Mixtures excess functions

Similarly, molar excess functions have been determined for various thiazole-solvent binary mixtures (Table 1-46) (307-310). 

The extent of deviation from ideal solution behavior and hence, the magnitude and arithmetic sign of the excess function, depend upon the nature of the interactions in the mixture. We will now give some representative examples. 

Highly efficient syntheses of both hetero- and homoleptic diorganozinc compounds, such as 6 and 7, were achieved by the UV (X > 280 nm) irradiation of mixtures of functionalized organoiodides and diethyl- or diisopropylzinc (Scheme 7).34 Irradiation with ultraviolet light avoids the use of a large excess of diethylzinc, which often furnishes homoleptic diorganozincs rather than the desired heteroleptic ones. Reaction times rarely exceed 2 h and conversions from 55 to 95% were commonly achieved. 

From the quantitative point of view, the success of the cell model of solutions was more limited. For example, a detailed analysis of the excess functions of seven binary mixtures by Prigogine and Bellemans5 only showed a very rough agreement between theory and experiment. One should of course realize here that besides the use of the cell model itself, several supplementary assumptions had to be made in order to obtain numerical estimates of the excess functions. For example, it was assumed that two molecules of species and fi interact following the 6-12 potential of Lennard-Jones ... 

It has been stressed in Section I that it is essentially these four parameters d, p, 0, and a which determine the values of the excess properties of the mixture A + B should all these parameters be equal to zero then all excess functions vanish. 

We report in Table VII the signs of the excess functions reported in the literature for eight binary liquid mixtures of simple molecules the corresponding values of 8 and p for each mixture are given (first component = reference component A) as well as the temperature and TAA. These values of 8 andp have been deduced from Tables V and VI, and the reference component has been chosen in such a way that all the <5 s are positive. 

TABLE VII. Experimental Signs of the Excess Functions of Several Mixtures of Simple Molecules in Relation to the d and p Values... 

We notice first that gE and hE are positive for all mixtures in agreement with the predictions of the APM. To analyze the behavior of the other excess functions we proceed in the following... 

Renon, H., Prauznits, J.M., 1968, Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures, AIChE Journal, 14, 135. 

Various functions have been used to express the deviation of observed behavior of solutions from that expected for ideal systems. Some functions, such as the activity coefficient, are most convenient for measuring deviations from ideality for a particular component of a solution. However, the most convenient measure for the solution as a whole, especially for mixtures of nonelectrolytes, is the series of excess functions (1) (3), which are defined in the foUowing way. 

General Equations of Excess Functions for Nonideal Binary Mixtures... 

So far, we have seen that deviation from ideal behavior may affect one or more thermodynamic magnitudes (e.g., enthalpy, entropy, volume). In some cases, we are able to associate macroscopic interactions with real (microscopic) interactions of the various ions in the mixture (for instance, coulombic and repulsive interactions in the quasi-chemical approximation). In practice, it may happen that none of the models discussed above is able to explain, with reasonable approximation, the macroscopic behavior of mixtures, as experimentally observed. In such cases (or whenever the numeric value of the energy term for a given substance is more important than actual comprehension of the mixing process), we adopt general (and more flexible) equations for the excess functions. 

If 0 0 and A 0, the excess functions do not exhibit symmetrical properties over the compositional field. In this case, the mixture is defined... 

The critical unmixing condition is obtained from equations 3.201 and 3.202 by substituting for the left-side terms the algebraic form of the excess function appropriate to the case under consideration. For a simple mixture, we have, for instance (cf eq. 3.144),... 

Renon, H. and Prausnitz, J.M., Local composition in thermodynamic excess functions for liquid mixtures, AIChE J., 14,135,1968. 

The thermodynamic excess functions for the 2-propanol-water mixture and the effects of lithium chloride, lithium bromide, and calcium chloride on the phase equilibrium for this binary system have been studied in previous papers (2, 3). In this paper, the effects of lithium perchlorate on the vapor-liquid equilibrium at 75°, 50°, and 25°C for the 2-propanol-water system have been obtained by using a dynamic method with a modified Othmer still. This system was selected because lithium perchlorate may be more soluble in alcohol than in water (4). 

