Gibbs energy excess

Excess properties find application in the treatment of liquid solutions. Of primary importance for engineering calculations is the excess Gibbs energy G, because its canonical variables are T, P, and composition, the variables usually specified or sought in design calculations. Knowing as a function of T, P, and composition, one can in principle compute from it all other excess properties. 

The excess volume for liquid mixtures is usually small, and in accord with Eq. (4-249) the pressure dependence of is usually ignored. Thus, engineering efforts to model G center on representing its composition and temperature dependence. Eor binary systems at constant T, C becomes a function of just Xi, and the quantity most conveniently represented by an equation is G /xpc RT. The simplest procedure is to express this quantity as a power series in Xi  

An equivalent power series with certain advantages is the Redlich-Kister expansion [Redlich, Kister, and Turnquist, Chem. Eng. Progr. Symp. Ser. No. 2, 48 49-61 (1952)]  

In application, different truncations of this expansion are appropriate, and for each truncation specific expressions for In Yi and In Y2 result from application of Eq. (4-251). When all parameters are zero, G /RT = 0, and the solution is ideal. If B = C = = 0, then 

The symmetric nature of these relations is evident. The infinite dilution values of the activity coefficients are In yf = In y2 = A. 

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Since the accuracy of experimental data is frequently not high, and since experimental data are hardly ever plentiful, it is important to reduce the available data with care using a suitable statistical method and using a model for the excess Gibbs energy which contains only a minimum of binary parameters. Rarely are experimental data of sufficient quality and quantity to justify more than three binary parameters and, all too often, the data justify no more than two such parameters. When data sources (5) or (6) or (7) are used alone, it is not possible to use a three- (or more)-parameter model without making additional arbitrary assumptions. For typical engineering calculations, therefore, it is desirable to use a two-parameter model such as UNIQUAC. 

Null (1970) discusses some alternate models for the excess Gibbs energy which appear to be well suited for systems whose activity coefficients show extrema. 

In most cases only a single tie line is required. When several are available, the choice of which one to use is somewhat arbitrary. However, our experience has shown that tie lines which are near the middle of the two-phase region are most useful for estimating the parameters. Tie lines close to the plait point are less useful, since no common models for the excess Gibbs energy can adequately describe the flat region near the... 

Guffey and Wehe (1972) used excess Gibbs energy equations proposed by Renon (1968a, 1968b) and Blac)c (1959) to calculate multicomponent LLE. They concluded that prediction of ternary data from binary data is not reliable, but that quarternary LLE can be predicted from accurate ternary representations. Here, we carry these results a step further we outline a systematic procedure for determining binary parameters which are suitable for multicomponent LLE. 

A liquid-phase model for the excess Gibbs energy provides... 

VLE data are correlated by any one of thirteen equations representing the excess Gibbs energy in the liquid phase. These equations contain from two to five adjustable binary parameters these are estimated by a nonlinear regression method based on the maximum-likelihood principle (Anderson et al., 1978). 

The excess Gibbs energy is of particular interest. Equation 160 may be written for the special case of species / in an ideal solution, with replaced by xj in accord with the Lewis-RandaH rule ... 

The difference on the left is the partial excess Gibbs energy G y the dimensionless mXio J on the right is called the activity coefficient of species i in solution, y. Thus, by definition. 

A.ctivity Coefficients. Activity coefficients in Hquid mixtures are directiy related to the molar excess Gibbs energy of mixing, AG, which is defined as the difference in the molar Gibbs energy of mixing between the real and ideal mixtures. It is typically an assumed function. Various functional forms of AG give rise to many of the different activity coefficient models found in the Hterature (1—3,18). Typically, the Hquid-phase activity coefficient is a function of temperature and composition expHcit pressure dependence is rarely included. 

The excess Gibbs energy is of particular interest. Equation (4-77) may be written ... 

The heat of mixing (excess enthalpy) and the excess Gibbs energy are also experimentally accessible, the heat of mixing by direcl measurement and G (or In Yi) indirectly as a prodiicl of the reduction of vapor/hqiiid eqiiihbriiim data. Knowledge of H and G allows calculation of by Eq. (4-13) written for excess properties. 

