Fraction of occupied sites

We can think of a heterogeneous catalyst as a collection of active sites (denoted by ) located at a surface. The total number of sites is constant and equal to N (if there is any chance of confusion with N atoms, we will use the symbol N ). The adsorption of the reactant is formally a reaction with an empty site to give an intermediate I (or more conveniently R if we explicitly want to express that it is the reactant R sitting on an adsorption site). All sites are equivalent and each can be occupied by a single species only. We will use the symbol 6r to indicate the fraction of occupied sites occupied by species R, making N6r the number of occupied sites. Hence, the fraction of unoccupied sites available for reaction will be 1 - 0r The following equations represent the catalytic cycle of Fig. 2.7 ... 

Figure 5.11 Ordered superstructure arrangements of vacant sites in cation layers of chalcogenides of NiAs structure. Fraction of occupied sites (a) 1 (b) (c)...

Fig.20. Order parameter profiles m(z)=([pA(z)-pB(z)])/([pA(z)+pB(z)]), where pA(z), pB(z) are densities of A-monomers or B-monomers at distance z from the left wall, for LxLx20 films confining a symmetric polymer mixture, polymers being described by the bond fluctuation model with N=32, ab=- aa=- bb=8 and interaction range 6. Four inverse temperatures are shown as indicated. In each case two choices of the linear dimension L parallel to the film are included. While for e/kBT>0.02 differences between L=48 and L=80 are small and only due to statistical errors (which typically are estimated to be of the size of the symbols), data for e/kBT=0.018 clearly suffer from finite size effects. Broken straight lines indicate the values of the bulk order parameters mb in each case [280]. Arrows show the gyration radius and its smallest component in the eigencoordinate system of the gyration tensor [215]. Average volume fraction of occupied sites was chosen as 0.5. From Rouault et al. [56].

At the point where the initial rate is a maximum, what is the fraction of vacant sites What is the fraction of occupied sites ly both A and B Wliat generalizations can you make from studying this problem ... 

A problem arises here in the definition of surface coverage since C" is not a single molecular weight, It cannot be equated with the fraction of occupied sites. Furthermore, It is assumed that two surface species are in equilibrium, which is not likely. 

The adsorption isotherm for the chemisorbed species, i.e. the equation relating the fraction of occupied sites to the partial pressure in the vapour, p, at a fixed temperature, may be written in terms of the rate constants for adsorption and desorption, kads and /y, v respectively. Then the Langmuir isotherm adsorption equation can be rewritten as... 

The simulation is performed in a grand canonical ensemble (GCE) where all microstates have the same volume (V), temperature and chemical potential under the periodic boundary condition to minimize a finite size effect [30, 31]. For thermal equilibrium at a fixed pu, a standard Metropolis algorithm is repetitively employed with single spin-flip dynamics [30, 31]. When equilibrium has been achieved, the lithium content (1 — 5) in the Li, 3 11204 electrode at a given pu is determined from the fraction of occupied sites. The thermodynamic partial molar quantities oflithium ions are theoretically obtained by fluctuation method [32]. The partial molar internal energy Uu at constant Vand T in the GCE is readily given by [32, 33]... 

Figure 6.31 Dependence of the fraction of occupied sites as a function of a (low affinity binding)...

If 6 be the fraction of occupied sites, assumption (a) leads to the linear relationship found in Fig. 4. 

Let us assume that step ibj is rate determining. Define 6 as the fraction of occupied sites ... 

In Equation 3, is the surface concentration of adsorbed 111 TCA. If gaseous 111 TCA is in equilibrium with adsorbed 111 TCA, and the fraction of occupied sites is small,... 

The frequency of collisions is proportional to the partial pressure, and the probability of adsorption is incorporated in the rate constant k, which has an exponential dependence on temperature. The concentration of vacant sites is expressed as (1 — ff), where 6 is the fraction of occupied sites, and the total site concentration is included in the rate constant kj. Molecules already adsorbed are assumed to have no effect on the rate of adsorption for nearby vacant sites ... 

Formal thermodynamic approaches can be elaborated to describe complex formation using different complexation schemes (see, for instance [14]), where Cm, c l, Cml are the molar concentrations of free metal ions, and the formal species free and bound sites, s is the maximum number of metal ions bound with one macromolecule, Cj is the molar concentration of the real species M,P (/ = 1,. .. s), and 0 is the coverage or fraction of occupied sites in a macromolecule (Eq. 3-5). 

The physical meaning of the function p k) is the density of probability of the stability constant k and is connected with experimental data, i.e. the change in the concentration of metal ions in solution and unreacted functional groups, or the concentration of surface sites occupied by metal ionsy([M]). At p( ) > 0 the equilibrium constant can be presented as an average equilibrium function depending on the molar concentration, Cj, of the real species M,P (/ = 1,. .. s) and on the fraction of occupied sites in a macromolecule, 0. 

The quantity W = yV changes weakly with temperature. After the second equation of (8.8), the fraction of occupied sites, y, satisfies the following relationship in the limit 5 oo ... 

As shown in Figure 8.15, the fraction of occupied sites, y, plotted versus the reciprocal of reduced volume, l/V, follows a master curve for the reduced pressure... 

In conclusion, the compressibility factor for SAN and EVOH copolymers obeys the principle of corresponding states over a wide range of pressure. Deviations are observed only for PS at P > 120MPa (outside the scale in Figure 8.14). Also, the fraction of occupied sites follows a master curve in the range of low reduced pressure. It varies linearly with inverse reduced volume. At high pressures, the quantity y deviates from linearity and approaches unity. Concomitantly, the free volume reduces with increasing pressure. At P odd x/roo n,o. -otir, in-4... 

However, simultaneously occurs desorption of adsorbed A on surface sites. The desorption rate of A is directly proportional to the fraction of occupied sites or adsorbed A molecules, and kd is the desorption constant. Thns,... 

Rearranging, we determine the fraction of occupied sites due to dissociation, thus,... 

Therefore, we can determine the fraction of occupied sites of each component from Equation 10.14, since... 

The fraction of occupied sites f and empty site fraction /o are related to the mass density, p, and the close packed mass density, p, by... 

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