Quasichemical approximation

To gain some qualitative insight into the elfect of lateral interactions it is useful to employ simple analytical approximations in the calculation of the chemical potential, of which the quasichemical approximation is the best suited. We split the chemical potential into a non-interacting part, Eq. (10), and a term due to lateral interactions, and get for the... 

The lattice gas has been used as a model for a variety of physical and chemical systems. Its application to simple mixtures is routinely treated in textbooks on statistical mechanics, so it is natural to use it as a starting point for the modeling of liquid-liquid interfaces. In the simplest case the system contains two kinds of solvent particles that occupy positions on a lattice, and with an appropriate choice of the interaction parameters it separates into two phases. This simple version is mainly of didactical value [1], since molecular dynamics allows the study of much more realistic models of the interface between two pure liquids [2,3]. However, even with the fastest computers available today, molecular dynamics is limited to comparatively small ensembles, too small to contain more than a few ions, so that the space-charge regions cannot be included. In contrast, Monte Carlo simulations for the lattice gas can be performed with 10 to 10 particles, so that modeling of the space charge poses no problem. In addition, analytical methods such as the quasichemical approximation allow the treatment of infinite ensembles. 

The mean-field approximation neglects correlations between the particles and is therefore not very accurate. The quasichemical approximation improves upon this by accounting for the correlations between pairs of neighboring particles. We briefly give the essence of this method, since it will be used in a few studies reported below. Detailed accounts can be found in the literature [6]. 

Within the quasichemical approximation the number of pairs can be expressed through the numbers Ni of cations Ki and solvent molecules Ni and N2-... 

In Section IV.B the energy of an ion was calculated by a simple version of the quasichemical approximation. The same procedure can be used to calculate the potential of mean force 0pinfW of an ion [18], which is the average potential that the ion experiences as a function of its position x in the direction perpendicular to the surface. This consists of two... 

Equation (9.50) is the primitive quasichemical approximation that was used to obtain the results of Figs. 9.1 and 9.2. Primitive emphasizes that the equilibrium constants are obtained with initial neglect of the effects of the outer-shell material, as (9.50)... 

The first point from this development and example is that, although the quasichemical approach is directed towards treating strong attractive - chemical - interactions at short range, it can describe traditional packing problems accurately. The second point is that this molecular-field idea permits us to go beyond the primitive quality noted above of the primitive quasichemical approximation, and specifically to account approximately for the influence of the outer-shell material on the equilibrium ratios Km required by the general theory. This might help with cases of delicate structures noted above with anion hydrates. 

B( n) is the contribution to Bn of all the configurations in which n defects are assigned to m sites. The summation over m in Eq. (205) is essentially similar in character to that employed in 0. Retention of only m = 1, 2 corresponds to the familiar quasichemical approximation of order-disorder theory.33... 

Equations (7) naturally generalize the well-known equations of the quasichemical approximation for adsorbed molecules of the same type. 

Angle brackets in Eqs. (31a) and (31b) denote averaging over the whole spectrum of the states of slow variables. Generally, the states of rapid variables depend on those of slow variables, so that SFj(a) and 8F (r a) cannot be taken out of the angle brackets. An expression in the quasichemical approximation for the free energy that are needed for calculation of the 8Fy(d) and 8F (r d) is given in Appendix A. 

After the seminal work of Guggenheim on the quasichemical approximation of the lattice statistical-mechanical theory[l], various practical thermodynamic models such as excess Gibbs energies[2-3] and equations of state[4-5] were proposed. However, the quasichemical approximation of the Guggenheim combinatory yields exact solution only for pure fluid systems. Therefore one has to resort to numerical procedures to find the solution that is analytically applicable to real mixtures. Thus, in this study we present a new unified group contribution equation of state[GC-EOS] which is applicable for both pure or mixed state fluids with emphasis on the high pressure systems[6,7]. 

The model described is far too elaborate to use in detail to calculate a partition function. However, the average resonance energy can be assigned to pairs of neighboring hydrogen atoms. The calculation of the equilibrium pressure now becomes that of the quasichemical approximation. Lacher (4) has used the quasichemical approximation to calculate the isotherms without justifying the attractive potential between the hydrogen atoms. The partition function assumed lacks the detail to explain either the heat capacities or the hysteresis. 

Skolnick, J., Jaroszewski, L, Kolinski, A. Godzik, A. (1997). Derivation and testing of pair potentials for protein folding. When is the quasichemical approximation correct Prot. Sci. 6, 676-688. 

An explanation of this fact was offered by Pereira et al. [11] and by Huber et al. [12] the overlap of the two solvents at the interface entails an overlap of the two space-charge regions. Thus the average separation between the opposing charges is reduced, and the capacity enhanced. Both groups performed explicit calculations the former employed the quasichemical approximation, while the latter performed Monte Carlo simulations. To some extent these two techniques complement each other the quasichemical approximation should be good at low electrolyte concentrations, while the Monte Carlo simulations, because of the finite ensemble size, are performed for more concentrated solutions. 

Lateral pairwise interactions between nearest neightbour (n.n.) occupied sites can be treated in the quasichemical approximation [45, 46, 64] by considering the equilibrium... 

Miyazawa, S., and Jernigan, R. L. [1985]. Estimation of effective interresidue contact energies from protein crystal structures Quasichemical approximation. Macromolecules, 18, pp. 534-552. 

The idea of treating in one dimension a system which has proved intractable in three is of course not new, and a number of authors have treated one-dimensional solutions. The results have proved to be semi-quantitatively applicable to the corresponding three-dimensional situation, and constitute the justification of the so-called quasichemical approximation. In view of this it might be hoped that the one-dimensional plasma would also serve as a model for the actual three-dimensional system, but it will appear that the equations of state for the two cases are quite different. 

Fig. 5 Thermodynamic functions for the hydration of apolar molecules, quasichemical approximation lattice with orientation-dependent interactions after Besseling (1993)...

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