Mixing, entropy

In addition to the calculations of changes in the free entropy mixing, Flory introduced the interaction parameter, %, to account for the intermolecular interactions between polymer and solvent molecules, thus giving [53]... 

The same result is obtained for mixtures of two perfect gases. Because there are no interactions between the individual particles of perfect gases (atoms or molecules) the decrease in the free enthalpy during mixing can be traced back to the increase in entropy. Mixing increases the disorder of the system. 

Lin, S.K. Molecular diversity assessment logarithmic relations ofinformation and species diversity and logarithmic relations of entropy and indistinguishability after rejection ofGibbs paradox of entropy mixing. Molecules, 1996,1, 57-67. 

The first term in Eq. (24) represents the ideal term. If we consider an athermal system, it is sufficient to take into account the entropy mixing alone so that = 0 And we have an important implication with the second virial coefficient A2l ... 

Owing to low values of the combinatorial entropy mixing, miscibility in polymer-polymer systems requires the existence of strong specific interactions between the components, such as hydrogen bonding [Olabisi et al., 1979 Sole, 1982 Walsh and Rostami, 1985 Utracki, 1989]. The thermodynamic characterization of the interactions in miscible polymer blends has been the subject of extensive studies [Deshpande et al., 1974 Olabisi, 1975 Mandal et al., 1989 Lezeano et al., 1992, 1995, 1996 Farooque and Deshpande, 1992 Juana et al., 1994]. 

The mixing process is, of course, irreversible, and is accompanied by an irreversible increase in entropy. Mixing with evaporation is particularly difficult to model because, in addition to the thermal mixing and release of thermal overfill, there is the compositional heat of mixing with irreversible entropy production which is sig-nificantiy path dependent. A considerable volume of vapour is produced by the usual positive heat of compositional mixing and by thermal contact between colder and hotter components (no homogeneous nucleate boiling has been observed) before the final equilibrium state of the mixture is achieved. 

The broken bond approach has been extended by Nason and co-workers (see Ref. 85) to calculate as a function of surface composition for alloys. The surface free energy follows on adding an entropy of mixing term, and the free energy is then minimized. 

For example, the expansion of a gas requires the release of a pm holding a piston in place or the opening of a stopcock, while a chemical reaction can be initiated by mixing the reactants or by adding a catalyst. One often finds statements that at equilibrium in an isolated system (constant U, V, n), the entropy is maximized . Wliat does this mean ... 

For those who are familiar with the statistical mechanical interpretation of entropy, which asserts that at 0 K substances are nonnally restricted to a single quantum state, and hence have zero entropy, it should be pointed out that the conventional thennodynamic zero of entropy is not quite that, since most elements and compounds are mixtures of isotopic species that in principle should separate at 0 K, but of course do not. The thennodynamic entropies reported in tables ignore the entropy of isotopic mixing, and m some cases ignore other complications as well, e.g. ortho- and para-hydrogen. 

The entropy of mixing of very similar substances, i.e. the ideal solution law, can be derived from the simplest of statistical considerations. It too is a limiting law, of which the most nearly perfect example is the entropy of mixing of two isotopic species. 

By the standard methods of statistical thermodynamics it is possible to derive for certain entropy changes general formulas that cannot be derived from the zeroth, first, and second laws of classical thermodynamics. In particular one can obtain formulae for entropy changes in highly di.sperse systems, for those in very cold systems, and for those associated, with the mixing ofvery similar substances. 

If the entropy and the enthalpy for the separate mixing in each of the half-mole superlattices are calculated and then combined, the following equation is obtained ... 

The entropy of a solution is itself a composite quantity comprising (i) a part depending only on tire amount of solvent and solute species, and independent from what tliey are, and (ii) a part characteristic of tire actual species (A, B,. ..) involved (equal to zero for ideal solutions). These two parts have been denoted respectively cratic and unitary by Gurney [55]. At extreme dilution, (ii) becomes more or less negligible, and only tire cratic tenn remains, whose contribution to tire free energy of mixing is... 

Solubility in Water A familiar physical property of alkanes is contained m the adage oil and water don t mix Alkanes—indeed all hydrocarbons—are virtually insoluble m water In order for a hydrocarbon to dissolve m water the framework of hydrogen bonds between water molecules would become more ordered m the region around each mole cule of the dissolved hydrocarbon This increase m order which corresponds to a decrease m entropy signals a process that can be favorable only if it is reasonably... 

The values of S° represent the virtual or thermal entropy of the substance in the standard state at 298.15 K (25°C), omitting contributions from nuclear spins. Isotope mixing effects are also excluded except in the case of the H—system. 

The thermodynamic probability is converted to an entropy through the Boltzmann equation [Eq. (3.20)] so we can write for the entropy of the mixture (subscript mix)... 

Although the right-hand sides of Eqs. (8.27) and (8.28) are the same, the former applies to the mixture (subscript mix), while the latter applies to the mixing process (subscript m). The fact that these are identical emphasizes that in Eq. (8.27) we have calculated only that part of the total entropy of the mixture which arises from the mixing process itself. This is called the configurational entropy and is our only concern in mixing problems. The possibility that this mixing may involve other entropy effects—such as an entropy of solvation-is postponed until Sec. 8.12. 

We concluded the last section with the observation that a polymer solution is expected to be nonideal on the grounds of entropy considerations alone. A nonzero value for AH would exacerbate the situation even further. We therefore begin our discussion of this problem by assuming a polymer-solvent system which shows athermal mixing. In the next section we shall extend the theory to include systems for which AH 9 0. The theory we shall examine in the next few sections was developed independently by Flory and Huggins and is known as the Flory-Huggins theory. 

Since the 0 s are fractions, the logarithms in Eq. (8.38) are less than unity and AGj is negative for all concentrations. In the case of athermal mixtures entropy considerations alone are sufficient to account for polymer-solvent miscibility at all concentrations. Exactly the same is true for ideal solutions. As a matter of fact, it is possible to regard the expressions for AS and AGj for ideal solutions as special cases of Eqs. (8.37) and (8.38) for the situation where n happens to equal unity. The following example compares values for ASj for ideal and Flory-Huggins solutions to examine quantitatively the effect of variations in n on the entropy of mixing. 

We express the calculated entropies of mixing in units of R. For ideal solutions the values of are evaluated directly from Eq. (8.28) ... 

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