Enthalpies of mixing

Because most of you are probably too lazy to go and review stuff, we ll briefly mention a couple of pertinent points. First, the interaction between two molecules is described in terms of the potential energy (P.E.). When the molecules are too close they strongly repel, when they are too far apart they don t feel each other. There is some optimum distance apart where their interaction is a maximum, hence P.E. is a minimum, or has the largest negative or attractive value. Second, there are advanced theoretical models that deal with potential functions like this, which we will not consider, but for dispersion and weak polar forces the attractive energy varies as 1/r6, where r is the distance between the molecules. 

36 See, M. M. Coleman, J. F. Graf, P. C. Painter, Specific Interactions and the Miscibility of Polymer Blends, CRC Press, Boca Raton, FL (1995). 

FIGURE 11-9 Schematic representation of nearest neighbor interactions. 

The enthalpy of mixing AH is the heat consumed (AH 0) or generated (AH 0) during the mixing, at constant pressure. If the mixing is exothermic, then the enthalpic term will drive the system towards miscibility. As the entropy provides a measure of disorder or randomness, and the systems always incline to a uniform distribution of energy, ASm is always positive and therefore the entropic term is favorable for mixing. 

the Flory-Huggins interaction parameter, Xiii defined below, is usually employed. This can be determined experimentally using the solubility parameter concept [2,3], which is described later  

The interaction parameter is dimensionless and, in principle, it characterizes the change of enthalpy for each interacting molecule or segment. In polymer blends, Xii was found to depend not only on temperature but also on the composition and molecular weight of the components. Inserting Eq. (3.4) into Eq. (3.3), another expression for AH is obtained  

The interaction parameter is accessible through the determination of Hildebrand solubility parameters d of the blend components [3], using the following 

The enthalpy (heat) of mixing expression for Eq. 2.1 is derived from the relationship  

For dispersive and non-polar (or modest polar) interactions, 12 can be estimated by a geometric mean  

This leads to solubility parameter concepts (discussed later in this chapter), used by Hildebrand [3] to show that  

A discussion of the enthalpy of mixing and derivation of the above relationships can be found in [2,3]. 

A necessary condition for an ideal solid-solution behaviour is that there be zero heat of mixing in forming the solution from its components, eq. (7.6). This condition cannot be fulfilled when differences exist between CFSE s of cations in the end-member components and in the solid-solutions. 

This differential CFSE factor is demonstrated by the formation of hortono-lite, (Mgo5Fe05)2Si04, by the mixing of forsterite and fayalite components. To a close approximation, Mg2+ and Fe2+ ions may be assumed to be randomly distributed in the olivine structure eq., (7.12), so that 0.5 Fe2+ ions per formula unit occupy each of the Ml and M2 positions. The CFSE s of the Fe2+ ion in the Ml and M2 sites of hortonolite are approximately -53.2 and -52.0 kJ/mole, respectively, and in fayalite the corresponding CFSE are -50.9 kJ/mole (Ml site) and -51.2 kJ/mole (M2 site) (eqs (5.2) and (5.3) table 5.16). The formation of hortonolite may be represented as follows  

Therefore, according to this calculation, there is an excess CFSE (enthalpy) of mixing of about -1.6 kJ/(mole of olivine). Similarly, the formation of all intermediate olivines by mixing of Mg2Si04 and Fe2Si04 components results in an excess CFSE of mixing. This is illustrated in fig. 7.3. These results imply that there is a heat of mixing term and that the olivine series is not an ideal solid-solution of forsterite and fayalite. 

This calculation shows that the formation of intermediate liebenbergite by mixing of Mg2Si04 and Ni2Si04 components is accompanied by an excess CFSE of mixing of -5.65 kJ/mole. These results suggest that Mg2+-Ni2+ olivines depart considerably from ideal solution behaviour (Bish, 1981). This is further demonstrated in fig. 7.4 by the compositional variation of excess CFSE of mixing for the suite of synthetic Mg2+-Ni2+ olivines for which site occupancy and CFSE data are available (table 7.2). 

