In deriving the entropy of mixing ideal gases in Eqn. 4.8 on page 138, we applied the notion that ideal gases are point masses and have no volume. We considered the entropy of mixing to be determined by the total volume of the mixture. When we consider the entropy of mixing liquids, however, we realize that the volume occupied by the molecules themselves is a significant part of the total liquid volume. The volume occupied by one molecule is not accessible to the other molecules, and therefore, our assumptions regarding entropy may be inaccurate. One simple way of correcting for this effect is to subtract the volume occupied by the molecules from the total volume and treat the resultant “free volume” in the same way we treated ideal gas volume.
Example 12.3. Deriving activity models involving volume fractions
The derivation of Scatchard-Hildebrand theory shows that volume fraction arises naturally as a characterization of composition, rather than mole fraction. This observation turns out to be true for many theories. Show that you can derive the relevant activity model from a Gibbs excess model involving volume fraction by deriving Eqn. 12.24 from Eqn. 12.28.
Solution
GE = RT(δ2 – δ1)2 (n1V1 + n2V2) Φ1 Φ2
Taking the derivative of the equation for GE involves applying the chain rule to the three compositional factors: (n1V1 + n2V2), Φ1, and Φ2.
The derivative of the first term is simply V1 and,
RTlnγ1 = ∂GE /∂n1 = RT(δ2 – δ1)2 [V1Φ1Φ2 + (n1V1 + n2V2) (Φ2∂Φ1/∂n1 + Φ1∂Φ2/∂n1)].
It is helpful to maintain dimensional consistency in order to provide a quick check as we proceed. This can be achieved by multiplying and dividing by n, resulting in:
RTlnγ1 = RT(δ2 – δ1)2 [V1 Φ1 Φ2 + (x1V1 + x2V2) (Φ2 n∂Φ1/∂n1 + Φ1 n∂Φ2/∂n1)]
A key strategy in these derivations is to replace all compositional quantities with expressions depending only on {ni}, not {xi}, and not {Φi}. For {Φi}, this is easily achieved by multiplying the numerator and denominator by n, noting that xi = ni/n.
Φ1 = x1V1/(x1V1 + x2V2) = n1V1/(n1V1 + n2V2); Φ2 = x2V2/(x1V1 + x2V2) = n2V2/(n1V1 + n2V2).
Taking the derivative involves product rule for Φ1 and a simple reciprocal for Φ2:
∂Φ1/∂n1 = V1/(n1V1 + n2V2) – n1V12/(n1V1 + n2V2)2; ∂Φ2/∂n1 = – n2V2V1/(n1V1 + n2V2)2.
Multiplying by n and simplifying gives:
Substituting and noting that (x1V1 + x2V2) cancels between numerator and denominator,
RTlnγ1 = RT(δ2 – δ1)2 [V1Φ1Φ2 + (Φ2 V1(1 – Φ1) – Φ1V1(Φ2)]
The first and last terms in the brackets cancel. Also, for a binary mixture, Φ2 = (1 – Φ1). So,
RTlnγ1 = RT(δ2 – δ1)2 [V1Φ22]
This procedure is extremely similar for all GE models. In particular, Eqns. 12.27 and 12.28 can be easily adapted to any GE model involving {Φi}, if you first apply the chain rule thoughtfully.
Free volume is the difference between the volume of a fluid and the volume occupied by its molecules.
To use the concept, we assume that there is a fractional free volume, ϖ, globally applicable to all liquids and liquid mixtures. Let us further assume that the entropy change for a component is given by the change in free volume available to that component.
Example 12.4. Scatchard-Hildebrand versus van Laar theory for methanol + benzene
Fit the Scatchard-Hildebrand and van Laar models to the methanol + benzene azeotrope. Match the azeotropic pressure (and the composition in the case of the van Laar two-parameter model). The azeotrope appears at 58.3°C and xm = 0.614. The vapor pressures at 58.3°C are 591.3 mmHg for methanol, 368.7 mmHg for benzene.
Solution
The van Laar parameters and binary interaction parameter are determined by matching the azeotropic pressure (and composition for the van Laar case) as described in previous examples. The resultant calculations are described in the worksheet REGULAR in the workbook Actcoeff.xlsx and the MATLAB file Ex12_04.m. Though not apparent from the figure below, the Scatchard-Hildebrand theory incorrectly predicts LLE until the binary interaction parameter is adjusted. See the supporting computer files. Fig. 12.1 illustrates the results of the fitting.
Figure 12.1. T-x-y diagram for methanol and benzene for Example 12.4. The compositions are plotted in terms of mole fractions of methanol.
Actcoeff.xlsx, sheet REGULAR, Ex12_04.m.
