Quasi-surface equilibrium approximation

The Quasi-Surface Equilibrium Approximation If we suppose that the adsorption and desorption processes are fast compared to the surface reaction, we can estimate the surface concentrations from the equilibrium constants. With the Langmuir adsorption isotherm, the following relations result for the simple monomolecular reaction presented in Equation 2.107. 

The Most Abundant Reaction Intermediate (MARI) approximation is a further development of the quasi-equilibrium approximation. Often one of the intermediates adsorbs so strongly in comparison to the other participants that it completely dominates the surface. This intermediate is called the MARI. In this case Eq. (156) reduces to... 

The quasi-equilibrium approximation relies on the assumption that there is a single rate-determining step, the forward and reverse rate constants of which are at least 100 times smaller than those of all other reaction steps in the kinetic scheme. It is then assumed that all steps other than the rds are always at equilibrium and hence the forward and reverse reaction rates of each non-rds step may be equated. This gives simple potential relations describing the varying activity of reaction intermediates in terms of the stable solution species (of known and potential-independent activity) that are the initial reactants or final products of the reaction. The variation of the activities of reaction intermediates is, however, restricted by either the hypothetical solubility limit of these species or, in the case of surface-confined reactions and adsorbed intermediates, the availability of surface sites. In both these cases, saturation or complete coverage conditions would result in a loss of the expected... 

Quasi-equilibrium approximations can be applied to the adsorption steps and a total balance gives a relation between the concentrations of the surface species ... 

The Four Kinetic Regimes of Adsorption from Micellar Solutions In the theoretical model proposed in Refs. [149,150], the use of the quasi-equilibrium approximation (local chemical equilibrium between micelles and monomers) is avoided. The theoretical problem is reduced to a system of four nonlinear differential equations. The model has been applied to the case of surfactant adsorption at a quiescent interface [150], that is, to the relaxation of surface tension and adsorption after a small initial perturbation. The perturbations in the basic parameters of the micellar solution are defined in the following way ... 

In this approximation we assume that one elementary step determines the rate while all other steps are sufficiently fast that they can be considered as being in quasi-equilibrium. If we take the surface reaction to AB (step 3, Eq. 134) as the rate-determining step (RDS), we may write the rate equations for steps (1), (2) and (4) as ... 

It is important to realize that the assumption of a rate-determining step limits the scope of our description. As with the steady state approximation, it is not possible to describe transients in the quasi-equilibrium model. In addition, the rate-determining step in the mechanism might shift to a different step if the reaction conditions change, e.g. if the partial pressure of a gas changes markedly. For a surface science study of the reaction A -i- B in an ultrahigh vacuum chamber with a single crystal as the catalyst, the partial pressures of A and B may be so small that the rates of adsorption become smaller than the rate of the surface reaction. 

In conclusion it should be noted that the indicated lowering of the dimension of the system of equations in the quasi-chemical approximation can be used not only in problems describing the equilibrium and kinetics of surface processes for the rapid surface mobility of particles in steady-state conditions, but also in non-steady conditions. In the latter case, the derivatives of the functions Y j(r) or Y j(r) °n the left-hand sides of the equations are linearly related to one another, and for integration of the system of equations with respect to time they must be determined preliminarily from the relevant system of equations. Notwithstanding this circumstance, the indicated replacement of the variables noticeably diminishes the calculation difficulties in solving the problem. 

In these approximations, as well as in higher ones, one finds that when w < 0 (attraction) there exists a critical temperature below which a first order phase change will be observed—a sudden condensation, as the equilibrium gas pressure is increased, from a dilute localized monolayer to a relatively condensed localized monolayer. For a plane square surface lattice of sites, the Bragg-Williams approximation gives — w/kTc = 1 and the quasi-chemical approximation — w/kTc = 1.386. 

Quasi -chemical approximation of the lattice gas model assumes that the adsorbate maintains an equilibrium distribution on the surface. The lattice gas model with this approximation has been used for the description of TPD spectra as well as for reaction of gases on metal surfaces in some instances. 

It is often assumed that the quasi-Fermi level of electrons in the space charge region does not deviate substantially from the dark Fermi level, but this is only an approximation because, as shown above, the equilibrium concentration of electrons can fall to very low values near the interface. As the local electron density can be increased significantly by illumination, the local quasi-Fermi level of electrons will increase towards the surface (not shown in Figure 18.3). If surface recombination occurs, there will be a flux of electrons into the surface states, casing the quasi-Fermi level to pass through a maximum. The change in... 

From all that was said above, it follows that the polymer alloy is a comph-cated midtiphase system with properties which are determined by the properties of constituent phases. It is very important to note that if, on the macrolevel, the thickness of the interphase regions is low, as compared with the size of the polymer species, for small sizes of the microregions of phase separation such approximation is not vahd. In comparison with the size of the microphase regions, the thickness of the interphase may be of the same order of magnitude. Therefore, they should be taken into accoiuit as an independent quasi-phase in calculation of properties of polymer alloys. We say quasi-phase because these region are not at equilibrium and are formed as a result of the non-equilibrium, incomplete phase separation. The interphase region may be considered as a dissipative structure, formed in the coiu-se of the phase separation. Although it is impossible to locate its position in the space (the result of arbitrary choice of the manner of its definition), its representation as an independent phase is convenient for model calculations (compare the situation with calculations of the properties of filled polymer systems, which takes into account the existence of the surface layer). 

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