Activity coefficients in liquids

In Section HI, we discussed the relation between fugacities and activity coefficients in liquid mixtures, and we emphasized that we have a fundamental choice regarding the way we wish to relate the fugacity of a component to the pressure and composition. This choice follows from the freedom we have in choosing a convention for the normalization of activity coefficients. 

A.ctivity Coefficients. Activity coefficients in liquid mixtures are direcdy related to the molar excess Gibbs energy of mixing, AGE, 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 AGE give rise to many of the different activity coefficient models found in the literature (1—3,18). Typically, the liquid-phase activity coefficient is a function of temperature and composition explicit pressure dependence is rarely included. 

Pressure is more directly connected to the concept of explosion nevertheless, it is less directly connected to the reactor status, since, for liquid-phase reactors, pressure nonlinearly depends on temperature (trough the vapor pressure relationship) and concentration (through the activity coefficients in liquid phase). Moreover, since pressure measurements are usually less accurate than temperature measurements, they are to be considered in particular for gassy reactions, i.e., when the runaway produces small temperature effects but large amounts of incondensable products in gaseous phase. 

A modern set of methods which can be used to estimate partition coefficients is the group contribution method. These methods were developed to allow chemical engineers to estimate activity coefficients in liquid and polymeric systems. Of the numerous methods developed, UNIFAC, the oldest and most thoroughly tested method, is probably the most universally applicable to a wide variety of substances and sytems despite its known weaknesses (Baner, 1999). The use of UNIFAC and example calculations will be described later in this chapter. 

As we have already seen, activity coefficients in liquid solutions may be derived from partial vapour pressure curves. As an example we give in fig. 21.6 the activity coefficients of methylal and carbon disulphide in mixtures of these substances, calculated from the data in fig. 21.5. 

Today, an impressive number of activity coefficients of electrolytes are known. Not much real innovation is to be expected in the field of experiments, except in the context of special systems such as biological or other confined ones. The general behavior of ions and their consequences on activity coefficients is mainly understood. However, the one problem will remain the underlying interactions and their balance are very subtle and consequently it will always be difficult to predict quantitatively the values of activity coefficients in liquid solutions. 

An extension of the Maier and Saupe theory to two component systems provides a basis for the theoretical interpretation of the solute activity coefficients in liquid crystals. Consider for this purpose a two-component system of rod-like molecules containing the mole fractions x of solute ( ) and (1 — x) of solvent (s). By generalising Equ. (24) one obtains for the average anisotropic dispersion energy of the solute in the mixed solute—solvent system [129]... 

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

Figure 4-7. Vapor-liquid equilibria and activity coefficients in a binary system showing a weak minimum in the activity coefficient of methanol.

Table 13-1, based on the binary-system activity-coefficient-eqnation forms given in Table 13-3. Consistent Antoine vapor-pressure constants and liquid molar volumes are listed in Table 13-4. The Wilson equation is particularly useful for systems that are highly nonideal but do not undergo phase splitting, as exemplified by the ethanol-hexane system, whose activity coefficients are snown in Fig. 13-20. For systems such as this, in which activity coefficients in dilute regions may...

These equations when combined with Eq. (13-5) lead to the following equations for liquid-phase activity coefficients in terms of measurable quantities ... 

Examples of this procedure for dilute solutions of copper, silicon and aluminium shows the widely different behaviour of these elements. The vapour pressures of the pure metals are 1.14 x 10, 8.63 x 10 and 1.51 x 10 amios at 1873 K, and the activity coefficients in solution in liquid iron are 8.0, 7 X 10 and 3 X 10 respectively. There are therefore two elements of relatively high and similar vapour pressures, Cu and Al, and two elements of approximately equal activity coefficients but widely differing vapour pressures. Si and Al. The right-hand side of the depletion equation has the values 1.89, 1.88 X 10- , and 1.44 X 10 respectively, and we may conclude that there will be depletion of copper only, widr insignificant evaporation of silicon and aluminium. The data for the boundaty layer were taken as 5 x lO cm s for the diffusion coefficient, and 10 cm for the boundary layer thickness in liquid iron. 

Some organic compounds can be in solution with water and the mixture may still be a flammable mixture. The vapors above these mixtures such as ethanol, methanol, or acetone can form flammable mixtures with air. Bodurtha [39] and Albaugh and Pratt [47] discuss the use of Raoult s law (activity coefficients) in evaluating the effects. Figures 7-52A and B illustrate the vapor-liquid data for ethyl alcohol and the flash point of various concentrations, the shaded area of flammability limits, and the UEL. Note that some of the plots are calculated and bear experimental data verification. 

