Surface fraction

Surface diffusivities are generally strongly dependent on the fractional surface coverage and increase rapidly at surface coverage greater than 80 percent [see for example Yang et al., AIChB J., 19, 1052 (1973)]. For estimation purposes, the correlation of Sladek et al. [Ind. Bng. Chem. Fundam., 13, 100 (1974)] can be used to predict surface diffusivities for gas-phase adsorption on a variety of adsorbents. 

Multiplying through by provides these expressions in terms of the fractional surface concentrations ... 

Conventional bulk measurements of adsorption are performed by determining the amount of gas adsorbed at equilibrium as a function of pressure, at a constant temperature [23-25], These bulk adsorption isotherms are commonly analyzed using a kinetic theory for multilayer adsorption developed in 1938 by Brunauer, Emmett and Teller (the BET Theory) [23]. BET adsorption isotherms are a common material science technique for surface area analysis of porous solids, and also permit calculation of adsorption energy and fractional surface coverage. While more advanced analysis methods, such as Density Functional Theory, have been developed in recent years, BET remains a mainstay of material science, and is the recommended method for the experimental measurement of pore surface area. This is largely due to the clear physical meaning of its principal assumptions, and its ability to handle the primary effects of adsorbate-adsorbate and adsorbate-substrate interactions. 

The measured NMR signal amplitude is directly proportional to the mass of adsorbate present, and the NMR signal versus pressure (measured at a fixed temperature) is then equivalent to the adsorption isotherm (mass of adsorbate versus pressure) [24-25]. As in conventional BET measurements, this assumes that the proportion of fluid in the adsorbed phase is significantly higher than the gaseous phase. It is therefore possible to correlate each relaxation time measurement with the calculated number of molecular layers of adsorbate, N (where N = 1 is monolayer coverage), also known as fractional surface coverage. 

Fig. 3.5.4 Spin-lattice relaxation (normalized to the bulk gas f at the same pressure) as a function of fractional surface coverage (i.e., the number of adsorbate monolayers).

Based on the assumption that hydrogen atoms are adsorded in pairs, instead of the expression of dissociative hydrogen atom adsorption, it s adequate to use H2 adsorption as the fractional surface coverage for the rate equation. 

The two major theories of flocculation, the bridging model (1) and the electrostatic patch model (2, 3 ), provide the conceptual framework for the understanding of polymer-aided flocculation, but they do not directly address the kinetics of the process. Smellie and La Mer (4) incorporated the bridging concept into a kinetic model of flocculation. They proposed that the collision efficiency in the flocculation process should be a function of the fractional surface coverage, 0. Using a modified Smoluchowski equation, they wrote for the initial flocculation rate... 

The flocculation rate dependency on the fractional surface coverage 0 in Equation (1) has been qualitatively confirmed (13, 14), although the maximum rate appears to occur for a surface coverage of less than 50%. The adsorption rate is also a function of 0, and it has been shown (15) for adsorption onto a smooth solid surface that the rate is proportional to the fraction of polymer-free surface area, 1-0. This approach has not... 

It has been assumed that the total surface area of a floe is proportional to the number of singlets in that floe, a reasonable assumption for small floes and open floe structures. The effective fractional surface coverage, 0k, is not necessarily equal to the fractional surface coverage at equilibrium for a given amount of adsorbed polymer. This will be discussed in some detail below. 

Table 8.3 gives their results for the present-day fractional surface densities and their metallicities at the time of formation of the Sun using different assumed mixing times and Fig. 8.36 shows the effect of finite mixing times on the relation... 

The conservation of adsorbed CO on the Pt surface can be expressed, in terms of its fractional surface coverage 0, as... 

Fig. 9.11 The calculated response of a 1.5 mm long silicon PWEF sensor in a Mach Zehnder interferometer configuration, as a film of final thickness d 2 nm and index n 1.5 grows on the waveguide surface. The normalized fractional surface coverage is also shown...

In this case, 4tt r D is the Smoluchowski rate constant, which is useful for describing radial diffusion to a cell surface totally covered with transport sites. The rate corresponding with fractional surface coverages of carriers has been derived both analytically [35,237,348] and numerically [349] by assuming that the radius of the cell is large compared to the carrier radii, a, and that the NR carriers are sufficiently dispersed so as to be independent and noncompeting. The result ... 

Eq. (4.29) can be extended to obtain a fractional surface coverage, 6 time t... 

Figure 2. Adsorption heat ( 1 SD) as a function of fractional surface coverage for phosphate ( ) and salicylate (x) at 10 g goethite/L in 0.05 M NaNO at pH 4.0.

Thus, the surface area A in Eqs. (10.13) and (10.14) must be replaced by the available (free, unoccupied) surface area. This surface area, which is free for the electrode reaction, can be expressed in terms of the surface coverage 9. The fractional surface coverage 6 of the electrode surface by an adsorbed substance is defined as... 

The fractional surface coverage 6 can be calculated from the experimental values of To, T, and t using the equation... 

Galvanostatic Transient Technique Double-Layer Capacitance Measurements. The value of the fractional surface coverage 9 may be inferred by means of doublelayer capacitance data. As discussed in Section 6.9, the double-layer capacitance C may, in turn, be determined by means of a transient technique. In the galvanostatic transient technique (as in Fig. 6.18), the duration of the constant-current (density) pulse is on the order of microseconds. In the microsecond time range the only process taking place at the electrode is charging of the double layer. Flence, in this case, Eq. (6.96) reduces to... 

The relationship between the fractional surface coverage 9 and the double-layer capacitance C may be better understood in terms of the following model. The doublelayer capacitance at the electrode in the presence of adsorption can be viewed as consisting of two capacitors connected in parallel. One capacitor corresponds to the electrode areas that are unoccupied (free) and the other to the electrode areas that are occupied (covered) with adsorbate (13-15). These two condenser-capacitors have different dielectrics and thus different capacitances. The capacitance of a parallel combination of capacitors is equal to the sum of the individual capacitances. 

Electrodeposition of lead-tin alloy films is usually performed in the presence of peptone as an additive. Peptone is adsorbed on the metal surface during the electrodeposition process. The fractional surface coverage Q of the lead-tin electrode may be determined from the double-layer capacitance C measurements, and/or chronopotentiometric measurements. For a solution containing 9.0 g/L of tin and 13.0 g/L of lead, the following relationship between the concentration of peptone, the double-layer capacitance C, and the transition time At is observed (8). 

This is the reason why, for example, the zero order formic acid dehydrogenation may easily be measured on bulk metal catalysts at 200-300°C. whereas the approximately first order ethanol dehydrogenation requires highly activated porous metals of large specific surface in order to become measurable under the same conditions. The same has been shown for the decomposition of formaldehyde, acetic acid, and hydrazine hydrate. In these cases, the fractional surface coverage is simply 1000 times lower than in the case of a zero order reaction. 

The dissociative adsorption of oxygen on tungsten described by the Langmuir isotherm with K = 0.35 KPa l. The fractional surface coverage at a pressure of 1 KPa. Calculated ... 

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