Chapter 13. Local Composition Activity Models

I have constructed three thousand different theories in connection with the electric light... Yet, in only two cases did my experiments prove the truth of my theory.

Thomas A. Edison

It is evident from Chapter 12 that describing solution nonideality with van der Waals’ mixing rules is imprecise. With a single parameter like k₁₂ or A₁₂, we can match the magnitude of the excess energy, but not the skewness. The Margules two-parameter and van Laar models address this problem in an ad hoc fashion, but there is no physical basis for extending these to multiple components. A rational basis for extending the analysis of mixtures should seek a legitimate explanation for the source of varied skewness. One such explanation is that molecules do not distribute themselves randomly. Instead, they may tend to form clusters. You might witness this kind of distribution at a dinner party, where the children are in one area, the men are discussing sports, and the ladies are discussing anything but sports. There is sufficient mixing among the clusters that you cannot consider it a phase separation. Some children and ladies like sports. Some men do not. Nevertheless, the distribution is not entirely random. In other words, the local composition around a child at a dinner party, for example, may differ from the bulk composition. We examine this prospect graphically in Example 13.2. Local composition models recognize this possibility and account for the local composition enhancement in terms of two parameters, just enough to characterize both the magnitude and the skewness of the excess energy. Careful analysis of the molecular scale energies in terms of the local compositions facilitates straightforward extension to multicomponent mixtures.

Chapter Objectives: You Should Be Able to...

1. Characterize adjustable parameters in local composition activity models using experimental data.

2. Comment critically on the merits and limitations of the following activity models: Wilson, UNIQUAC, UNIFAC, and NRTL, including the ability to identify the most appropriate model for a given mixture.

A Preliminary Glimpse of UNIFAC — A Predictive Method

This chapter is densely packed with theoretical details. In the interest of “telling you what we are going to tell you,” it is useful to see how the final equations are applied before being concerned with their derivations. One popular activity coefficient model is the predictive model of UNIFAC. It is the closest thing to a universally applicable predictive model that we currently have, so it makes sense to get right to the point and introduce the rudiments of implementing this model at an early stage. It is a rather complicated model and deriving it must await several other derivations. Nevertheless, the availability of a computer program for applying the method makes it possible to apply it as a “black box” at this stage, and the utility of the model should inspire us to learn more about it. Detailed calculations are illustrated in the UNIFAC spreadsheet in the file Actcoeff.xlsx introduced in Chapter 11, and in the MATLAB file Ex13_01.m.

a. Perry, R.H., Chilton, C.H. 1973. Chemical Engineers’ Handbook, 5 ed, New York NY: McGraw-Hill, Chapter 13.

b. The functional groups for a given molecule are often determined automatically by process design software. Several examples of group assignments are given in Table 13.2 on page 512.

13.1. Local Composition Theory

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 x_AB = 9/10 and x_BB = 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, Ω_ij, 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

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 Ω_ij 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 (x_ij) by the local interaction energy (ε_ij). 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–M^ig)⁽¹⁾, and the local composition environment of the type 2 molecules determines (M–M^ig)⁽²⁾. Note that (M–M^ig)⁽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 N_c,j is the coordination number (total number of atoms in the neighborhood of the j^th 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 – U^ig)_pure1 = (N_A/2)N_c,1ε₁₁. Similarly, when x₂ approaches unity, (U – U^ig)_pure2 = (N_A/2)N_c,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 Ω_ij on temperature, which is what distinguishes the local composition theories from each other.

