Equation quasi-chemical

When there is a large difference between ys(A) and ys(B) in the equation above, there must be signihcant deparmres from dre assumption of random mixing of the solvent atoms around tire solute. In this case tire quasi-chemical approach may be used as a next level of approximation. This assumes that the co-ordination shell of the solute atoms is hlled following a weighting factor for each of tire solute species, such that... 

In liquid metal solutions Z is normally of the order of 10, and so this equation gives values of Ks(a+B) which are close to that predicted by the random solution equation. But if it is assumed that the solute atom, for example oxygen, has a significantly lower co-ordination number of metallic atoms than is found in the bulk of die alloy, dieii Z in the ratio of the activity coefficients of die solutes in the quasi-chemical equation above must be correspondingly decreased to the appropriate value. For example, Jacobs and Alcock (1972) showed that much of the experimental data for oxygen solutions in biiiaty liquid metal alloys could be accounted for by the assumption that die oxygen atom is four co-ordinated in diese solutions. 

These models are semiempirical and are based on the concept that intermolecular forces will cause nonrandom arrangement of molecules in the mixture. The models account for the arrangement of molecules of different sizes and the preferred orientation of molecules. In each case, the models are fitted to experimental binary vapor-liquid equilibrium data. This gives binary interaction parameters that can be used to predict multicomponent vapor-liquid equilibrium. In the case of the UNIQUAC equation, if experimentally determined vapor-liquid equilibrium data are not available, the Universal Quasi-chemical Functional Group Activity Coefficients (UNIFAC) method can be used to estimate UNIQUAC parameters from the molecular structures of the components in the mixture3. 

While a random distribution of atoms is assumed in the regular solution case, a random distribution of pairs of atoms is assumed in the quasi-chemical approximation. It is not possible to obtain analytical equations for the Gibbs energy from the partition function without making approximations. We will not go into detail, but only give and analyze the resulting analytical expressions. 

Analyses of the defect chemistry and thermodynamics of non-stoichiometric phases that are predominately ionic in nature (i.e. halides and oxides) are most often made using quasi-chemical reactions. The concentrations of the point defects are considered to be low, and defect-defect interactions as such are most often disregarded, although defect clusters often are incorporated. The resulting mass action equations give the relationship between the concentrations of point defects and partial pressure or chemical activity of the species involved in the defect reactions. 

By combining these expressions for defect chemical potentials and coefficients with the relations between the chemical potentials at equilibrium (for example Eqs. (74)) explicit expressions are obtained for the defect concentrations at equilibrium which are quite analogous to the quasi-chemical results (Section IV- A) apart from the presence of the activity coefficients. We consider examples of these equations in later sections. 

So far, we have seen that deviation from ideal behavior may affect one or more thermodynamic magnitudes (e.g., enthalpy, entropy, volume). In some cases, we are able to associate macroscopic interactions with real (microscopic) interactions of the various ions in the mixture (for instance, coulombic and repulsive interactions in the quasi-chemical approximation). In practice, it may happen that none of the models discussed above is able to explain, with reasonable approximation, the macroscopic behavior of mixtures, as experimentally observed. In such cases (or whenever the numeric value of the energy term for a given substance is more important than actual comprehension of the mixing process), we adopt general (and more flexible) equations for the excess functions. 

The Lagrange multipliers have vanished because there are equals numbers of A s and s in the numerator and denominator. The equation is identical to what we have obtained in Sections 3.1.3 for the quasi-chemical approximation. There are however two differences. First, we have another way to determine the 2-site probabilities. Instead of using eqn. (48) or (15) we can use the equation we get by substituting (47) in the restriction (25). Second, this way of determining the 2-site probabilities insures automatically that the solutions fulfill the sum rules. 

The defect structure of wiistite can be discussed from the view of a quasi-chemical equilibrium among defects, similar to the case of Nij O. Assuming that the predominant defects are iron ion vacancies, we obtain the following equations ... 

