Mixing and Excess Functions

Deviation functions for mixtures are concerned mainly with variation in composition rather than pressure or density. Consequently, it is convenient to use molar quantities. Molar excess functions are defined by  

Again an important aspect in this definition is the choice of the independent variables. From eq 2.76 two different definitions of excess functions can be obtained  

A required feature of the model is that eq 2.92 can be solved for the pressure. For eq 2.91, Abbott and Mass proposed an approximate relation suitable for practical purposes  

Based on eq 2.89, Abbott and Mass have summarized expressions for the thermodynamic properties. 

Excess functions and residual functions are related. From eqs 2.37 and 2.90 it can be shown the following equality holds  

A perfect solution is one for which the chemical potential of the form p (p, T, Xk) = Pfc(p, T) + RT InjCfe is valid for all values of the mole fraction 

Xk- The molar Gibbs free energy of such a solution is 

If each of the components were separated, the total Gibbs free energy for the components would be the sum G = i i which we have 

the enthalpy of the solution is the same as the enthalpy of the pure components there is no change in the enthalpy of a perfect solution due to mixing. Similarly, by noting that Vm = Gta/ p)r it is easy to see (exc. 8.16) that there is no change in the molar volume due to mixing, i.e. AV ix = 0. Furthermore, since AU = AH - pAV, we see also that AGmix = 0. Thus, for a 

In a perfect solution the irreversible process of mixing of the components at constant p and T is entirely due to the increase in entropy no heat is evolved or absorbed. 

---

Thermodynamic Description. As shown by several authors some knowledge of the thermodynamic mixing and excess functions can be deduced from the demixing behaviour. According to a classical description the following relations hold for critical solution points ... 

Once the species present in a solution have been chosen and the values of the various equilibrium constants have been determined to give the best fit to the experimental data, other thermodynamic quantities can be evaluated by use of the usual relations. Thus, the excess molar Gibbs energies can be calculated when the values of the excess chemical potentials have been determined. The molar change of enthalpy on mixing and excess molar entropy can be calculated by the appropriate differentiation of the excess Gibbs energy with respect to temperature. These functions depend upon the temperature dependence of the equilibrium constants. 

Fig. 48 Excess free energies of mixing as a function of varying film composition at surface pressures of 2.5,5.0,10.0, and 15.0 dyn cm "1 for mixtures of ( )- and meso- (a) C-12 6,6 (b) C-15 6,6 (c) C-18 6,6 (d) C-15 9,9 ketodiacids at 25°C. Reprinted with permission from Arnett et al, 1988b. Copyright 1988 American Chemical Society.

The average interactions and 3X6 evaluated under the assumption of a random distribution of the A and B molecules in space (random mixing). This point has been discussed by Rice13 using the quasi-chemical approximation the corrections to the various excess functions appear to be of the order of 5-10%. 

As for direct emulsions, the presence of excess surfactant induces depletion interaction followed by phase separation. Such a mechanism was proposed by Binks et al. [ 12] to explain the flocculation of inverse emulsion droplets in the presence of microemulsion-swollen micelles. The microscopic origin of the interaction driven by the presence of the bad solvent is more speculative. From empirical considerations, it can be deduced that surfactant chains mix more easily with alkanes than with vegetable, silicone, and some functionalized oils. The size dependence of such a mechanism, reflected by the shifts in the phase transition thresholds, is... 

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. 

Grover J. (1977). Chemical mixing in multicomponent solutions An introduction to the use of Margules and other thermodynamic excess functions to represent non-ideal behaviour. In Thermodynamics in Geology, D. G. Fraser, ed. D. Reidel, Dordrecht-Holland. 

The relative partial molar enthalpies of the species are obtained by using Eqs. (70) and (75) in Eq. (41). When the interaction coefficients linear functions of T as assumed here, these enthalpies can be written down directly from Eq. (70) since the partial derivatives defining them in Eq. (41) are all taken at constant values for the species mole fractions. Since the concept of excess quantities measures a quantity for a solution relative to its value in an ideal solution, all nonzero enthalpy quantities are excess. The total enthalpy of mixing is then the same as the excess enthalpy of mixing and a relative partial molar enthalpy is the same as the excess relative partial molar enthalpy. Therefore for brevity the adjective excess is not used here in connection with enthalpy quantities. By definition the relation between the relative partial molar entropy of species j, Sj, and the excess relative partial molar entropy sj is... 

J. Grover, Chemical mixing in multicomponent solutions. An introduction to the use of Margules and other thermodynamics excess functions to represent nonideal behavior, pp. 67-97 in Thermodynamics in Geology, ed. by D. G. Fraser, D. Reidel, Dordrecht, The Netherlands, 1977. It follows from Eqs. 5.17 and 5.19 that, in general, In fA = Xjat and In fBA = Xjbj. Equation 5.20a is a special case of this relation for a third-order Margules expansion. 

The concept of free volume varies on how it is defined and used, but is generally acknowledged to be related to the degree of thermal expansion of the molecules. When liquids with different free volumes are mixed, that difference contributes to the excess functions (Prausnitz et al., 1986). The definition of free volume used by Bondi (1968) is the difference between the hard sphere or hard core volume of the molecule (Vw= van der Waals volume) and the molar volume, V ... 

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... 

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). 

---