Fraction of sites

If the fraction of sites occupied is 0, and the fraction of bare sites is 0q (so that 00 + 1 = 0 then the rate of condensation on unit area of surface is OikOo where p is the pressure and k is a constant given by the kinetic theory of gases (k = jL/(MRT) ) a, is the condensation coefficient, i.e. the fraction of incident molecules which actually condense on a surface. The evaporation of an adsorbed molecule from the surface is essentially an activated process in which the energy of activation may be equated to the isosteric heat of adsorption 4,. The rate of evaporation from unit area of surface is therefore equal to... 

Each lattice site is defined to have z nearest neighbors, and 0i and 02 > respectively, can be used to describe the fraction of sites which are occupied by solvent molecules and polymer segments. The following inventory of interactions can now be made for the mixture ... 

Density The simplest statistical measure that can be used to characterize a configuration is the density pi t), defined to be the average fraction of sites with value a = I at time t. For a disordered configuration, for example, pi = p. 

As an example, consider a system of size 20 x 20, and take N = 100, px = 2 and Pj = 3. What happens if we vary the parameter u between the value 1 and, say, 20 Gerhardt and Schuster found that when this system evolves from a random initial state (using the Moore neighborhood for updates), some combination of four basic behavioral types emerges [gerh89]. Behavioral types - which appear to depend most strongly on the value of the parameter u - are characterized by both the manner in which the fraction of sites that are infected (= / ) varies as a function of time and the kind of transition-wave spatial patterns that develop ... 

The fraction of sites occupied by A, designated a, is known as the Langmuir adsorption isotherm. It is given by... 

At low values of the bulk concentration Bcy surface coverage is proportional to this concentration, but at high values it tends toward a limit of unity. This equation was derived by Irving Langmuir in 1918 with four basic assumptions (1) the adsorption is reversible (2) the number of adsorption sites is limited, and the value of adsorption cannot exceed A° (3) the surface is homogeneous aU adsorption sites have the same heat of adsorption and hence, the same coefficient B and (4) no interaction forces exist between the adsorbed particles. The rate of adsorption is proportional to the bulk concentration and to the fraction 1-9 of vacant sites on the surface = kjil - 9), while the rate of desorption is proportional to the fraction of sites occupied Vj = kjd. In the steady state these two rates are equal. With the notation kjk = B, we obtain Eq. (10.14). 

Figure 3.37. Relative reaction rates as a function of the fraction of sites poisoned.

Obviously, the diffusion coefficient of molecules in a porous medium depends on the density of obstacles that restrict the molecular motion. For self-similar structures, the fractal dimension df is a measure for the fraction of sites that belong... 

If all partial pressures except that of one reactant, say species A, are held constant and the partial pressure of species A is varied, tfye rate will go through a maximum. The rate increases initially with increasing partial pressure of reactant A as the fraction of sites occupied by species A increases. However, as this fraction increases, the fraction occupied by species B declines as B molecules are displaced by A molecules. Eventually one reaches a point where the decline ip the value of 0B more than offsets the increase in 0A and the product 6A6B goes through a maximum. 

The sum of the fractions of sites occupied by the various species and the fraction vacant must be unity. 

Poisoning is caused by chemisorption of compounds in the process stream these compounds block or modify active sites on the catalyst. The poison may cause changes in the surface morphology of the catalyst, either by surface reconstruction or surface relaxation, or may modify the bond between the metal catalyst and the support. The toxicity of a poison (P) depends upon the enthalpy of adsorption for the poison, and the free energy for the adsorption process, which controls the equilibrium constant for chemisorption of the poison (KP). The fraction of sites blocked by a reversibly adsorbed poison (0P) can be calculated using a Langmuir isotherm (equation 8.4-23a) ... 

Kittel and Abrahams 12S) have predicted an approximately Lorentzian magnetic resonance line shape for a system of spins which are randomly distributed over a small fraction of a large number of possible sites. This effect has been observed in electron spin resonance (124)- Kittel and Abrahams estimate that appreciable deviations from Gaussian shape will occur when the fraction of sites occupied, f, is less than 0.1, in the case of spins of / = H iu a simple cubic lattice with the magnetic field directed... 

The occupancy represents the fraction of sites that are occupied. The number of atoms in a position, then, is given by the occupancy, times the multiplicity of the position. 

Experimentally, one does not measure the fraction of sites containing adsorbed solute directly instead, either the number of moles of solute adsorbed per unit weight of adsorbent ns2/w or the number of moles per unit area of adsorbent ns2/A is measured. These quantities are related by the equation... 

This makes a convenient point of contact with theory since models for adsorption inevitably subdivide the surface into an array of adsorption sites that gradually fill as the pressure increases. If 6 is defined as the fraction of sites filled, then 6 = 1 corresponds to monolayer coverage, with 8 < 1 or 6 > 1 to submonolayer and multilayer coverages, respectively. Theoretical isotherms predict how 6 varies with p in terms of some particular model for adsorption. It turns out that a set of experimental points can often be fitted by more than one theoretical isotherm, at least over part of the range of the data that is, theoretical isotherms are not highly sensitive to the model on which they are based. A comparison between theory and experiment with respect to the temperature dependence of adsorption is somewhat more discriminating than the isotherms themselves. 

Figure 4. Typical surface functions—fractions of sites covered...

The influence of diffusion anisotropy on the reaction rate constant k is illustrated in Figure 5-10. In accordance with the fact that lower dimensionality increases the fraction of sites that are encountered more than once by the diffusing particles, the... 

If the fraction of sites randomly occupied by the vacancies is Xy, the jump rate of the host atoms must be... 

The site fraction is the number of species of a particular component that occupy a particular site divided by the total number of sites of that type. For example, in sodium chloride (NaCl) there is a distinction between cation and anion sites. Impurity species and vacancies may also be present. If there is a total of s distinct types of sites (s = 2 in NaCl) and there is a total number, ATjot, of sites of type j on which are distributed N- atoms (molecules) of component i, the fraction of sites of type j occupied by component i is... 

Here (3 is a function of vp, the number of sites active toward recombination in the recombination sphere. (In [110] the concentration and recombination volumes were expressed in units of the volume v0 of the unit cell, and n0 coincides with the fraction of sites or interstiatial sites occupied respectively by vacancies or interstitial atoms.) We note that, in the model being discussed, the cell itself in which a vacancy occurs is considered inactive with respect to recombination of an interstitial on it. 

At equilibrium, the above rates of adsorption and desorption are equated, and an expression is obtained for the fraction of sites occupied /, which appears identical to Equation 5.22a for simple hydrates of one component ... 

Figure 1 Fraction of sites with co-ordination number 4(C4), 6(C6), 7 (C7), 8(Cg) or 9 (C9)for (a) a f.c.c. octahedron and (b) a fc.c. cubo-octahedron, as a function of particle diameter i (after van Hardeveld and Hartog9)...

When the crystal surface contains sites of different adsorption energies the adsorbate will condense area by area. According to Predali and Cases89) the adsorption isotherm will be steplike in relation to condensation of molecules on areas i, and each step i is compared with sites having the same energy 0 j. Developing the isotherm with condensation on inhomogenous surfaces they start with the definition of the fraction of sites a and /S on which condensation occurs between undersaturation Afi and dAft. When all the... 

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