Only in calcic clinopyroxenes, in which Ca2+ ions completely fill the M2 sites and Fe2+ and other transition metal ions occur in the Ml sites alone, is ideal solution behaviour to be expected. This is because cation ordering is not possible in one-site atomic substitution in the pyroxene Ml site. Furthermore, there is an insignificant variation of the CFSE of Fe2+ across the diopside-hedenbergite series ( 5.5.3). 

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Literature references for vapor-liquid equilibria, enthalpies of mixing and volume change for binary systems. 

For an ideal vapor mixture of m components, there is no enthalpy of mixing. The enthalpy of such a mixture is then... 

In Equation (15), the third term is much more important than the second term. The third term gives the enthalpy of the ideal liquid mixture (corrected to zero pressure) relative to that of the ideal vapor at the same temperature and composition. The second term gives the excess enthalpy, i.e. the liquid-phase enthalpy of mixing often little basis exists for evaluation of this term, but fortunately its contribution to total liquid enthalpy is usually not large. 

The second term in Equation (15a) gives the enthalpy of mixing of the condensable components. It is difficult to estimate that enthalpy but fortunately it ma)ies only a small contri-... 

Figure 1 gives an enthalpy-concentration diagram for ethanol(1)-water(2) at 1 atm. (The reference enthalpy is defined as that of the pure liquid at 0°C and 1 atm.) In this case both components are condensables. The liquid-phase enthalpy of mixing... 

Table 1 indicates that the enthalpy of mixing in the liquid phase is not important when calculating enthalpies of vaporization, even though for this system, the enthalpy of mixing is large (Brown, 1964) when compared to other enthalpies of mixing for typical mixtures of nonelectrolytes. 

Significance of Enthalpy of Mixing for the System Ethanol(1)-n-Hexane(2)... 

LIQUID ENTHALPY IS CALCULATED WITH EXCESS ENTHALPY OF MIXING TAKEN... 

Figure Bl.27.6. A calorimeter for enthalpies of mixing in the absence of a vapour space. (Reproduced with pennission from Larkin J A and McGlashan ML 1961 J. Chem. Soc. 3245.)...

Various flow calorimeters are available connnercially. Flow calorimeters have been used to measure heat capacities, enthalpies of mixing of liquids, enthalpy of solution of gases in liquids and reaction enthalpies. Detailed descriptions of a variety of flow calorimeters are given in Solution Calorimetry by Grolier [17], by Albert and Archer [18], by Ott and Womiald [H], by Simonson and Mesmer [24] and by Wadso [25]. 

Enthalpies of mixing have their origin in the forces that operate between individual molecules. Intermolecular forces drop off rapidly with increasing distance of separation between molecules. This means that only nearest neighbors need be considered in the model. 

This approach to solution chemistry was largely developed by Hildebrand in his regular solution theory. A regular solution is one whose entropy of mixing is ideal and whose enthalpy of mixing is nonideal. Consider a binary solvent of components 1 and 2. Let i and 2 be numbers of moles of 1 and 2, 4>, and 4>2 their volume fractions in the mixture, and Vi, V2 their molar volumes. This treatment follows Shinoda. ... 

According to Hildebrand and Scott [5], the enthalpy of mixing per unit volume A/im is related to the volume fraction

solubility parameter 5 of the two components ... 

The ideal enthalpy of mixing is easily obtained from equations (7.7) and (7.8) and the relationship... 

The enthalpy of mixing is the same as the enthalpy of solution, AH ,, but AHmix is used more commonly for the mixing of two liquids. 

Solubility occurs where the free energy of mixing, AG , is negative. This value is related to the enthalpy of mixing, AH, and the entropy of mixing, AiSjjj, by the Gibbs equation ... 

Entropy of mixing is usually (though not always) positive, hence the sign of AG is generally determined by the size and magnitude of AH. For nonpolar molecules, AH is found to be positive and closely similar to the enthalpy of mixing of small molecules. In such a case, the enthalpy of mixing per unit volume can be approximated to ... 

A particular type of nonideal solution is the regular solution which is characterized by a nonzero enthalpy of mixing but an ideal entropy of mixing. Thus, for a regular solution,... 

Thus the integral molar excess free energy of mixing as well as the enthalpy of mixing are independent of temperature for a regular solution. 

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