13.2. Wilson’s Equation

Wilson² made a bold assumption regarding the temperature dependence of Ω_ij. Wilson’s original parameter used in the literature is Λ_ji, but it is related to Ω_ij in a very direct way. Wilson assumes³

(note: Λ_ii = Λ_jj = 1, and A_ij ≠ A_ji even though ε_ij = ε_ji), and integration with respect to T becomes very simple. Assuming N_c,j = 2 for all j at all ρ,

A convenient simplifying assumption before proceeding further is that G^E ~ A^E. This corresponds to neglecting the excess volume of mixing relative to the other contributions and is quite acceptable for liquids. The customary way of interpreting G^E/RT is to separate it into an energetic part known as the residual contribution, (G^E/RT)^RES, that vanishes at infinite temperature or when ε₁₂ – ε₂₂ = 0 and ε₂₁ – ε₁₁ = 0, and a size/shape part known as the combinatorial contribution, (G^E/RT)^COMB, that represents the infinite temperature limit at the liquid density. For Wilson’s equation, the first two terms vanish at high T, so

For the combinatorial contribution, Wilson used Flory’s equation,

It should be noted that the assumption of the temperature dependence of Eqn. 13.18 has been made for convenience, but there is some justification for it, as we show in Section 13.7. Wilson’s equation becomes

Algebraic rearrangement of Wilson’s equation results in the form that is usually cited,

For a binary system, the activity coefficients from the Wilson equation are:

Noting that Λ₁₁ = Λ₂₂ = 1, and looking back at Eqn. 13.6, we can also see that for the first equation . We can rearrange this expression to obtain the slightly more compact relation:

Similar rearrangement of the second expression gives:

One limitation of Wilson’s equation is that it is unable to model liquid-liquid equilibria, but it is reasonably accurate for modeling the liquid phase when correlating the vapor liquid equilibria.

Extending Eqn. 13.23 to a multicomponent solution,

To determine activity coefficients, the excess Gibbs energy is differentiated. Differentiating the last term,

and letting “sum” stand for the summation of Eqn. 13.28, and combining,

combining,

13.3. NRTL

The NRTL model⁴ (short for Non-Random Two Liquid) equates U^E from Eqn. 13.16 directly to G^E, ignoring the proper thermodynamic integration. At the same time, it introduces a third binary parameter that generates an extremely flexible functional form for fitting activity coefficients.

For a binary mixture, the activity equations become

For a binary mixture, the activity equations become

When α_ij = 0, the binary model simplifies to the one-parameter Margules model,

The NRTL model is not very appealing from a theoretical perspective, but its flexibility has led to a broad range of applications including combinations with electrolyte models. As a practical matter, a value of α_ij = 0.3 is taken as a default and the equation works much like the Wilson equation, except that it is able to model LLE. The parameter α_ij is adjusted for additional flexibility when necessary, such as when modeling LLE where the value is commonly increased. The multicomponent form of NRTL is

13.4. UNIQUAC

UNIQUAC⁵ (short for UNIversal QUAsi Chemical model) builds on the work of Wilson by making three primary refinements. First, the temperature dependence of Ω_ij is modified to depend on surface areas rather than volumes, based on the hypothesis that the interaction energies that determine local compositions are dependent on the relative surface areas of the molecules. If the parameter q_i is proportional to the surface area of molecule i,

where z = 10. The intermediate parameter τ_ij = exp(– a_ij/T) is used for compact notation where τ_ii = τ_jj = 1, a_ii = a_jj = 0.⁶ In addition, when the energy equation 13.16 is written with N_c,j = zq_j = 10q_j for all j at all ρ, the different sizes and shapes of the molecules are implicitly taken into account. Qualitatively, the number of molecules that can contact a central molecule increases as the size of the molecule increases. Using surface fractions attempts to recognize the branching and overlap that can occur between segments in a polyatomic molecule. The inner core of these segments is not accessible, only the surface is accessible for energetic interactions. Therefore, the model of the energy is proposed to be proportional to surface area. Unfortunately, it is not straightforward to construct a more rigorous argument in favor of surface fractions from the energy equation itself. Inserting Eqns. 13.31 and 13.16 into Eqn. 13.17, the excess Helmholtz for a binary solution becomes

where θ_i is the surface area fraction, and θ_i = x_iq_i/(x₁q₁ + x₂q₂) for a binary. Analogous to Wilson’s equation, G^E is calculated as A^E, a good approximation. The first two terms represent (G^E/RT)^RES,