Grover, J., Chemical mixing in multicomponent solutions An introduction to the use of Margules and other thermodynamic excess functions to represent non-ideal behavior, pp. 67-97 in Thermodynamics in Geology, ed. by D. G. Fraser, D. Reidel, Dordrecht, The Netherlands, 1977. This review article provides a fine introduction to the thermodynamic theory of mixtures underlying the Margules expansion for adsorbate-species activity coefficients. 

Considerable information concerning structural effects on aqueous salt solutions has been provided by studies of the properties of mixed solutions (Anderson and Wood, 1973). In a mixed salt solution prepared by mixing YAm moles of a salt MX (molality m) with Yhm moles of a salt NX (molality m) to yield m moles of mixture in 1 kg of solvent, if W is the weight of solvent, the excess Gibbs function of mixing Am GE is given by (19) where GE is the excess function for... 

Before surveying aqueous mixtures, it is informative to examine briefly the thermodynamic excess functions for two particular non-aqueous mixtures, (a) acetone + chloroform (Fig. 27) and (b) methyl alcohol + carbon tetrachloride (Fig. 28). 

Figure 27. Thermodynamic excess functions for acetone + chloroform mixtures at 298 K x j = mole fraction of chloroform (Franks and Ives, 1966).

The importance of the excess entropy of mixing in aqueous mixtures explains why many of these systems show phase separation with a lower critical solution temperature (LCST). This phenomenon is rarer—though not unknown—in non-aqueous mixtures (for an example, see Wheeler, 1975). The conditions for phase separation at a critical temperature can be expressed in terms of the excess functions of mixing (Rowlinson, 1969 Copp and Everett, 1953). 

Thus the enthalpy of mixing is a key quantity for a system to show a UCST. For the other excess functions, e.g. VE and Cp, there are no restrictions, but generally at a UCST, Cf< 0 and VE > 0, while at a LCST, Cp > 0 and VE < 0 (Rowlinson, 1969). If Cf is negative at an LCST and remains so as the temperature increases, then HE and SE may change in such a way that the conditions for a UCST are met. Such systems show a closed solubility loop. The mixture water + nicotine is a classic example of such a system. The behaviour of another example, the mixture water + 2-butoxyethanol, is shown in Fig. 29 (Ellis, 1967). 

In practice AG is known for a given T, p and x2 so that the other quantities based on the ratio a must be calculated from the excess functions for the mixture. Differentiation of eqn (26) with respect to yields, using the Gibbs-Duhem equation, In (fi/f2), and hence a. A second differentiation yields d In a/dxj. If eqn (27) is used to fit the Ge data then these quantities can be calculated from the /-coefficients. The arithmetic is tedious but a computer program can be used to advantage here (Blandamer et al., 1975b). Because the... 

In these mixtures, the excess function GE is positive while T SE is large and negative such that T SE > //E. Co-solvents forming such mixtures include monohydric alcohols, acetone, tetrahydro-furan and dioxan which are often used in kinetic studies. 

The mixture dimethyl sulphoxide + water has attracted a great deal of interest. The excess function HE is negative for this mixture at 298 K (Clever and Piggott, 1971 Fox and Whittingham, 1975), as also are GE (Lam and Benoit, 1974 Philippe and Jambon, 1974) and FE-quantities (Lau et al., 1970). A set of smoothed thermodynamic excess functions is shown in Fig. 54 (Kenttamaa and Lindberg, 1960). The dependence on x2 of the isothermal compressibilities of DMSO + water mixtures is quite different from that for the TA monohydric alcohols + water mixtures. The curves for the latter systems show... 

Hexamethylphosphoramide + water also belongs to this class of mixtures, the negative GE and other excess functions show maximum deviations from ideality near x2 = 0-3 consistent with intense intercomponent interaction (Jose et al., 1975 Jambon and Philippe, 1975). 

The model used combines two equations of state and an excess function. It has been already developed to represent the thermodynamic properties of carbon dioxide-hydrocarbons mixtures [1]. 

Briefly, we recall the method described previously [1]. The model excess function- equation of state explained elsewhere [2], allows us to write the molar Helmoltz energy at the mixture... 

The "Excess Function-Equation of State" model is based on the relation between the excess Helmholtz energy of the mixture and the equations of state of its components. The excess Helmholtz energy is calculated in Guggenheim s quasi-reticular theory. We chose van der Waals - like equations of state for the constituents of the mixture ... 

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