The formal Galvani potential, described by Eq. (22), practically does not depend on the concentration of ions of the electrolyte MX. Since the term containing the activity coefficients of ions in both solutions is, as experimentally shown, equal to zero it may be neglected. This results predominantly from the cross-symmetry of this term and is even more evident when the ion activity coefficients are replaced by their mean values. A decrease of the difference in the activity coefficients in both phase is, in addition, favored by partial hydration of the ions in the organic phase [31 33]. Thus, a liquid interface is practically characterized by the standard Galvani potential, usually known as the distribution potential. 

Several equations have been developed to represent the dependence of activity coefficients on liquid composition. Only those of most use in the design of separation processes will be given. For a detailed discussion of the equations for activity coefficients and their relative merits the reader is referred to the book by Reid et al. (1987), Walas (1984) and Null (1970). 

For systems that are only partially miscible in the liquid state, the activity coefficient in the homogeneous region can be calculated from experimental values of the mutual solubility limits. The methods used are described by Reid et al. (1987), Treybal (1963), Brian (1965) and Null (1970). Treybal (1963) has shown that the Van-Laar equation should be used for predicting activity coefficients from mutual solubility limits. 

It must be emphasised that extreme caution needs to be exercised when using predicted values for liquid activity coefficients in design calculations. 

Fredenslund, A., Jones, R. L., Prausnitz, J. M. (1975) Group-contribution estimation of activity coefficients in nonideal liquid mixtures. AIChEJ. 21, 1086-1099. 

About the same time Beutier and Renon (11) also proposed a similar model for the representation of the equilibria in aqueous solutions of weak electrolytes. The vapor was assumed to be an ideal gas and < >a was set equal to unity. Pitzer s method was used for the estimation of the activity coefficients, but, in contrast to Edwards et al. (j)), two ternary parameters in the activity coefficient expression were employed. These were obtained from data on the two-solute systems It was found that the equilibria in the systems NH3+ H2S+H20, NH3+C02+H20 and NH3+S02+H20 could be represented very well up to high concentrations of the ionic species. However, the model was unreliable at high concentrations of undissociated ammonia. Edwards et al. (1 2) have recently proposed a new expression for the representation of the activity coefficients in the NH3+H20 system, over the complete concentration range from pure water to pure NH3. it appears that this area will assume increasing importance and that one must be able to represent activity coefficients in the region of high concentrations of molecular species as well as in dilute solutions. Cruz and Renon (13) have proposed an expression which combines the equations for electrolytes with the non-random two-liquid (NRTL) model for non-electrolytes in order to represent the complete composition range. In a later publication, Cruz and Renon (J4J, this model was applied to the acetic acid-water system. 

The local compostion model is developed as a symmetric model, based on pure solvent and hypothetical pure completely-dissociated liquid electrolyte. This model is then normalized by infinite dilution activity coefficients in order to obtain an unsymmetric local composition model. Finally the unsymmetric Debye-Huckel and local composition expressions are added to yield the excess Gibbs energy expression proposed in this study. 

Other references in Table in discuss applications in precipitation of metal.compounds, gaseous reduction of metals from solution, equilibrium of copper in solvent extraction, electrolyte purification and solid-liquid equilibria in concentrated salt solutions. The papers by Cognet and Renon (25) and Vega and Funk (59) stand out as recent studies in which rational approaches have been used for estimating ionic activity coefficients. In general, however, few of the studies are based on the more recent developments in ionic activity coefficients. 

Group-Contribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures", AIChE Journal, 21(6), 1975... 

We have already seen that, within the range of Nernst s law, the solid/liquid partition coefficient differs from the thermodynamic constant by the ratio of the Henry s law activity coefficients in the two phases—i.e.,... 

Thus, Eq. (KKK) and the analogous logarithmic form of Eq. (Ill) in Box 9.3 predicts that a plot of log Kp against log pL for the partitioning of a series of compounds into liquid particles or into a liquid layer on particles should be a straight line with a slope of 1 if the activity coefficients in the liquid phase, yom, remain constant. 

Xj>0y0pL), where the symbols have their usual meaning of mole fraction, activity coefficient, and liquid vapor pressure, respectively, all in octanol. The mole fraction can be replaced in this expression, since Xj 0 C0(MW0) 103p(), where C0 is the... 

When = 0.90 this gives x2 = xCd = 3.1(10-4) while Eq. (A14) gives Xj = xHg = 0.9922. This is in agreement with the more exact computer calculations whose results are shown in Fig. 31. The activity coefficients in the metal-rich liquid in equilibrium with a solid solution with = 0.9 and at 673°K obtained from a computer calculation also agree to within 3% with the approximate values listed above. Thus the relative stability of CdTe(s) compared to HgTe(s) is a major factor in the tie-lines converging toward the Hg corner. The smallness of Q2/k5, which is determined in part by the large value of 244 for the activity coefficient of the CdTe liquid species, also enters and is less transparent. 

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