that can be compared with Eqn. 13.20 and the final term represents (G^E/RT)^COMB. The (G^E/RT)^COMB term is attributed to the entropy of mixing hard chains, and an approximate expression for this contribution is applied by Maurer and Prausnitz.⁷ This representation of the entropy of mixing traces its roots back to the work of Staverman⁸ and Guggenheim⁹ and was discussed more recently by Lichtenthaler et al.¹⁰ It is very similar to the Flory term, but it corrects for the fact that large molecules are not always large balls, but sometimes long “strings.” By noting that the ratio of surface area to volume for a sphere is different from that for a string, Guggenheim’s form (the form actually applied in UNIFAC and UNIQUAC) provides a simple but general correction, giving an indication of the degree of branching and nonsphericity. Nevertheless, the Staverman-Guggenheim term represents a relatively small correction to Flory’s term. As shown in Fig. 13.1, the extra correction of including the “surface to volume” parameter serves to decrease the excess entropy to some value between zero and the Flory-Huggins estimate. The combinatorial part of UNIQUAC for a binary system takes a form that can be compared with 13.21

Figure 13.1. Excess entropy according to the Flory-Huggins equation versus Guggenheim’s equation at V₂/V₁ = 1695 for a polymer solvent mixture.

The Guggenheim form of the excess entropy is based on the molecular volume fractions, Φ_j, and the surface fractions, θ_j. Instead of using macroscopic property data to calculate volume fractions and surface fractions they are based on relative molecule volumes, r, and relative molecule surface areas, q, for each type of molecule.

The relative molecular parameters r and q may be calculated from group size and surface area parameters using the concept of group contributions. The size/area parameters are ratios to the equivalent size/area for the -CH₂- group in a long chain alkane.¹¹ The group parameters are added in the same manner as the UNIFAC method discussed in the next section and as given in Table 13.2 on page 512, except the UNIQUAC r and q values for alcohols are typically not calculated by group contributions (see the footnote to Table. 13.2). In the table, the uppercase R_k parameter is for the group volume, and the uppercase Q_k parameter is for group surface area. From these values, the molecular size (r_j) and shape (q_j) parameters may be calculated by multiplying the group parameter by the number of times each group appears in the molecule, and summing over all the groups in the molecule,

Table 13.2. Selected Group Parameters for the UNIFAC and UNIQUAC Equations^a

a. AC in the table means aromatic carbon. The main groups serve as categories for similar subgroups as explained in the UNIFAC section.

b. Alcohols are usually treated in UNIQUAC without using the group contribution method. Accepted UNIQUAC values for the set of alcohols [MeOH, EtOH, 1-PrOH, 2-PrOH, 1-BuOH] are r = [1.4311, 2.1055, 2.7799, 2.7791, 3.4543], q = [1.4320, 1.9720, 2.5120, 2.5080, 3.0520]. See Gmehling, J., Oken, U. 1977-Vapor-Liquid Equilibrium Data Collection. Frankfort, Germany: DECHEMA.

where is the number of groups of the kth type in molecule j. The subdivision of the molecule into groups is sometimes not obvious because there may appear to be more than one way to subdivide, but the conventions have been set forth in examples in the table and these conventions should be followed. The large number of possible functional groups is divided into main groups and further subdivided into structurally similar subgroups. Usually the functional groups include a nearest neighbor atom as part of the group. The group parameters are calculated from the van der Waals volume and van der Waals surface area. Note that the van der Waals volume and van der Waals area are not calculated from the van der Waals EOS. They are inferred from x-ray and other property data.¹².

The parameters to characterize the volume and surface area fractions have already been tabulated, so no more adjustable parameters are really introduced by writing it this way. The only real problem is that including all these group contributions into the formulas makes hand calculations extremely tedious. Fortunately, computers and spreadsheets make this task much simpler. As such, we can apply the UNIQUAC method almost as easily as the van Laar method.

For a binary mixture, the activity equations become

Like the Wilson equation, the UNIQUAC equation requires that two adjustable parameters be characterized from experimental data for each binary system. The inclusion of the excess entropy in UNIQUAC by Abrams et al. (1975) is more correct theoretically, but Wilson’s equation can be as accurate as the UNIQUAC method for many binary vapor-liquid systems, and much simpler to apply by hand. UNIQUAC supersedes the Wilson equation for describing liquid-liquid systems, however, because the Wilson equation is incapable of representing liquid-liquid equilibria as long as the λ_ij parameters are held positive (as implied by their definition as exponentials, and noting that exponentials cannot take on negative values).

