especially those for high polymers (large n) that, even with a simple model, this presents considerable difficulty. The problems can be seen clearly by consideration of some typical polymer adsorption data. Experimental isotherms for the adsorption of commercial polymer flocculants on a kaolin clay are shown in Figure 4. These data were obtained, in the usual way, by determination of residual polymer concentrations after equilibration with the solid. In general, such methods are limited at both extremes of the concentration scale. Serious errors arise at low concentration due to loss in precision of the analytical technique and at high concentration because the amount adsorbed is determined by the difference between two large numbers. 

Figure 9,4 (A) Reduced isotherms of N2, CO2, and H2O vapor according to reduced pressure. (B) Comparison between experimental isotherms of H2O and f predicted by principle of corresponding states.

The experimental isotherms were fitted to the BET and Langmuir isotherms, selecting the best fit. The values of the monolayer capacity, the specific surface and the net heats of adsorption were calculated. 

FIGURE 10.14 Single-component experimental isotherms foriJ- and 5-1-indanol on cellulose tribenzoate fitted to three different models. The inset shows the low-concentration data. (Reproduced from Zhou, D., Chem. Eng. Sci., 58, 3257, 2003. With permission from Elsevier.)... 

Rarely, if ever, does the BET theory exactly match an experimental isotherm over its entire range of relative pressures. In a qualitative sense, however, it does provide a theoretical foundation for the various isotherm shapes. Of equal significance is the fact that in the region of relative pressures near completed monolayers (0.05 P/Pq 0.35) the BET theory and experimental isotherms do agree very well, leading to a powerful and extremely useful method of surface area determination. 

Using equations (9.1) and (9.2) both V and E can be calculated from an experimental isotherm. Therefore,... 

Mikhail, Brunauer, and Bodor proposed an extension of deBoer s r-method for the analysis of micropores which offers several advantages. These include the ability to obtain the micropore volume, surface, and their distributions from one experimental isotherm. Data for the MP (micropore analysis) method need not be measured at the very low pressures needed for the Dubinin and Kaganer theories. The method... 

It has been suggested that the experimental isothermal kinetic rate constants of some reactions at near and supercritical conditions could not be explained solely by the thermodynamic pressure effect, but from the combination of local composition enhancement and density augmentation around reactants. 

To effectively determine the start-of-cycle reforming kinetics, a set of experimental isothermal data which covers a wide range of feed compositions and process conditions is needed. From these data, selectivity kinetics can be determined from Eq. (12). With the selectivity kinetics known, Eqs. (17) and (18a)-(18c) are used to determine the activity parameters. It is important to emphasize that the original definition of pseudomonomolecular kinetics allowed the transformation of a highly nonlinear problem [Eq. (5)] into two linear problems [Eqs. (12) and (15)]. Not only are the linear problems easier to solve, the results are more accurate since confounding between kinetic parameters is reduced. 

Many systems that definitely do not conform to the Langmuir assumptions —the adsorption of polymers, for example —nevertheless display experimental isotherms that resemble Figure 7.16. Although these can be fitted to Equation (67), the significance of the constants is dubious. Therefore the Langmuir equation is often written as... 

The adsorption of various aliphatic alcohols from benzene solutions onto silicic acid surfaces has been studied.t The experimental isotherms have an appearance consistent with the Langmuir isotherm. Both the initial slopes of an n/w versus c plot and the saturation value of n/w decrease in the order methanol > ethanol > propanol > butanol. Discuss this order in terms of the molecular structure of the alcohols and the physical significence of the initial slope and the saturation intercept. Which of these two quantities would you expect to be most sensitive to the structure of the adsorbed alcohol molecules Explain. 

It is convenient to divide the extent of adsorption into three categories submonolayer, monolayer, and multilayer. We discuss them in this order. The thermodynamics of adsorption may be developed around experimental isotherms or around calorimetric data. We begin with the definition of adsorption isotherms and how they are determined experimentally (Section 9.2). 

If the experimental isotherm (n/w as a function of p) is known, then Equation (7) may be integrated either analytically or graphically to give the two-dimensional pressure as a function of coverage. This relationship therefore establishes the connection between the two-and three-dimensional pressures that characterize the surface and bulk phases. This is how adsorption data could be used to determine the film pressure in equilibrium with a drop of bulk liquid on a solid surface as discussed in Section 6.6b. 

In principle, it is possible to correlate an adsorption isotherm and a two-dimensional equation of state by working from either direction that is, we may start with an experimental isotherm and develop the associated equation of state (as in going from Equation (8) to Equation (10)), or we may proceed from the equation of state to the isotherm (as illustrated in Example 9.2). 

Relatively small increases in complexity for the equation of state result in considerably more complex equations for the adsorption isotherms. The gross features of the more complex isotherms are also given by simpler isotherms. This means that it is very difficult to choose among various isotherms in terms of the goodness of fit to experimental data. Therefore it is difficult to conclude from an experimental isotherm what the two-dimensional surface phases are like. 

FIG. 9.11 A schematic illustration of hysteresis between the adsorption and desorption branches of an experimental isotherm. 

The CB-MC method has been used to simulate the adsorption isotherms of various alkanes in silicalite (170, 171). Using potential parameters that were fitted to obtain good agreement with experimental Henry s law coefficients, Smit and Maesen (170,171) have simulated the adsorption isotherms of straight-chain alkanes in silicalite. Good agreement was obtained for ethane and propane in comparison with the different type-I curves measured experimentally. The overall agreement with experimental isotherms was found to be satisfactory with hexane and heptane, and a kink is seen... 

For all of the systems analyzed so far, the effect of the exponential terms in Equation 8 is small, and an almost equally good representation of the experimental isotherms can be obtained by considering only the effect... 

Virial Isotherm Equation. No isotherm equation based on idealized physical models provides a generally valid description of experimental isotherms in gas-zeolite systems (19). Instead (6, 20, 21, 22) one may make the assumption that in any gas-sorbent mixture the sorbed fluid exerts a surface pressure (adsorption thermodynamics), a mean hydrostatic stress intensity, Ps (volume filling of micropores), or that there is an osmotic pressure, w (solution thermodynamics) and that these pressures are related to the appropriate concentrations, C, by equations of polynomial (virial) form, illustrated by Equation 3 for osmotic pressure ... 

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