Regular solution model defined

Depending on the type of the alloy, different models can be used to calculate the activity coefficient. For electrodeposited CoPt and FePt alloys which show strong negative departure from ideality,an asymmetric regular solution model is used to define XB (B = Fe or Co) ... 

Here, Wg and Wq2 represent temperature and composition independent constants called Margules parameters. The substitution of Eq. (11) into Eq. (7) yields the equilibrium potential of the less noble component in the alloy. If this expression is subtracted from the equilibrium potential of the elemental phase defined by Eq. (1), the relation between the underpotentially co-deposited alloy composition and corresponding value of underpotential can be obtained. The example of this approach is shown in Fig. 6 where the composition of UPCD CoPt and FePt is measured as a function of deposition underpotential. The solid lines in the plot indicate the fit of the asymmetric regular solution model. It is important to note that Eq. (11) combined with Eqs. (7) and (1) suggest that composition of UPCD AB alloys (CoPt and FePt) is not dependent on A and B (Co and Fe) deposition kinetics (concentrations in the solutions). 

We can see that the regular solution model does not give us the variations of the activity coefficient with the amounts of substance. In fact, it is a family of models which includes several possible subsets - each one defined by a relation 9tj(x). We shall come across two of these subsets in our study the Van Laar equation (see Table 3.3) and the Hildebrand-strictly-regular solution model, which we shall touch on. 

The solubility parameter 5 of a pure solvent defined initially by Hildebrand and Scott based on a thermodynamic model of regular solution theory is given by Equation 4.4 [13] ... 

A simple thermodynamical model derived from the theory of regular solutions [26], can account for the main features of both continuous and discontinuous transitions. In this theoretical framework, so-called Slitcher-Drickamer model, the interaction term in equation (2), r( HsX defined as ) hs(J — hs) where y is an interaction parameter which reflects the tendency for molecules of one type to be surrounded by like molecules (y > 0). So, equation (5) becomes ... 

Natural systems rarely contain perfectly uniform, regular voids. A regular shape model may be unrealistic for macropores and fractures with irregular walls. Thus, it is useful to examine the impact of systematic variations in channel diameter on solute dispersion. Variable shape models are attractive for simulating pore scale dispersion because a single unit cell is often able to capture a wide range of transport processes, from convection in the center of the channel to diffusion in backwater zones near the apex (see Fig. 3-2B). Furthermore, the macroscopic behavior of such models can be predicted from well-defined geometric parameters. 

Most of the recent theories of liquid solution behavior have been based on well-defined thermodynamic or statistical mechanical assumptions, so that the parameters that appear can be related to the molecular properties of the species in the mixture, and the resulting models have some predictive ability. Although a detailed study of the more fundamental approaches to liquid solution theory is beyond the scope of this book, we consider two examples here the theory of van Laar, which leads to regular solution theory and the UNIFAC group contribution model, which is based on the UNIQUAC model introduced in the previous section. Both regular solution theory and the UNIFAC model are useful for estimating solution behavior in the absence of experimental data. However, neither one is considered sufficiently accurate for the design of a chemical process. 

This new, non-ideal model is only one step removed from an ideal solution. All restrictions remain the same except that the intermolecular forces are no longer uniform (hence G and H will be non-ideal, but S should remain ideal, and V nearly so). The special conditions we have just described define what is called a regular solution. 

All systems shown in these figures can be perfectly described by the model defined by Eqs. (3.28)-(3.31) which supports this theoretical model based on Butler s equation for the chemical potentials of the surface layer, and the regular solution theory. In addition, this agreement is due to the certain choice of the dividing surface after Lucassen-Reynders, and to the fact that Eq. (3.31) was used to calculate the mean molar area of the surfactants mixture. It is important to note that in some cases (for mixtures of normal alcohols. Fig. 3.62, and mixtures of sodium dodecyl sulphate (Ci2S04Na) with 1-butanol and 1-nonanol, Figs. 3.63 and... 

Hildebrand has defined a regular solution in which deviations from ideality are attributed only to the enthalpy of mixing, the intermolecular forces being dispersion forces. The equation describing this model is... 

Equation (5.1) includes only the ideal, combinatorial entropy of mixing and the simplest conceivable regular solution type estimate of the enthalpy of mixing based on completely random mixing of monomers mm ( ) = 1 in the liquid state language i referred to as the bare chi parameter since it ignores all aspects of polymer architecture and Interchain nonrandom correlations. For these reasons, the model blend for which Eq. (5.1) is thought to be most appropriate for is an interaction and structurally symmetric polymer mixture. The latter is defined such that the only difference between A and B chains is a v (r) tall potential, which favors phase separation at low temperatures. The closest real system to this idealized mixture is an isotopic blend, where the A and B... 

The second chapter describes the tools used for macroscopic modeling of solutions. The use of limited expansions of the activity coefficient logarithm is presented, before we define simple solution models such as the ideal dilute solution, regular solutions and athermal solutions, on the basis of macroscopic properties. 

AH variables are assumed to be nonnegative. LU, is a slack variable denoting the amount of excess labor available in period t. Note that LU, and LO, cannot both be positive in the same period. The first set of constraints ensures that the inventory levels are consistent across periods, where NI is the net inventory of product i in period t. The second set of constraints defines the net inventory to be either positive or negative, since both variables on the right-hand side cannot be positive in any optimal solution. The third set of constraints models the evolution of the labor level over time, while the fourth set indicates the relationship among overtime, undertime, and the amount of regular labor on hand. 

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