Extending Eqn. 13.40 to a multicomponent solution, the UNIQUAC equation becomes

Note that the leading sum is simply Flory’s equation. The first two terms are the combinatorial contribution and the last is the residual contribution. The parts can be individually differentiated to find the contributions to the activity coefficients,

13.5. UNIFAC

UNIFAC¹³ (short for UNIversal Functional Activity Coefficient model) is an extension of UNIQUAC with no user-adjustable parameters to fit to experimental data. Instead, all of the adjustable parameters have been characterized by the developers of the model based on group contributions that correlate the data in a very large database. The assumptions regarding coordination numbers, and so forth, are similar to the assumptions in UNIQUAC. The same strategy is applied:

The combinatorial term is therefore identical and given by Eqn. 13.54. The major difference between UNIFAC and UNIQUAC is that, for the residual term, UNIFAC considers interaction energies between functional groups (rather than the whole molecule). Interactions of functional groups are added to predict relative interaction energies of molecules. Examples are shown in Table 13.2. Each of the subgroups has a characteristic size and surface area; however, the energetic interactions are considered to be the same for all subgroups with a particular main group. Thus, representative interaction energies (a_ij) are tabulated for only the main functional groups, and it is implied that all subgroups will use the same energetic parameters. An illustrative sample of values for these interactions is given in Table 13.3. Full implementations of the UNIFAC method with large numbers of functional groups are typically available in chemical engineering process design software. A subset of the parameters is provided on the UNIFAC spreadsheet in the Actcoeff.xlsx spreadsheet included with the text. Knowing the values of these interaction energies permits estimation of the properties for a really impressive number of chemical solutions. The limitation is that we are not always entirely sure of the accuracy of these predictions.

Table 13.3. Selected VLE Interaction Energies a_ij for the UNIFAC Equation in Units of Kelvin

Although UNIFAC is closely related to UNIQUAC, keep in mind that there is no direct extension to a correlative equation like UNIQUAC. If you want to fit experimental data that might be on hand, you cannot do it within the defined framework of UNIFAC; UNIQUAC or NTRL is the preferred choice when adjustable parameters are desired. Although it is tedious to estimate the a_ij parameters of UNIQUAC or NRTL from UNIFAC, some implementations of chemical engineering process design software have included facilities for estimating UNIQUAC or NRTL parameters from UNIFAC. This approach can be useful for estimating interactions for a few binary pairs in a multicomponent mixture when most of the binary pairs are known from experimental data specific to those binary interactions.

The basic approach to understanding UNIFAC is the generalization of the methods for calculating the residual activity coefficient. Imagine the interactions of a CH₃ group in a mixture of isopropanol (1) and component (2). The isopropanol consists of 2(CH₃) + 1(CH) + 1(OH). Therefore, in the mixture, a CH₃ will encounter CH₃, CH, OH groups, and the groups of component (2), and the interaction energies depend on the number of each type of group available in the solution. Therefore, the interaction energy of CH₃ groups can be calculated relative to a hypothetical solution of 100% CH₃ groups. The mixture can be approximated as a solution of groups (SOG)¹⁴ (rather than a solution of molecules), and the interaction energies can be integrated with respect to temperature to arrive at chemical potential in a manner similar to the development of Eqn. 13.40.

Therefore, it is possible to calculate where is the chemical potential in a hypothetical solution of 100% CH₃ groups and Γ_CH3 is the activity coefficient of CH₃ in the solution of groups. The chemical potential of CH₃ groups in pure isopropanol (1), given by , will differ from because even in pure isopropanol CH₃ will encounter a mixture of CH₃, CH, and OH groups in the ratio that they appear in pure isopropanol, and therefore the activity coefficient of CH₃ groups in pure isopropanol, , is not unity, where the superscript (1) indicates pure component (1). The difference that is desired is the effect of mixing the CH₃ groups in isopropanol with component (2), relative to pure isopropanol,

Fig. 13.2 provides an illustration of the differences that we seek to calculate, with water as a component (2).

