Now that we see the capabilities of the predictions, we have motivation to understand the model. One of the major assumptions of van der Waals mixing was that the mixture interactions were independent of each other such that quadratic mixing rules would provide reasonable approximations as shown in Eqn. 12.3 on page 467. But in some cases, like radically different strengths of attraction, the mixture interaction can be strongly coupled to the mixture composition. That is, for instance, the cross parameter could be a function of composition: a12 = a12(x). One way of treating this prospect is to recognize the possibility that the local compositions in the mixture might deviate strongly from the bulk compositions. As an example, consider a lattice consisting primarily of type A atoms but with two B atoms right beside each other. Suppose all these atoms were the same size and that the coordination number was 10. Then the local compositions around a B atom are xAB = 9/10 and xBB = 1/10 (notation of subscripts is AB ⇒ “A around B”). Specific interactions such as hydrogen bonding and polarity might lead to such effects, and thus, the basis of the hypothesis is that energetic differences lead to the nonrandomness that causes the quadratic mixing rules to break down. To develop the theory, we first introduce nomenclature to identify the local compositions summarized in Table 13.1.
Table 13.1. Nomenclature for Local Composition Variables
We assume that the local compositions are given by some weighting factor, Ωij, relative to the overall compositions.
Therefore, if Ω12 = Ω21 = 1, the solution is random. Before introducing the functions that describe the weighting factors, let us discuss how the factors may be used.
Local Compositions around “1” Molecules
Let us begin by considering compositions around “1” molecules. We would like to write the local mole fractions x21 and x11 in terms of the overall mole fractions, x1 and x2. Using the local mole balance,
Rearranging Eqn. 13.1,
Substituting Eqn. 13.4 into Eqn. 13.3,
Rearranging,
Substituting Eqn. 13.6 into Eqn. 13.4,
Local Compositions around “2” Molecules
Similar derivations for molecules of type “2” results in
Example 13.2. Local compositions in a two-dimensional lattice
The following lattice contains x’s, o’s, and void spaces. The coordination number of each cell is 8. Estimate the local composition (xxo) and the parameter Ωxo based on rows and columns away from the edges.
Solution
There are 9 o’s and 13 x’s that are located away from the edges. The number of x’s and o’s around each o are as follows:
Note: Fluids do not really behave as though their atoms were located on lattice sites, but there are many theories based on the supposition that lattices represent reasonable approximations. In this text, we have elected to omit detailed treatment of lattice theory on the basis that it is too approximate to provide an appreciation for the complete problem and too complicated to justify treating it as a simple theory. This is a judgment call and interested students may wish to learn more about lattice theory. Sandler presents a brief introduction to the theory which may be acceptable for readers at this level.a
a. Sandler, S.I. 1989. Chemical Entineering Thermodynamic, 2nd ed. Hoboken NJ: Wiley.
Local Composition and Gibbs Energy of Mixing
We need to relate the local compositions to the excess Gibbs energy. The perspective of representing all fluids by the square-well potential lends itself naturally to the local composition concept. Then the intermolecular energy is given simply by the local composition times the well-depth for that interaction. We simply ignore all but the nearest neighbors because they are outside the square-well. In equation form, the energy equation for mixtures can be reformulated in terms of local compositions. The local mole fraction can be related to the bulk mole fraction by defining a quantity Ωij as follows:
The next step in the derivation requires scaling up from the molecular-scale local composition to the macroscopic energy in the mixture. The rigorous procedure for taking this step requires integration of the molecular distributions times the molecular interaction energies, analogous to the procedure for pure fluids as applied in Section 7.11. This rigorous development is presented below in Section 13.7. On the other hand, it is possible to simply present the result of that derivation for the time being. This permits a more rapid exploration of the practical implications of local composition theory. The form of the equation is not so difficult to understand from an intuitive perspective, however. The energy departure is simply a multiplication of the local composition (xij) by the local interaction energy (εij). The departure properties are calculated based on a general model known as the two-fluid theory.1 According to the two-fluid theory, any intensive departure function in a binary is given by
Where the local composition environment of the type 1 molecules determines (M–Mig)(1), and the local composition environment of the type 2 molecules determines (M–Mig)(2). Note that (M–Mig)(i) is composition-dependent and is equal to the pure component value only when the local composition is pure i.
Using the concept of a square-well model and thus counting only nearest neighbors, noting that ε12 = ε21, and recalling that the local mole fractions must sum to unity, we have for a binary mixture
where Nc,j is the coordination number (total number of atoms in the neighborhood of the jth species), and where we can identify
When x1 approaches unity, x2 goes to zero, and from Eqn. 13.1 x21 goes to zero, and x11 goes to one. The limits applied to Eqn. 13.11 result in (U – Uig)pure1 = (NA/2)Nc,1ε11. Similarly, when x2 approaches unity, (U – Uig)pure2 = (NA/2)Nc,2ε22. For an ideal solution,
The excess energy is obtained by subtracting Eqn. 13.13 from Eqn. 13.11,
Collecting terms with the same energy variables, and using the local mole balance from Table 13.1 on page 502, (x11–1)ε11 = –x21ε11, and (x22–1)ε22 = –x12ε22, resulting in
Substituting Eqn. 13.7 and Eqn. 13.8,
At this point, the traditional local composition theories deviate from regular solution theory in a way that really has nothing to do with local compositions. Instead, the next step focuses on one of the subtleties of classical thermodynamics. Example 6.7 shows that the derivative of Helmholtz energy is related to internal energy. Therefore, we can integrate energy to find the change in Helmholtz energy,
where is the infinite temperature limit at the given liquid density, independent of temperature but possibly dependent on composition or density. We need to insert Eqn. 13.16 into Eqn. 13.17 and integrate. We need to have some algebraic expression for the dependence of Ωij on temperature, which is what distinguishes the local composition theories from each other.
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