Heats of mixing

Next we consider the heat of mixing using a simple nearest neighbor model. Let m a be the nearest neighbor binding energy per atom for pure A, Mbb for pure B, and Mab for an A atom next to a B atom. For mole fraction x of B atoms, the probability of atom B next to atom A is X and the total energy for the A atoms is 

The total free energy can be written as F(T,x) = Utot — ES ix- From Equations 12.3 and 12.8, 

Free energy plot for various values of T/Tc, where the consolute temperature Tq is defined by Equation 12.1Z The mixed phase is stable for all x if T= Tc (solid line) but the system will become segregated if T Tc as will be explained in Section 12.4.1. 

The curvature of the free energy will continue to be positive (curve upwards) so long as 

From a purely thermodynamic standpoint there is nothing to prevent us examining all the properties characteristic of the deviation of a solution from ideality in terms of the activity coefficients. Experimentally, however, the derivatives above those of the second order become more and more difficult to measure precisely. In fact, in the discussion of this chapter we shall only consider and  

The above excess functions have been defined for one mole of solution. For the whole system we could write 

It is then easily verified that, taking account of (20.58), 

The heat of mixing is defined as the heat absorbed by the system when Ui moles of component 1 and moles of component 2 are mixed at constant temperature and pressure. 

The heat of mixing is thus equal to the variation, H, in the enthalpy which accompanies the mixing is, by definition, the integral heat of mixing, and is an extensive thermodynamic variable. 

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The special appeal of this approach is that it allows the heat of mixing to be estimated in terms of a single parameter assigned to each component. This considerably simplifies the characterization of mixing, since m components (with m 6 values) can be combined into m(m - l)/2 binary mixtures, so a considerable data reduction follows from tabulating 6 s instead of AH s. Table 8.2 is a list of CED and 6 values for several common solvents, as well as estimated 6 values for several common polymers. 

DMPPO and polystyrene form compatible blends. The two components are miscible in all proportions (59). Reported dynamic—mechanical results that indicate the presence of two phases in some blends apparendy are caused by incomplete mixing (60). Transition behavior of thoroughly mixed blends indicates that the polymers are truly compatible on a segmental level (61). CompatibiUty may be attributed to a %— % interaction between the aromatic rings of the two polymers sufficient to produce a negative heat of mixing. However, the forces are very small, ie, = ca40 J/mol (9.6 cal/g), and any... 

M refer to the density and molecular weight of /, and R is the gas constant. For simplicity, we assume each component to be monodisperse mote complex expressions result when polydispersity is considered (6). This model also assumes the heat of mixing pet unit volume follows a van Laar-type relation where B is... 

The heat of reaction AH is given in joules per gram mole of ammonia formed. Equation 22 ignores the heat of mixing which must be taken into account for apphcation to industrial situations (21,22). The heat evolved in synthesis at 370—540°C is approximately 54,430 kj/mol (23, 400 Btu/lb-mol). 

Figure 14 shows the heat of mixing of sulfuric acid and water (83). Additional data are ia Reference 84. 

Fig. 14. Heat of mixing of sulfuric acid from H2O and H2SO4 at 25°C (83). To convert to cal, divide by 4.184.

For the volume change of mixing and the enthalpy change (heat) of mixing, which are direcdy measurable, AV = and AH =. ... 

When both phases form ideal thermodynamic solutions, ie, no heat of mixing, no volume change on mixing, etc, Raoult s law apphes ... 

Linking this molecular model to observed bulk fluid PVT-composition behavior requires a calculation of the number of possible configurations (microstmctures) of a mixture. There is no exact method available to solve this combinatorial problem (28). ASOG assumes the athermal (no heat of mixing) FIory-Huggins equation for this purpose (118,170,171). UNIQUAC claims to have a formula that avoids this assumption, although some aspects of athermal mixing are still present in the model. 

The heat of mixing (excess enthalpy) and the excess Gibbs energy are also experimentally accessible, the heat of mixing by direcl measurement and G (or In Yi) indirectly as a prodiicl of the reduction of vapor/hqiiid eqiiihbriiim data. Knowledge of H and G allows calculation of by Eq. (4-13) written for excess properties. 

The Wilson parameters A,, NRTL parameters G,, and UNIQUAC parameters X all inherit a Boltzmann-type T dependence from the origins of the expressions for G, but it is only approximate. Computations of properties sensitive to this dependence (e.g., heats of mixing and liquid/hquid solubihty) are in general only qualitatively correct. 

The constant-molar-overflow assumption represents several prior assumptions. The most important one is equal molar heats of vaporization for the two components. The other assumptions are adiabatic operation (no heat leaks) and no heat of mixing or sensible heat effects. These assumptions are most closely approximated for close-boiling isomers. The result of these assumptions on the calculation method can be illustrated with Fig. 13-28, vdiich shows two material-balance envelopes cutting through the top section (above the top feed stream or sidestream) of the column. If L + i is assumed to be identical to L 1 in rate, then 9 and the component material balance... 

When solid alloys have a positive heat of mixing, AHm, complete solid solution occurs above the temperature where TASm is more negative than the heat... 

The drermodynamic data for CuaS-FeS (Krivsky and Schuhmann, 1957) show that drese sulphides mix to form approximately ideal ionic liquids. These are molten salts in which the heat of mixing is essentially zero, and die entropy of mixing is related to the ionic fractions of die cations and anions. In the present case die ionic fractions yield values for the activities of the two sulphides... 

An important element that must be recovered from zinc is cadmium, which is separated by distillation. The alloys of zinc with cadmium are regular solutions with a heat of mixing of 8300 Xcd fzn J gram-atom and the vapour pressures of the elements close to the boiling point of zinc (1180K) are... 

The less the heat of mixing the greater the likelihood of mixing. 

In the first category of solutions ( regular solutions ), it is the enthalpic contribution (the heat of mixing) which dominates the non-ideality, i.e. In such solutions, the characteristic intermolecular potentials between unlike species differ significantly from the average of the interactions between Uke species, i.e. 

Other properties that are influenced by H bonding are solubility and miscibility, heats of mixing, phase-partitioning properties, the... 

This is another way of expressing that the heat load from tray to tray is varying in the column to such an extent as to make the usual simplifying assumption of equal molal overflow invalid. The relations to follow do not include heats of mixing. In general they apply to most hydrocarbon systems. 

Table 15. Physico-mechanical characteristics and heats of mixing with chloroform (dH3) of poly(butyl methacrylate) polymerization-filled with aerosil (10% by mass) [333, 334]...

According to Flory-Huggins theory, the heat of mixing of solvent and polymer is proportional to the binary interaction parameter x in equation (3). The parameter x should be inversely proportional to absolute temperature and independent of solution composition. 

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