Figure 13.2. Illustration relating the chemical potential of CH₃ groups in pure 2-propanol, a real solution of groups where water is component (2), and a hypothetical solution of CH₃ groups. The number of groups sketched in each circle is arbitrary and chosen to illustrate the types of groups present. The chemical potential change that we seek is . We calculate this difference by taking the difference between the other two paths.

If the chemical potential of a molecule consists of the sum of interactions of the groups,

Therefore, we arrive at the important result that is utilized in UNIFAC,

where the sum is over all function groups in molecule (1) and is the number of occurrences of group m in the molecule. The activity coefficient formula for any other molecular component can be found by substituting for (1) in Eqn. 13.56. Note that Γ_m is calculated in a solution of groups for all molecules in the mixture, whereas is calculated in the solution of groups for just component (1). Note that we use uppercase letters to represent the group property analog of the molecular properties, with the following exceptions: Uppercase τ looks too much like T, so we substitute Ψ, and the a_ij for UNIFAC is understood to be a group property even though the same symbol is represented by a molecular property in UNIQUAC. The relations are shown in Table 13.4.

Table 13.4. Comparison of Group Variables and Molecular Variables for UNIFAC

lnΓ_m is calculated by generalizing the UNIQUAC expression for . Generalizing Eqn. 13.55 and supporting equations,

where is the number of groups of type k in molecule i. Fortunately, the spreadsheet and programs provided with the textbook save us from doing the tedious calculations for UNIFAC, although an understanding of the principles is important.

a. Perry, R.H., Chilton, C.H. 1973. Chemical Engineers’ Handbook, 5 ed. New York, NY: McGraw-Hill, Chapter 13.

13.6. COSMO-RS Methods

In principle, all electronic and molecular interactions are described by quantum mechanics, so you may wonder why we have not considered computing mixture properties from this fundamental approach. In practice, two considerations limit the feasibility of this approach. First, quantum mechanical computations tend to be time consuming. Precise computations can require days for a single molecule the size of naphthalene and the computation time increases as N⁷, where N is the number of atoms in the molecule. Second, the intermolecular interaction energy, which affects the mixing properties, is at least an order of magnitude smaller than the intramolecular energy. Therefore, high precision is required to compute the intermolecular interactions directly. Circumventing these limitations has been the focus of the “COSMO-RS” approach.

COSMO-RS is an abbreviation for “COnductor-like Screening MOdel for Real Solvents.” It refers to a method of performing quantum mechanical calculations as if the simulated molecule were in a conductive bath rather than a vacuum. The method was developed originally by Klamt and coworkers¹⁵ as an extension of previous work on a continuum solvation model (CSM).¹⁶ The implementation of Klamt and coworkers is marketed commercially as COSMOtherm and updated continually. A free educational version with graphical user interface is available that includes capability for about 350 compounds. The implementation of Klamt and coworkers is currently based on the TURBOMOLE package for quantum mechanical simulations, and a few other packages also provide consistent results. Later, Lin and Sandler developed an implementation based on the DMOL³ simulation package,¹⁷ which is included in the Accelrys Materials Studio. We refer to this implementation as COSMO-RS/SAC. It is available as a free download including capability for about 1500 compounds from a web site maintained by Y.A. Liu at Va. Tech.¹⁸ The method includes several empirical parameters that have been characterized by fitting experimental data. The specific parameters depend on the quantum simulation method. In the example below, we have applied the SAC parameters.

The key idea of the COSMO-RS approach has been to focus on the polarization of the surface surrounding a molecule. The significance of the surface polarization can be likened to the acidity and basicity of the SSCED model. If the surrounding surface is positively charged, then the molecular surface must be basic; if the surrounding surface is negatively charged, then the molecular surface must be acidic. If we imagine coloring these acidic interactions as blue and the basic interactions as red, we could represent the molecular surfaces in the manner of the pictures on the cover of the textbook. The overall surface energy, including positive, negative, and neutral influences, can be correlated with activity coefficients and other properties like the heat of vaporization.