The free volume available to any pure component is
If we assume that there is no volume change on mixing, the resultant free volume in the mixture is given by the same fraction, ϖ, and the mixture volume is
When two components mix, each component’s entropy increases according to how much more space it has by an modification of Eqn. 4.6 using the free volume rather than the total volume:
Note that Eqn. 12.32 reduces to the ideal solution result, Eqn. 10.63, when V1 = V2. The excess entropy is
This expression provides a simplistic representation of deviations of the entropy from ideal mixing. The entropy of mixing given by Eqn. 12.33 is frequently called the combinatorial entropy of mixing because it derives from the same combinations and permutations that we discussed in the case of particles in boxes. If entropy is the dominant factor in mixing, this formula can be used to find the excess Gibbs energy. When the excess enthalpy is zero, the mixture is called athermal.
Flory’s equation.It can also be combined with the Scatchard-Hildebrand solution theory6 to derive the predictive theory of Blanks and Prausnitz7 or the more common “Flory-Huggins” theory. These expressions are particularly important for solutions containing large molecules like polymers.
For a binary solution,
Frequently, for mixtures of polymer and solvent, the enthalpic term is fitted empirically to experimental data by adjusting the form of the equation to be the Flory-Huggins model,
where component 1 is always the solvent, and component 2 is always the polymer. The variable r = V2/V1 denotes the ratio of volume of the polymer to the solvent. Similarly, χ ≡ V1(δ1 – δ1)2/RT. A solvent for which χ = 0 is an athermal mixture. Plotting the result for SE versus mole fraction for several size ratios, Fig. 12.2 shows that it is always positive, and it becomes larger and more skewed as the size ratio increases. Thus, the size ratio has a large effect on the phase stability when the ratio is large.
Figure 12.2. Illustration of excess entropy according to Flory’s equation for various pure component volume ratios.
One of the major problems with recycling polymeric products is that different polymers do not form miscible solutions with one another; rather, they form highly non-ideal solutions. To illustrate, suppose 1g each of two different polymers (polymer A and polymer B) is heated to 127°C and mixed as a liquid. Estimate the activity coefficients of A and B using the Flory-Huggins model.
Solution
xA = (1/10,000)/(1/10,000 + 1/12,000) = 0.546; xB = 0.454
ΦA = 0.546(1.54)/[0.546(1.54) + 0.454(1.68)] = 0.524; ΦB = 0.476
lnγA = ln (0.5238/0.5455) + (1 – 0.5238/0.5455) + (1.54E6(19.4 – 19.2)2(0.4762)2)/(8.314(400))
= –0.0008 + 4.200 ⇒ γA = 66
lnγP = ln (0.4762/0.4545) + (1 – 0.4762/0.4545) + (1.68E6(19.4 – 19.2)2(0.5238)2)/(8.314(400))
= +0.0008 + 5.544 ⇒ γB = 258
Several important implications can be interpreted from Example 12.5. It is often noted that the Flory-Huggins model is especially appropriate for polymer solution models. While the excess entropy is most significant for polymer-solvent mixtures, it is not so important for polymer-polymer mixtures. The key to polymer-polymer mixtures is noting that the activity coefficient is proportional to the exponential of the molar volume of the polymer. Therefore, even tiny differences in solubility parameter are amplified. Furthermore, the large activities computed for these components mean that the fugacities of these components would be greatly enhanced if intermingled at this composition. This means that they show a strong tendency to escape from each other. On the other hand, polymer compounds are too non-volatile to escape to the vapor phase. The only alternative is to escape into separate liquid phases. In other words, the liquids become immiscible. Computations of activity coefficients like those above play a major role in the liquid-liquid phase equilibrium calculations detailed in Chapter 14.
If you think creatively for a moment, you can imagine staggering possibilities for this amplification principle. To begin, we could synthesize molecules of just the right size to generate phase behavior that precisely measures the magnitude of the molecular interactions between, say, polyethylene and polypropylene. But suppose we would like to homogenize an immiscible blend of high molecular weight polymers. Then perhaps a polymer that was half of each type could help. Next comes a consideration that we generally avoid throughout this text. Would the intramolecular structure make a difference? In other words, it is possible to synthesize one (random) copolymer that alternates randomly between monomer types and another (block) copolymer that has a long section of one monomer type followed by another long section of a different monomer type. The properties resulting from these different intramolecular structures are very different. If the blocks are large enough, they can aggregate similar to phase separation. This is not exactly a phase separation, however, because the blocks might be part of the same molecule. Repeating this theme on a grand scale with 20 particular monomer types (amino acids) is called protein engineering. Modern science is just beginning to manipulate these kinds of interactions to synthesize self-assembled structures with specific design objectives. Exploring these possibilities would take us beyond the introductory level, however.
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