Entropy of mixing, excess

The ideal solution approximation is well suited for systems where the A and B atoms are of similar size and in general have similar properties. In such systems a given atom has nearly the same interaction with its neighbours, whether in a mixture or in the pure state. If the size and/or chemical nature of the atoms or molecules deviate sufficiently from each other, the deviation from the ideal model may be considerable and other models are needed which allow excess enthalpies and possibly excess entropies of mixing. 

At 42°C the enthalpy of mixing of 1 mole of water and 1 mole of ethanol is — 343.1 J. The vapor pressure of water above the solution is 0.821 p and that of ethanol is 0.509 P2, in which p is the vapor pressure of the corresponding pure liquid. Assume that the vapors behave as ideal gases. Compute the excess entropy of mixing. 

The regular solution approximation is introduced by assuming definition) that the excess entropy of mixing is zero. This requires that the excess free energy equal the excess enthalpy of mixing. For binary mixtures the excess enthalpy of mixing is ordinarily represented by a function of the form... 

This conclusion implies that the excess entropy of mixing is non-zero and that the mixed micelles presumably acquire more internal order than they would by random mixing. An examination of the magnitude of the deviations from the regular solution approximation shows that there must be a large TS contribution to the excess free energy of mixing. 

Haselton H. T, Hovis G. L., Hemingway B. S., and Robie R. A. (1983). Calorimetric investigation of the excess entropy of mixing in analbite-sanidine solid solutions Lack of evidence for Na, K short range order and implications for two feldspar thermometry. Amer. Mineral, 68 398-413. 

The data are then compared to predictions based on a new model which includes an excess enthalpy of mixing and two contributions to the excess entropy of mixing. 

Osborne-Lee et al. (12) have accounted for the additional contribution to the excess entropy of mixing and found the following excess free energy of mixing per amphiphile... 

Thermodynamic definitions show that the first term of Eq. (1) is the enthalpy of mixing, AHu, while the second term is the negative of the excess entropy of mixing, ASm, multiplied by T. When all four parameters are zero, the liquid is ideal with a zero enthalpy and excess entropy of mixing. What has been called the quasiregular model, a = b = 0, has been used by Panish and Ilegems (1972) to fit the liquidus lines of a number of III—V binary compounds. The particular extension of this special case of Eq. (1) to a ternary liquid given by... 

Solid-Solution Models. Compared with the liquid phase, very few direct experimental determinations of the thermochemical properties of compound-semiconductor solid solutions have been reported. Rather, procedures for calculating phase diagrams have relied on two methods for estimating solid-solution model parameters. The first method uses semiem-pirical relationships to describe the enthalpy of mixing on the basis of the known physical properties of the binary compounds (202,203). This approach does not provide an estimate for the excess entropy of mixing and thus... 

As a demonstration of quantitative LLE calculations, we now consider in more detail some of the binary mixtures that we have discussed qualitatively in section 6.3. In Fig. 6.13 we see the excess Gibbs free energy of mixing Gex, the heat of mixing Hex, and the excess entropy of mixing Sex for mixtures of acetone and... 

The situation is quite different for a mixture of 1-propanol and n-hexane (see Fig. 6.15). Here, we find a significant positive excess heat of mixing. The excess entropy of mixing shows a change of sign at about 10% n-hexane. This feature has also been found experimentally in many alcohol-alkane mixtures. 

Figure 8.4. Excess entropy of mixing in a binary system.

For an ideal mixture, the enthalpy of mixing is zero and so a measured molar enthalpy of mixing is the excess value, HE. The literature concerning HE -values is more extensive than for GE-values because calorimetric measurements are more readily made. The dependence of HE on temperature yields the excess molar heat capacity, while combination of HE and GE values yields SE, the molar excess entropy of mixing. The dependences of GE, HE and T- SE on composition are conveniently summarized in the same diagram. The definition of an ideal mixture also requires that the molar volume is given by the sum, Xj V + x2 V2, so that the molar volume of a real mixture can be expressed in terms of an excess molar volume VE (Battino, 1971). 

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). 

Activity coefficients can be related to the partial enthalpies and partial excess entropies of mixing of A1 and O at infinite dilution in M as follows ... 

This reaction stops when the liquid composition reaches point J, where SiC becomes stable in contact with the liquid and precipitated C. At this point, the equilibrium molar fraction of Si dissolved in M, X (Figure 7.2), is related to the partial enthalpy of mixing of Si in M, AHsi(M) (neglecting the partial excess entropy of mixing), and to the molar Gibbs energy of formation of SiC, AGf(SiC), by the equation ... 

AS 00 partial excess entropy of mixing of solute i at infinite dilution in solvent j... 

AGme = excess free energy of mixing of water with the co-solvent AHme = excess enthalpy of mixing water with the co-solvent ASme = excess entropy of mixing water with the co-solvent... 

An improvement that has recently been introduced 21.22 is to treat the particles as moving in cells rather than restrict them to lattice sites. This improved model does lead to an excess entropy of mixing because the vibrational motions of particles in their cells may differ in the mixture and pure liquids. Quantitative estimates of this entropy vary according to the type of cell field used, but the more realistic fields lead to a value of A S which is of the same sign as A F. 

Both these expressions lead to an excess entropy of mixing which has the same sign as the excess free energy and which is such that... 

The most important qualitative conclusion to be drawn is that the excess entropy of mixing due to directional force differences is approximately equal to the excess free... 

The above method can be apphed to a calculation of other excess functions in terms of intermolecular forces. We may note that the excess entropy of mixing is closely related to the excess volume and to the change of the free volume of the solution with composition. We shall not, however, go into any further details here. 

Equation (25.37) enables us to interpret the existence of a positive excess entropy of mixing in the solutions mentioned in paragraph 3. Similarly, for solutions of 7i-heptane + 7i-hexadecane (c/. chap, XXIV, 4) or benzene + diphenyl II the order of magnitude of the excess entropy is in agreement with (25.37). 

In order to calculate the other excess functions one must know the temperature and pressure derivatives of the activity coefficients ya and Yg. The excess entropy of mixing is given by... 

The term regular solutions was first coined by Hildebrand (1929). It was characterized phe-nomenoligically in terms of the excess entropy of mixing. It was later used in the context of lattice theory of mixtures mainly by Guggenheim (1952). It should be stressed that in both the phenomenological and the lattice theory approaches, the regular solution concept applies to deviations from SI solutions, (see also Appendix M). 

Perhaps the most important term in Eq. (5.2-3) is the liquid-phase activity coefficient, and mathods for its prediction have been developed in many forms and by many workers. For binery systems die Van Laar [Eq. (1.4-18)]. Wilson [Eq. (1.4-23)]. NRTL (Eq. (1.4-27)], and UNIQUAC [Eq. (1.4-3 )] relationships are useful for predicting liquid-phase nonidealities, but they require some experimental data. When no data are available, and an approximate nonideality correction will suffice, the UNiFAC approach Eq-(1.4-31)], which utilizes functional group contributions, may be used. For special cases Involving regular solutions (no excess entropy of mixing), the Scatchard-Hiidebmod mathod provides liquid-phase activity coefficients based on easily obtained pane-component properties. 

---