Regular solutions chemical potential

These simple expressions may also be obtained from the chemical potentials according to Eqs. (XII-26) and (XII-32) by appropriately changing subscripts and recalling that x in these equations represents the ratio of the molar volumes, which in the present case is unity. Owing to the identity of volume fractions with mole fractions in this case, Eqs. (18) and (19) are none other than the chemical potentials for a regular binary solution in which the heat of dilution can be expressed in the van Laar form. The critical conditions (see Eqs. 2)... 

The behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

For regular solutions [i , the partial free enthalpy (chemical potential) of substance B in solution, calculated with respect to the solid state, also contains the partial heat of mixing Au (the entropy is equal to that of the ideal case). From the expression for A U on p. 358 we have for the partial heat of mixing ... 

Often the liquid and solid are not the same metal and sometimes they are mutually insoluble. When the equilibrium value of the molar fraction of solid component A in a liquid B X < 1, by equalling the chemical potentials of A in the solid and liquid phases, X is related to the regular solution parameter X given by equation (4.3) by ... 

Since, a22IX22 = (p )(, then by combining Equations (377) and (401) we obtain the excess chemical potential of the component, 2, for a regular solution... 

We proceed with illustrative examples for application of the proposed up-scaling scheme to seven soil types with properties listed in Table 1-2. The closed-form solution for degree of saturation (Eq. [23]) was fitted to measured data by optimizing parameters p, go, X, and the chemical potential pd at air entry point (that defines Lmax). Note that the Hamaker constant was estimated beforehand, as described in Estimation of the Effective Hamaker Constant for Solid-Vapor Interactions for Different Soils above. The estimated parameters were then used to calculate the liquid-vapor interfacial area for each soil (Eq. [28]). We used square shaped central pores for all soil types except the artificial sand mixture, where triangular pores were applied to emphasize capillaiy processes over adsorption in sand. I lowcver, the closed-form solutions for retention and interfacial area were derived lo accommodate any regular polygon-shaped central pore. Constants for various shapes are described in Table I-1. The values of primary physical constants employed in (he calculations and (heir units are shown in Table 1-3. 

In this chapter, we apply some of the general principles developed heretofore to a study of the bulk thermodynamic properties of nonelectrolyte solutions. In Sec. 11-1 we discuss conventions for the description of chemical potentials in nonelectrolyte solutions and introduce the concept of an ideal component. In Sec. 11-2, we demonstrate how the concept of solution molecular weight can be introduced into thermodynamics in a natural fashion. Section 11-3 is devoted to a study of the properties of ideal solutions. In Sec. 11-4, we discuss the properties of solutions that can be considered to be ideal when they are dilute but are not necessarily ideal when they are more concentrated. In Sec. 11-5, regular solutions are defined and some of their properties are derived. Section 11-6 is devoted to a study of some of the approximations that prove useful in the derivation of the properties of real solutions. Finally, in Sec. 11-7, some of the experimental techniques utilized for the measurement of chemical potentials and activity coefficients of components in solution are described. 

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... 

The Floiy-Huggins theory generalizes the theory of regular solutions by taking into account the entropic effects associated with the dissimilarity of sizes in the mixing of molecules. The excess chemical potential with regard to the pure bodies is written ... 

The basic approach described here for ideal solutions can be extended to regular solutions, that is, those where mixing is random but where interaction energies differ significantly for the various species present. For a binary mixture of two species with equal molar volumes it is well known that the regular solution approximation leads to the following expression for the bulk phase chemical potentials ... 

Regular solution theory substitutes for the excess chemical potential of component 1 in a phase... 

Figure 4.3 Free energy as a function of composition (Xj) according to the regular solution model showing binodal (B) and spinodal (S) concentrations. The chemical potentials of the two binodal points obtained by the intercept method are shown.

In the case of the VCyNi.y compound. Equation (6) can be generalised. The chemical potentials //, are developed as a function of the molar fractions in solid solution Xf according to the regular matrix solid solution assumption, and the equilibrium fractions Xf are introduced. This leads to the expression ... 

The physical signiflcance of this result may be clearer if we return to the case of a regular solution defined by Eq. 6.3-6. Using the chemical potential derivative in Eq. 6.3-7, we find from Eq. 6.3-19 that... 

As for the non-uniqueness of the solution, there is no method that can bypass this inherent problem. In inverse problems, one of the common practices to overcome the stability and non-uniqueness criteria is to make assumptions about the nature of the unknown function so that the finite amount of data in observations is sufficient to determine that function. This can be achieved by converting the ill-posed problem to a properly posed one by stabilization or regularization methods. In the case of groundwater pollution source identification, most of the time we have additional information such as potential release sites and chemical fingerprints of the plume that can help us in the task at hand. 

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