As usual, the rate of dissociative adsorption (e.g. of 02 on various metals [92, 95, 99, 100]) rapidly decreases with increasing surface coverage. As a rule, this is attributed to the fact that dissociative adsorption requires two unoccupied cells, i.e. the sticking coefficient must be S(9) = S(60) Po (0). If a solid surface adsorbs only molecules A, in the quasi-chemical approximation we will have the set of equations... 

Lowering the Dimension of a System of Equations in the Quasi-Chemical... 

The stages of migration of adsorbed A and B particles are written as (5) jZf+YZg<-+YZf+jZg, where j — A, B / and g are adjacent sites, V is a vacant site (a vacancy). The index a corresponds to the indicated stage numbers. It is enough to consider the interactions of the first and second neighbors in the quasi-chemical approximation. There are two possibilities of the equation constructions for the distributed two-dimensional model, and for point models. In the last subsection the next question will be discussed - How the form of the systems of equations alters for a great difference in the mobilities of the reactants ... 

A closed system of equations in the quasi-chemical approximation consists of next equations relative to the functions ffj- and (r), where 1 < i and j < s. ... 

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. 

Show that the equations for mean field approximation could be derived from equations of the quasi-chemical approximation under condition of jSe -> 0. 

Show that the one- and two-site rates of reactions taking into account a non-ideal behavior of the system in the quasi-chemical approximation at the small density (0 -> 0) transform to equations of the law of acting masses. 

Show that the equality of adsorption and desorption rates for dissociating molecules, derived in the mean field and chaotic approximations for interacting the nearest neighbors, do not satisfy the equations of isotherms in similar approximations (this means the absence of a self-consistency between description of the equilibrium and dynamic characteristics of the system). Check out, that the discussed self-consistency property is fulfilled for equations in the quasi-chemical approximation. 

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

Modern theoretical developments in the molecular thermodynamics of liquid-solution behavior are based on the concept of local composition. Within a liquid solution, local compositions, different from the overall mixture composition, are presumed to account for the short-range order and nonrandom molecular orientations that result from differences in molecular size and intermolecular forces. The concept was introduced by G. M. Wilson in 1964 with the publication of a model of solution behavior since known as the Wilson equation. The success of this equation in the correlation of VLE data prompted the development of alternative local-composition models, most notably the NRTL (Non-Random-Two Liquid) equation of Renon and Prausnitz and the UNIQUAC (UNIversal QUAsi-Chemical) equation of Abrams and Prausnitz. A further significant development, based on the UNIQUAC equation, is the UNIFAC method,tt in which activity coefficients are calculated from contributions of the various groups making up the molecules of a solution. 

Absuleme, J. A. Vera, J. H., "A Generalized Solution Method for the Quasi-Chemical Local Composition Equations," Can. J. Chem. Eng., 63, 845 (1985). 

If molecular densities were determined on the basis of Eq. (3.38), atomic densities might be evaluated by contraction of those results. Equation (3.38) provides a derivation of the previously mentioned conditional density of Eq. (3.4). This point hints at a physical issue that we note. As we have emphasized, the potential distribution theorem doesn t require simplified models of the potential energy surface. A model that implies chemical formation of molecular structures can be a satisfactory description of such molecular systems. Then, an atomic formula such as Eq. (3.35) is fundamentally satisfactory. On the other hand, if it is clear that atoms combine to form molecules, then a molecular description with Eq. (3.38) may be more convenient. These issues will be relevant again in the discussion of quasi-chemical theories in Chapter 7 of this book. This issue comes up in just the same way in the next section. 

The appellation quasi-chemical is acquired from Guggenheim (Guggenheim, 1935 1938). This is because central equations of the theory have a structure that is familiar from chemical considerations this is true of the developments below... 

Figure 7.12 Excess chemical potential of the hard-sphere fluid as a function of density. The open and filled circles correspond to the predictions of the primitive quasi-chemical theory and the self-consistent molecular field theory, respectively. The solid and dashed lines are the scaled-particle (Percus-Yevick compressibility) theory and the Carnahan-Starling equation of state, respectively (Pratt and Ashbaugh, 2003).

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