Chemical potential coverage-dependent

Fig. 5. Chemical potential pa dependence on surface coverage p is the equilibrium value of the chemical potential fi2,i the equilibrium values of surface coverage by phase 2 (vapor) and 1 (condensate) 9c is the critical coverage.

Transfer matrix calculations of the adsorbate chemical potential have been done for up to four sites (ontop, bridge, hollow, etc.) or four states per unit cell, and for 2-, 3-, and 4-body interactions up to fifth neighbor on primitive lattices. Here the various states can correspond to quite different physical systems. Thus a 3-state, 1-site system may be a two-component adsorbate, e.g., atoms and their diatomic molecules on the surface, for which the occupations on a site are no particles, an atom, or a molecule. On the other hand, the three states could correspond to a molecular species with two bond orientations, perpendicular and tilted, with respect to the surface. An -state system could also be an ( - 1) layer system with ontop stacking. The construction of the transfer matrices and associated numerical procedures are essentially the same for these systems, and such calculations are done routinely [33]. If there are two or more non-reacting (but interacting) species on the surface then the partial coverages depend on the chemical potentials specified for each species. 

The thermodynamic functions of fc-mers adsorbed in a simple model of quasi-one-dimensional nanotubes s adsorption potential are exactly evaluated. The adsorption sites are assumed to lie in a regular one-dimensional space, and calculations are carried out in the lattice-gas approximation. The coverage and temperature dependance of the free energy, chemical potential and entropy are given. The collective relaxation of density fluctuations is addressed the dependence of chemical diffusion coefficient on coverage and adsorbate size is calculated rigorously and related to features of the configurational entropy. 

The differences observed in the adsorption isotherm are also qualitatively and quantitatively significant for the entropy. It has been recently shown that the isotherm of adsorption of an ideal adsorbate on a heterogeneous surface can be appreciably improved by taking into account the exact form of S from Eq. (7) instead of the approximate one arising from F-H theory [22], Results for the coverage dependence of the chemical potential (adsorption isotherm) and entropy per site are shown in figs. 1-2 for various fc-mer s sizes and interaction energies [attractive (w<0) as well as repulsive (w>0)]. 

This gives a much weaker coverage dependence for D than the one predicted by using the chemical potential from the FH approximation,... 

Conversely, the correct approach to formulate the diffusion of a single component in a zeolite membrane is to use the MaxweU-Stefan (M-S) framework for diffusion in a nonideal binary fluid mixture made up of species 1 and 2 where 1 and 2 stands for the gas and the zeohtic material, respectively. In the M-S theory it is recognized that to effect relative motions between the species 1 and 2 in a fluid mixture, a force must be exerted on each species. This driving force is the chemical potential gradient, determined at constant temperature and pressure conditions [68]. The M-S diffiisivity depends on coverage and fugacity, and, therefore, is referred to as the corrected diffiisivity because the coefficient is corrected by a thermodynamic correction factor, which can be determined from the sorption isotherm. 

The brush chemical potential of each individual component psbrush or pI-brush depends only on the surface coverage of the same constituent, os or oL, respectively. This is due to the simple form of the overall free energy of the mixed brush layer Ftot (Eq. 72). Individual brush chemical potential is easily evaluated using Eq. (65) for the individual free energy per brush chain specified by Eq. (63). 

This model better reflects the physical reality of particle adsorption processes by considering a three-dimensional motion of the wandering particle within the adsorption layer (see Fig. 36a). Thus, the ASF (blocking function) depends not only on particle coverage and structure but also on the distance from the interface h [112]. One may, therefore, postulate that the ASF is connected with the activity coefficient occurring in the expression for the chemical potential, Eq. (130), by the simple relationship... 

Here f(0) expresses the dependence upon coverage with C HpOq, and /toH is the chemical potential of OH d- It follows from the assumed equilibrium of reaction 26 ... 

The extent of surface coverage (or simply surface coverage), reached as a result of adsorption, is usually denoted as 0. It is a ratio between adsorbed particles number (Nadi) and the number of adsorption sites available at a surface (usually denoted as active sites - Nsurf). 0 = Nads/ surf The chemical equilibrium between adsorbed species and gas phase particles is reached when chemical potentials of adsorbate particles in both phases are equal (the rates of adsorption and desorption are equal) and it is characterized by constant value of surface coverage 9. The temperature dependence of the gas pressurep required for equilibrium between the adsorption and desorption can be calculated from the Clausius-Clapeyron equation [6], Neglecting the volume of the condensed surface phase, this relation becomes ... 

The prefactor, k(0, 7), is discussed in detail by Kreuzer et al. and incorporates such physical processes as the flux of gas molecules to the surface, rotations and vibrations of the adsorbate species, and sticking probabilities for impinging gas molecules. Within this formalism the desorption rate depends on the relationship between chemical potential and coverage, which we have shown can... 

Figure 16. The temporal dependence of the coverage and the chemical potential upper panel), and the coverage-chemical potential loops lowerpanel), (a) Results obtained using lattices of side L = 100 and by taking fiQ = 0.3 eV and r = 10. (b) As in (a) but for r = 100. In (a) the coverage is amplified by a factor oflO for the sake of clarity. So, to properly obtain the actual coverage, the -scale has to be divided by a factor of 10. (Reprinted from Ref. [17], with permission from the the American Chemical Society.)...

These XPS results cause great uncertainty as to the chemical state of emersed Pb(UPD) films on Ag and validate questions raised about partial desorption of UPD films after potential control is lost (3,9). Nevertheless, XPS chemical shifts of the Pb(4f) peaks show a dependence on the emersion potential i.e. the Pb coverage before emersion. Some emersed films also displayed at least two different chemical states of the UPD Pb (Figures 4b an 5b). 

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