Effective cross-sectional areas

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. 

Consider a solution of concentration of Cw (mol/vol of fluid) entering at IT in a control volume of length Az and effective cross-sectional area A, with a volumetric flow rate Q (Figure 3.34). The reaction takes place with rate (-R) in (mol dissappearing/time vol of the reactor) and the exit concentration is CE (mol/vol of fluid). 

Impact tests may be performed with notched or unnotched specimens, but the results cannot be compared. The presence of a notch induces a great part of the crack initiation energy. The speed of the striker or of the crosshead may be varied from 1ms-1 to several kms-1 for ballistic tests strain rates may vary in the range of 10-1 up to 100 s-1. Very often the energy for failure is divided by the effective cross-sectional area or the deformed volume, in order to calculate, respectively, the surface or the volume resilience. Using the compliance formula and simple assumptions, GIc may be directly calculated (Williams, 1984). 

Effective cross-sectional areas of molecules adsorbed on to solid surfaces are catalogued in reference 72. 

Consequently, the fraction of the fuel surface area exposed at the base of the pores is equal to e. Since corrosion is confined to this exposed area, the rates of the interfacial anodic and cathodic reactions [i.e., Eqs. (7) and (8)] must be multiplied by this factor, e, to account for the reduction in effective cross-sectional area. Figure 17 shows a schematic representation of this model. 

The use of a single fixed value for the effective cross-sectional area of an adsorbed gas molecule sa assumes it is unaffected by the nature of the solid. For most adsorbates on most solids the effective value of s0 varies slightly from solid to solid. Errors due to different values of sa can be avoided by comparing the experimental adsorption isotherm with those for materials, for which the surface has been determined by different, independent methods [12]. 

In principle, isotherms at low partial pressures of the sorbate may be used to determine specific surface areas by the Brunauer-Emmett-Teller (BET) method (G64). In this method, it is assumed that molecules of the sorbate are adsorbed on surfaces that can include the walls of pores, provided that the distance between molecules on opposing walls is large compared with molecular dimensions. From a plot derived from the isotherm, and given the effective cross-sectional area of the sorbate molecule, the specific surface area of the sorbent and the net heat of adsorption are obtained. Using water as sorbate, specific surface areas of about 200 m per g of D-dried paste have typically been obtained for mature cement pastes of normal w/c ratios... 

For the runs on n-butanol solutions, plot the surface tension of the solution y versus the logarithm of the bulk concentration c and determine the slope. The surface concentration (in units of mol cm or mol rcC ) can be calculated from Eq. (14). Express this surface concentration in molecules per square Angstrom, and obtain the effective cross-sectional area per molecule of adsorbed butanol in A. ... 

Let us designate the average volume flux density of water across area A3 of component j by 7y, which is the average velocity of the water movement (Chapter 2, Section 2.4F). A7 can be the root surface area, the effective cross-sectional area of the xylem, or the area of one side of the leaves. In the steady state, the product J v A7 is essentially constant, because nearly all of the water taken up by the root is lost by transpiration that is, the same volume of water moves across each component along the pathway per unit time. We will represent the drop in water potential across component j by AT7 defining the resistance of component j ( ) as follows ... 

If A is the effective cross-sectional area of the quasi-one-dimensional flow and is the stagnation temperature, then for an ideal gas with constant specific heats [122] in variable-area, isentropic flow. 

Eyring pre-exponential factor Effective cross-sectional area of specimen Total cross-sectional area of specimen Area strain... 

Fig. 4 Critical packing parameters and characteristic structure typical to mesoporous silica films. The top shows a cross-section of a micelle, displaying the parameters used to calculate the CPP. The spherical head group for the surfactant represents the effective cross-sectional area that the head group occupies. (View this art in color at www.dekker.com.)...

It is noted that the collision cross-sectional area, also referred to as the total scattering cross section (2.163), can be approximated in different ways too. In one approach the target particles are approximated as point-like particles. Thus, the cross section of the target particles are not considered in calculating the effective cross-sectional area, so oap Trdg/d. To achieve an improved estimate of the effective cross-sectional area the size of the target particles should be taken into account as well, thus oap =... 

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