The Regular Solution

In real solutions, the thermodynamic magnitudes AG , AH , and AS , deviate from the ideal values, and the excess functions can be defined as the difference between the real values and the ideal values. For example, the excess enthalpy is [1]  

AH = no4 i(p2zwi2 +y2no P zwn +y2no(p zw22 -V2Wo iZWn -V2UO02ZW22 

According to Eq. (2.9), the formation of each 1-2 contact involves breaking half 1-1 and half 2-2 contacts on average. The interaction energy corresponding to the formation of each 1-2 contact, A1P12, is defined according to [2-5]  

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A key feature of this model is that no data for mixtures are required to apply the regular-solution equations because the solubiHty parameters are evaluated from pure-component data. Results based on these equations should be treated as only quaHtative. However, mixtures of nonpolar or slightly polar, nonassociating chemicals, can sometimes be modeled adequately (1,3,18). AppHcations of this model have been limited to hydrocarbons (qv) and a few gases associated with petroleum (qv) and natural gas (see Gas, natural) processiag, such as N2, H2, CO2, and H2S. Values for 5 and H can be found ia many references (1—3,7). 

When started with a smooth image, iterative maximum likelihood algorithms can achieve some level of regularization by early stopping of the iterations before convergence (see e.g. Lanteri et al., 1999). In this case, the regularized solution is not the maximum fikelihood one and it also depends on the initial solution and the number of performed iterations. A better solution is to explicitly account for additional regularization constraints in the penalty criterion. This is explained in the next section. 

The regularized solution is easy to obtain in the case of Gaussian white noise if we choose a smoothness prior measured in the Fourier space. In this case, the MAP penalty writes ... 

We briefly recall here a few basic features of the radial equation for hydrogen-like atoms. Then we discuss the energy dependence of the regular solution of the radial equation near the origin in the case of hydrogen-like as well as polyelectronic atoms. This dependence will turn out to be the most significant aspect of the radial equation for the description of the optimum orbitals in molecules. 

It should be emphasized that we are not interested here specifically by these particular values of e. On the contrary, what is useful here i.e. for the description of optimum orbitals in molecules is to study the variation of the regular solution when e varies continuously. 

A particular type of nonideal solution is the regular solution which is characterized by a nonzero enthalpy of mixing but an ideal entropy of mixing. Thus, for a regular solution,... 

The micellization and adsorption properties of industrial sulfonate/ ethoxylated nonionic mixtures have been assessed in solution in contact with kaolinite. The related competitive equilibria were computed with a simple model based on the regular solution theory (RST). Starting from this analysis, the advantage of adding a hydrophilic additive or desorbing agent to reduce the overall adsorption is emphasized. 

The corresponding monomer/micelle equilibria can be dealt with by the regular solution theory (RST), as shown in particular by Rubingh in 1979 (1). The application of this theory to numerous binary surfactant systems (2 - 4) has followed and led to a set of coherent results (5). 

Comparison Theoretical Equilibrium Calculations and Results of Circulation Tests in Porous-Media To make this interpretation more quantitative, the regular solution theory (RST) was applied to sulfonate/desorbent dynamic equilibria reached inside porous media by using the approach described above. In so doing, we assumed that the slugs injected were sufficiently large and that a new equilibrium was reached at the rear of micellar slug in the presence of desorbent. 

The simplest model beyond the ideal solution model is the regular solution model, first introduced by Hildebrant [9]. Here A mix, S m is assumed to be ideal, while A inix m is not. The molar excess Gibbs energy of mixing, which contains only a single free parameter, is then... 

The regular solution model can be extended to multi-component systems, in which case the excess Gibbs energy of mixing is expressed as... 

The regular solution model (eq. 3.68) is symmetrical about xA = xB =0.5. In cases where the deviation from ideality is not symmetrical, the regular solution model is unable to reproduce the properties of the solutions and it is then necessary to introduce models with more than one free parameter. The most convenient polynomial expression with two parameters is termed the sub-regular solution model. 

These expressions can be simplified since the activity coefficient in the particular case of a regular solution can be expressed by the regular solution constant 2 through eqs. (3.84) and (3.85) ... 

In the two-state model [20,21] the two different species interact and the interaction can be expressed using the regular solution model. Thus the Gibbs energy of the liquid is... 

Here Q is the regular solution constant and xSiB the fraction of Si atoms in silicon state B. By noting that xSiA =1 - xSi B, equation (5.31) becomes... 

We have now derived the phase boundary between the two liquids. By analogy with our earlier examples, the two phases may exist as metastable states in a certain part of the p,T potential space. However, at some specific conditions the phases become mechanically unstable. These conditions correspond to the spinodal lines for the system. An analytical expression for the spinodals of the regular solution-type two-state model can be obtained by using the fact that the second derivative of the Gibbs energy with regards to xsi)B is zero at spinodal points. Hence,... 

The regular solution model, originally introduced by Hildebrand [2] and further developed by Guggenheim [3], is the most used physical model beside the ideal... 

The Gibbs energy for the regular solution of an arbitrary number of A and B atoms follows... 

The excess Gibbs energy of the regular solution, as pointed out in Chapter 3, is a purely enthalpic term ... 

In the derivation of the regular solution model the vibrational contribution to the excess properties has been neglected. However, as a first approximation the vibrational contribution can be taken as independent of the interaction between the different atoms, and this contribution can be factored out of the exponential and taken into account explicitly. The partition function of the solution is then given as... 

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. 

If the second term in the configurational entropy of mixing, eq. (9.42), is zero, the quasi-chemical model reduces to the regular solution approximation. Here, Aab is given by (eq. (9.21). If in addition yAB =0the ideal solution model results. 

The regular model for an ionic solution is similarly analogous to the regular solution derived in Section 9.1. Recall that the energy of the regular solution model was calculated as a sum of pairwise interactions. With two sub-lattices, pair interactions between species in one sub-lattice with species in the other sub-lattice (nearest neighbour interactions) and pair interactions within each sub-lattice (next nearest neighbour interactions), must be accounted for. 

Let us first derive the regular solution model for the system AC-BC considered above. The coordination numbers for the nearest and next nearest neighbours are both assumed to be equal to z for simplicity. The number of sites in the anion and cation sub-lattice is N, and there are jzN nearest and next nearest neighbour interactions. The former are cation-anion interactions, the latter cation-cation and anion-anion interactions. A random distribution of cations and anions on each of... 

NA i [ l is easily derived when the cations are assumed to be randomly distributed on the cation sub-lattice. The probability of finding an AB (or BA) pair is 2Xa+Xb+ in analogy with the derivation of the regular solution in Section 9.1. iVA+B+ is then the product of the total number of cation-cation pairs multiplied by this probability... 

The first term is the ideal entropy of mixing while the second term is the enthalpy of mixing in the regular solution approximation ... 

The equations for the regular solution model for a binary mixture with two sublattices are quite similar to the equations derived for a regular solution with a single lattice only. The main difference is that the mole fractions have been replaced by ionic fractions, and that while the pair interaction is between nearest neighbours in the single lattice case, it is between next nearest neighbours in the case of a two sub-lattice solution. 

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