Gamma/Phi Approach For many XT E systems of interest the pressure is low enough that a relatively simple equation of state, such as the two-term virial equation, is satisfactoiy for the vapor phase. Liquid-phase behavior, on the other hand, may be conveniently described by an equation for the excess Gibbs energy, from which activity coefficients are derived. The fugacity of species i in the liquid phase is then given by Eq. (4-102), written... 

Thermodynamics gives limited information on each of the three coefficients which appear on the right-hand side of Eq. (1). The first term can be related to the partial molar enthalpy and the second to the partial molar volume the third term cannot be expressed in terms of any fundamental thermodynamic property, but it can be conveniently related to the excess Gibbs energy which, in turn, can be described by a solution model. For a complete description of phase behavior we must say something about each of these three coefficients for each component, in every phase. In high-pressure work, it is important to give particular attention to the second coefficient, which tells us how phase behavior is affected by pressure. 

Following the idea of van Laar, Chueh expresses the excess Gibbs energy per unit effective volume as a quadratic function of the effective volume fractions. For a binary mixture, using the unsymmetric convention of normalization, the excess Gibbs energy gE is found from6... 

We consider a binary liquid mixture of components 1 and 3 to be consistent with our previous notation, we reserve the subscript 2 for the gaseous component. Components 1 and 3 are completely miscible at room temperature the (upper) critical solution temperature Tc is far below room temperature, as indicated by the lower curve in Fig. 27. Suppose now that we dissolve a small amount of component 2 in the binary mixture what happens to the critical solution temperature This question was considered by Prigogine (P14), who assumed that for any binary pair which can be formed from the three components 1, 2 and 3, the excess Gibbs energy (symmetric convention) is given by... 

The activity coefficient y,fpr) is determined by differentiation of gE, the molar excess Gibbs energy at reference pressure Pr,... 

Therefore, if we have information on the partial molar volumes and the excess Gibbs energy of the ternary system, we can use Eqs. (119)—(122) to find the ends of the tie lines which comprise the coexistence curve. 

To illustrate such a calculation, Balder (Bl) considered a simple case wherein he assumed that the (symmetric) excess Gibbs energy of the ternary system is given by a two-suffix Margules expansion ... 

R. C. Pemberton and C. J. Mash. "Thermodynamic Properties of Aqueous Non-Electrolyte Mixtures II. Vapour Pressures and Excess Gibbs Energies for Water-)- Ethanol at 303.15 to... 

At constant temperature and pressure the excess Gibbs energy of the surface layer depends on surface area S and on the composition of the layer (i.e., on the excess amounts of the components). When there are changes in surface area and composition (which are sufficiently small so that accompanying changes in parameters a... 

Bissell, T.G., Williamson, A.G. (1975) Vapour pressures and excess Gibbs energies of n-hexane and of n-heptane + carbon tetrachloride and + chloroform at 298.15 K. J. Chem. Thermodyn. 7, 131-136. 

Harris, K.R., Dunlop, P.J. (1970) Vapor pressures and excess Gibbs energies of mixtures of benzene with chlorobenzene, -hexane and -heptane at 25°C. J. Chem. Thermodyn. 2, 801-811. 

Using eq. (3.34) the excess Gibbs energy of mixing is given in terms of the mole fractions and the activity coefficients as... 

For a large number of the more commonly used microscopic solution models it is assumed, as we will see in Chapter 9, that the entropy of mixing is ideal. The different atoms are assumed to be randomly distributed in the solution. This means that the excess Gibbs energy is most often assumed to be purely enthalpic in nature. However, in systems with large interactions, the excess entropy may be large and negative. 

Figure 3.5 The molar Gibbs energy of mixing and the molar excess Gibbs energy of mixing of molten Fe-Ni at 1850 K. Data are taken from reference [3].

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 entropy of mixing of many real solutions will deviate considerably from the ideal entropy of mixing. However, accurate data are available only in a few cases. The simplest model to account for a non-ideal entropy of mixing is the quasi-regular model, where the excess Gibbs energy of mixing is expressed as... 

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