Quasi-chemical approximation,

The zeroth approximation used in 2 and corresponding to complete randomness cannot be strictly exact. The differences in the interactions will tend to favorize distributions which lower the lattice eneigy and the true average value of Nab will be less or greater than the random value (3.2.10) for w positive or negative, respectively. In other words, non vanishing values of w will introduce a certain order in the mutual distribution of molecules. Before we go into details about the quantitative treament of this effect let us consider the order of magnitude of the correction which may be introduced by this ect. 

Let US assume the validity of the two parameter interaction laws (2.4.2) for each of the three interactions AA, AB, BB and suppose the sizes equal (rAA = rAB = bb )- We then have using the definition (2.7.7) of the parameter 6, 

We see that the parameter m is proportional to 6. The expression for the excess functions obtained in the zeroth approximation are 

Another limitation of w we have to keep in mind is related to the 

Hence we have the following upper limit for w (supposed positive) 

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This is the quasi-chemical approximation introduced by Fowler and Guggenlieim [98] which treats the nearest-neighbour pairs of sites, and not the sites themselves, as independent. It is exact in one dimension. The critical temperature in this approximation is... 

This is the simplest closure and corresponds, for the equilibrium solution, to the quasi-chemical approximation (14). If higher accuracy is required one must keep larger correlators and factor with larger overlap. As an example with 2-site overlap, one writes... 

Finally, we combine these simple approximations for the outer shell terms with the primitive quasi-chemical approximation for the inner shell term from Eq. (33) to obtain,... 

The theories of hydration we have developed herein are built upon the potential distribution theorem viewed as a local partition function. We also show how the quasi-chemical approximations can be used to evaluate this local partition function. Our approach suggests that effective descriptions of hydration are derived by defining a proximal... 

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. 

The average interactions and 3X6 evaluated under the assumption of a random distribution of the A and B molecules in space (random mixing). This point has been discussed by Rice13 using the quasi-chemical approximation the corrections to the various excess functions appear to be of the order of 5-10%. 

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 Quasi-Chemical Approximation. The mean-field approximation ignores all correlation in the occupation of neighboring sites. This is incorrect when there is a strong interaction between adsorbates at such sites. The simplest way to include some correlation is to work with probabilities of occupations of two sites (XY) instead of one site (X). Approximations that do this are generally called pair approximations (not to be confused with pair interactions). There are more possibilities to reduce multi-site probabilities as in eqn. (8) to 2-site probabilities than to 1-site probabilities. This leads to different types of pair approximations. The best-known approximation that is used for Ising models is the Kirkwood approximation, which uses for example ... 

The quasi-chemical approximation does not have this drawback. We will derive it here in a manner that has inspired its name, but there are other derivations possible. Suppose we have a pair of neighboring sites that each are either vacant or occupied by an adsorbate A. If we have two such pairs we can write... 

So far the quasi-chemical approximation has been shown as a way to deal with 2-site probabilities. For lateral interactions we are usually dealing with probabilities of many more sites (e.g., see the 5-site probability in eqn. (8)). The quasi-chemical approximation becomes then much more cumbersome and is hardly ever used. The approach in Section 3.1.4 presents a way that is straightforward to extend to larger numbers of sites, while it can be made to coincide with the quasi-chemical approximation for 2-site probabilities. 

The Maximum-Entropy Principle. The mean-field and the quasi-chemical approximations can be extended to larger clusters. Using the derivation of Section 3.1.3 to obtain a quasi-chemical approximation for a cluster with more sites is quite cumbersome. In this section we present an approach that is new and that unifies various approximations to deal with multi-site probabilities. It is based on the maximum-entropy principle. Suppose we have a cluster of n sites with occupations Xi, X2,. .., X . We define an entropy... 

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. 

According to the quasi-chemical approximation, different pairs are treated as being independent, hence the probability that, for example, n particles of A and m particles of B are localized near the particle A, is... 

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... 

The quasi-chemical approximation gives only qualitative results and appears to be particularly inaccurate at temperatures below the "order-disorder phase transition points of T = 0.567 EAA at 0 = 1/2. 

Here <5gy — s j — e(y C" is the statistical weight of the configuration with n particles of A around the central particle / having the z neighboring sites. The ways of an approximate calculation of the functions 0, n) were discussed in Appendix B. In the quasi-chemical approximation, they are expressed in terms of the function 6, and 0y, where 6, — Nt/N, N, is the number of particles of species / on N total sites, and Oy — (1 + A y)NyjzN (Ny is the number of pairs of particles if). 

To calculate the rate of a reaction, one must use an approximate expression for the concentrations of the quasi-particles AB( [nm] ). In the quasi-chemical approximation, the probabilities of all the multiparticle configurations are approximated in terms of the probabilities of two-body configurations. Here it is possible to use the same division into pairs that was employed in the first formula (19) for the energy of two-body... 

However this way could not give a self-consistent description between equilibrium and dynamical characteristics for elementary stages [80,89]. The lattice-gas model is the unique one that provides a self-consistent description of the equilibrium distribution of molecules as well as their dynamic behavior if the correlation effects are taken into account. Then the first correlator that provides such self-consistent description of elementary stages is a pair correlator 0j (r) (m — 2) in the quasi-chemical approximation. In the non-equilibrium conditions to calculate the unknown functions 6j (r) the kinetic Eq. (28) should be used. 

More exact than the quasi-chemical approximation (QCA) is the Kirkwood superposition approximation since if takes into account the indirect correlations. The form of the components in the right-hand side of... 

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 ... 

The stage of adsorption is the simplest elementary process among the other surface processes. It can be both a main process in adsorption and one of the stages of complex interface process. At least one of the adsorption or desorption stage is always presented in any surface process. In the theory of desorption process, the AC was introduced independently for the mono-and bimolecular desorption processes by different authors [107,108] in 1974. In both papers the quasi-chemical approximation has been used. Flowever, actual computations [107] have been performed at e — 0 (the collision model). They have shown that TDS slitting is caused even by a slight repulsion e <0.05 des. The expressions obtained for the desorption rates have been applied to TDS computations for H2/W(100), CO/W(210), and N2/W(100) [109,110]. 

As shown in previous subsection, the activation energy of desorption coincides with the adsorption heat if there is no lateral interaction of the activated desorption complex and its surrounding, e — 0. This is the model that Jones and Perry invoked [144,145] to describe thermal desorption in the Hg/W(100) system. They were able to reproduce the experimental data, taking K(l(d)=K() — const, and Eej(6)=Q(6) the adsorption heat Q(6) was calculated with the quasi-chemical approximation for a homogeneous surface. The calculated and experimental spectra were found to be in a... 

The function hh( h) should be calculated by using actual approximations. To calculate the thermodynamic and kinetic characteristics of the system in a self-consistent manner, it is necessary to use the quasi-chemical approximation, because it takes into account nearest-order correlations (which are absent in the mean-field approximation) [80]. 

Fig. 8.13. Concentration dependences of the relative change in the sheer modulus (1, 2) tj(6u) for the equilibrium state of the system and (3, 4) rj (9n) for the non-equilibrium state and (5) a relative change in the volume per palladium atom F(0H) at T = 300 K. Curves 1 and 3 are constructed in the quasi-chemical approximation curves 2 and 4 are in the mean field approximation [213].

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

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. 

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