Adsorbate cross-sectional areas

Using the BET equation to determine the monolayer weight, and with 

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The designer now needs to make some estimates of mass transfer. These properties are generally well known for commercially available adsorbents, so the job is not difficult. We need to re-introduce the adsorber cross-section area and the gas velocity in order to make the required estimates of the external film contribution to the overall mass transfer. For spherical beads or pellets we can generally employ Eq. (7.12) or (7.15) of Ruthven s text to obtain the Sherwood number. That correlation is the mass transfer analog to the Nusselt number formulation in heat transfer ... 

The success of kinetic theories directed toward the measurements of surface areas depends upon their ability to predict the number of adsorbate molecules required to exactly cover the solid with a single molecular layer. Equally important is the cross-sectional area of each molecule or the effective area covered by each adsorbed molecule on the surface. The surface area then, is the product of the number of molecules in a completed monolayer and the effective cross-sectional area of an adsorbate molecule. The number of molecules required for the completion of a monolayer will be considered in this chapter and the adsorbate cross-sectional area will be discussed in Chapter 6. 

Strong interactions with the surface lead to localized adsorption which constrains the adsorbate to a specific site. The effective adsorbate cross-sectional area will then reflect the spacing between sites rather than the actual adsorbate dimensions. 

For surface area determinations the ideal adsorbate should exhibit BET C values sufficiently low to preclude localized adsorption. When the adsorbate is so strongly tied to the surface as to be constrained to specific adsorption sites, the adsorbate cross-sectional area will be determined more by the adsorbent lattice structure than by the adsorbate dimensions. This type of epitaxial adsorption will lead to decreasing measured surface areas relative to the true BET value as the surface sites become more widely spaced. 

In either of the preceding cases, very high or very low C values, any attempt to calculate the effective adsorbate cross-sectional areas from the bulk liquid properties will be subject to considerable error. Nitrogen, as an adsorbate, exhibits the unusual property that on almost all surfaces its C value is sufficiently small to prevent localized adsorption and yet adequately large to prevent the adsorbed layer from behaving as a two-dimensional gas. 

Kiselev and Eltekov established that the BET C value influences the adsorbate cross-sectional area. They measured the surface area of a number of adsorbents using nitrogen. When the surface areas of the same adsorbents were measured using n-pentane as the adsorbate the cross-sectional areas of n-pentane had to be revised in order to match the surface areas measured using nitrogen. It was found that the revised areas... 

Since (C — 1)/ (C — 1), (0q )m > calculated from a BET plot, there exists a potential means of predicting the cross-sectional area variation relative to nitrogen. On surfaces that contain extensive porosity, which exclude larger adsorbate molecules from some pores while admitting smaller ones, it becomes even more difficult to predict any variation in the adsorbate cross-sectional area by comparison to a standard. 

Table 6.1 lists some approximate adsorbate cross-sectional areas. Because the adsorbates listed are used at various temperatures and on vastly different adsorbents the values are only approximate. 

A fractal surface of dimension D = 2.5 would show an apparent area A app that varies with the cross-sectional area a of the adsorbate molecules used to cover it. Derive the equation relating 31 app and a. Calculate the value of the constant in this equation for 3l app in and a in A /molecule if 1 /tmol of molecules of 18 A cross section will cover the surface. What would A app be if molecules of A were used ... 

To illustrate, consider the hmiting case in which the feed stream and the two liquid takeoff streams of Fig. 22-45 are each zero, thus resulting in batch operation. At steady state the rate of adsorbed carty-up will equal the rate of downward dispersion, or afV = DAdC/dh. Here a is the surface area of a bubble,/is the frequency of bubble formation. D is the dispersion (effective diffusion) coefficient based on the column cross-sectional area A, and C is the concentration at height h within the column. 

In general, the BET equation fits adsorption data quite well over the relative pressure range 0.05-0.35, but it predicts considerably more adsorption at higher relative pressures than is experimentally observed. This is consistent with an assumption built into the BET derivation that an infinite number of layers are adsorbed at a relative pressure of unity. Application of the BET equation to nonpolar gas adsorption results is carried out quite frequently to obtain estimates of the specific surface area of solid samples. By assuming a cross-sectional area for the adsorbate molecule, one can use Wm to calculate specific surface area by the following relationship ... 

Before moving on to true mass transport issues it is worthwhile to point out that the quantity vep(Y - Yp) is in fact the water load per unit of time (whichever units of time v is expressed in.) Further that with an a-priori design process we may not know the required cross sectional area for flow and hence it may be more convenient to multiply the above relationship by and thus obtain the adsorbable contaminant input rate ... 

The amount of oxygen adsorbed in a monolayer on 1 g of a sample of silica gel at low temperature, measured as the volume of oxygen adsorbed, was 105 cill at 300 K and 1 atm. If the cross-sectional area for oxygen is taken as 14 A, determine the surface area per gram for this silica sample. 

A reasonable approximation of the cross-sectional area of adsorbates was proposed by Emmett and Brunauer. They assumed the adsorbate molecules to be spherical and by using the bulk liquid properties they calculated the cross-sectional area from... 

The conclusion, based on the above factors, is that surface areas calculated from equation (4.13) usually give different results depending upon the adsorbate used. If the cross-sectional areas are arbitrarily revised to give surface area conformity on one sample, the revised values generally will not give surface area agreement when the adsorbent is changed. 

In those instances of very high C values, the fraction of surface uncovered by adsorbate again increases, as a result of epitaxial deposition on specific surface sites, which when widely spaced, would lead to high apparent cross-sectional areas. 

In summary, it may be concluded that the uncertainty in calculating absolute cross-sectional areas, the variation in cross-sectional areas with the BET C value and the fact that on porous surfaces less area is available for larger adsorbate molecules all point to the need for a universal, although possibly arbitrary, standard adsorbate. The unique properties of nitrogen have led to its acceptance in this role with an assigned cross-sectional area of 16.2 usually at its boiling point of —195.6 °C. 

Argon at liquid-nitrogen temperature exhibits an equilibrium pressure of 187 torrs. It offers the advantage of a lower vapor pressure than nitrogen, which will reduce the void volume error while retaining ease of pressure measurements. However, the cross-sectional area of argon is not well established and appears to vary according to the surface on which it is adsorbed. 

From the above equation, the variation of equilibrium disjoining pressure and the radius of curvature of plateau border with position for a concentrated emulsion can be obtained. If the polarizabilities of the oil, water and the adsorbed protein layer (the effective Hamaker constants), the net charge of protein molecule, ionic strength, protein-solvent interaction and the thickness of the adsorbed protein layer are known, the disjoining pressure II(x/7) can be related to the film thickness using equations 9 -20. The variation of equilitnium film thickness with position in the emulsion can then be calculated. From the knowledge of r and Xp, the variation of cross sectional area of plateau border Qp and the continuous phase liquid holdup e with position can then be calculated using equations 7 and 21 respectively. The results of such calculations for different parameters are presented in the next session. 

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