Calculating Site Fractions

In a steady-state calculation, it must be the case that the surface species concentrations (or site fractions) are not changing with time, that is,... 

The vacant-site fraction 0 / is calculated from the requirement that all surface fractions must sum to unity. Hence, the population balance is... 

Obviously, an expression for fractional surface coverage by the intermediate ab is required to calculate the vacant-site fraction v. If... 

Since surface coverage fractions for all species, including the intermediate, are finearly proportional to the vacant-site fraction, it is straightforward to calculate V by stipulating that all fractions must sum to unity. Hence,... 

All the Hongen-Watson models prior to this section have been presented analytically (i.e., in closed form) because solution of equation (14-121) for vacant-site fraction y is trivial when aU (Pi = I, which is consistent with the fact that each gas in the reactive mixture exhibits single-site adsorption. Numerical methods are required to calculate y if one or more gases adsorbs on several adjacent active sites without dissociation. 

Reactant equilibrium constants Kp and affect the forward kinetic rate constant, and all Ki s affect die adsorption terms in the denominator of the Hougen-Watson rate law via the 0, parameters defined on page 493. However, the forward kinetic rate constant does not appear explicitly in the dimensionless simulations because it is accounted for in Ihe numerator of the Damkohler number, and is chosen independently to initiate the calculations. Hence, simulations performed at larger adsorption/desorption equilibrium constants and the same intrapellet Damkohler number implicitly require that the forward kinetic rate constant must decrease to offset the increase in reactant equilibrium constants. The vacant-site fraction on the internal catalytic surface decreases when adsorption/desorption equilibrium constants increase. The forward rate of reaction for the triple-site reaction-controlled Langmuir-Hinshelwood mechanism described on page 491 is proportional to the third power of the vacant-site fraction. Consequently, larger T, s at lower temperature decrease the rate of reactant consumption and could produce reaction-controlled conditions. This is evident in Table 19-3, because the... 

Before the individual L-J parameters are computed from Eq. (14.5), the molar and site fractions (x , and X , respectively) must be calculated [Utracki et al., 2003] ... 

The number of contacts A 2 is calculated for random mixing. Then the probability that a cell will contain a segment of species 1 equals the site fraction, 0j ... 

For Feo.gsO, the lattice parameter a = 0.4352 nm and the density is 5.7 g/cm. Calculate the site fraction of iron vacancies and the number of iron vacancies per cubic centimeter. 

Calculate the deviation from stoichiometry and the site fractions of each structural element related to the point defect. 

From relation [17.5], we can calculate the iron(lll) site fraction and apply relation [17.1] to clarify the variation of conductivity with oxygen pressure. For this, we can use two methods ... 

The LDOS have been calculated using 10 exact levels in the continued fraction expansion of the Green functions. For clean surfaces the quantities A Vi are the same for all atoms in the same plane they have been determined up to p = 2, 4, 6 for the (110), (100) and (111) surfaces, respectively, and neglected beyond. The cluster C includes the atoms located at the site occupied by the impurity and at al the neighbouring sites up to the fourth nearest neighbour. 

Degrees of polymerization can be calculated from quantitative 13C NMR data by considering the number of substituted (reacted) relative to unsubstituted (not yet reacted) ortho and para phenolic carbons where [5] is the sum of substituted ortho and para carbons and [5] + [f/ is the total ortho and para carbons. The fraction of reacted ortho and para sites is denoted by fs [Eq. (7.2)]. Thus, the number-average number of phenol units per chain Or) can be calculated using Eq. (7.3). This leads to a simple calculation of Mn = x x 106 — 14 ... 

These parameters are used calculate the site and mass distribution functions assuming a Schulz-Zimm molecular weight distribution. The Schulz-Zimm parameters are calculated in lines 930-950. The weight fraction of diluent (as a fraction of the amount of polymer) is then sought. If there is no diluent enter 0. If there is a diluent, the functionality and molecular weight of the diluent is requested (line 1040). The necessary expectation values are computed (lines 1060-1150). 

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