To calculate the surface polarization an approximate quantum mechanical method can be applied, known as density functional theory (DFT). DFT computations typically require only a few hours for a molecule like naphthalene, and the increase in computation time scales as N³. If we integrate over the interactions between the surfaces of molecules, we can imagine how results similar to SSCED could be achieved. An additional advantage of COSMO-RS is that local composition effects are implicit in the integration of the local polarization interactions over all orientations between the two molecules. Furthermore, acidity and basicity are not limited to a single characteristic value per molecule, but are characterized by a range of polarizations over the entire surface of each molecule.

To implement the method, the molecular surface charges are calculated using DFT. The observed polarization is referred to as a σ-profile, where σ refers to the surface charge density (Coulombs/Ų). Typical σ-profiles are illustrated in Fig. 13.3. The p(σ) represents the amount of area per σ-interval (Ų/(Coulombs/Ų)) plotted against the surface charge density (Coulombs/Ų). In other words, p(σ) is proportional to a count of how many segments possess a given amount of surface polarization. It is analogous to the number of occurrences of a group in UNIFAC. The curve is normalized such that an integral of the σ-profile gives the total surface area of the molecule. The effective area per segment is divided out near the end of the calculation when computing the activity coefficient. The area under the curve in a particular region shows the total area of the molecule with the charge density. Fig. 13.3(a) shows how ethanol is a smaller molecule than octanol based on the total area under the curve.

Figure 13.3. Samples of σ-profiles for application to the COSMO-RS method, A_effn_i(σ). Dashed vertical lines show the threshold values for hydrogen bonding.

Since the charge density, σ, ranges over negative and positive values, all values less than –0.0084 are considered to contribute to acidity. A similar consideration applies to the β contribution except that all σ > 0.0084 contribute to basicity. In Fig. 13.3(a), note that the two alcohols have extremely similar contributions of total area for both acidity and basicity. Fig. 13.3(b) shows that water is a very small molecule (small area under the curve) with extensive hydrogen bonding (both acidic and basic outside the dispersion bounds) while chloroform is a relatively large molecule with strong acidity and no basicity. All of these behaviors make sense qualitatively, so what remains is the translation into a quantitative method.

Computing activity coefficients for a binary solution is similar to the UNIFAC method if you can imagine transforming the earlier summation over groups to a summation over discretized polarization segments of the σ-profile. Note that use of the term “segment” does not refer to a geometric segment, but to an “interval” of surface polarization. A particular amount of surface polarization may occur at various places over the surface of the molecule, but all would be added together to get p(σ). It may be helpful to think of this quantity in mathematical terms rather than as a physical entity. Typically, the σ-profile is discretized into 51 values ranging from –0.025 to 0.025 (Coulombs/Ų).¹⁹ Therefore, we can refer to σ_k where k = [1,51]. To simplify the computations, the integration ∫p(σ)dσ ~ ΔσΣp(σ_k) is represented as a sum, Σp(k). The larger size of a particular molecule is reflected in the Σp(k) in a comparable manner to the molecular volume in the Scatchard-Hildebrand or SSCED models. Discretized segments on molecule i interact with segments on all molecules, including the i^th molecule itself. In a mixture, p_i(k) designates p(k) for the i^th component. The activity coefficient contribution for a segment k(Γ_k) in a solution of polarization segments is analogous to Eqn. 13.57, but with a significantly different functional form.

The surface area fraction of polarization segments in a binary mixture of molecules type 1 + 2 is

where a_jk is defined below and the molecular surface area is

A practical difference between Eqns. 13.57 and 13.61 is that the activity coefficient contribution of a given σ-interval depends on itself through the summation over all interactions in Eqn. 13.61. Conceptually, the influence of a segment on its neighbors alters its own behavior as those neighbors respond to the local activity. This necessitates iteration, initiated with Γ_k = 1 for all k.

A fundamental difference between Eqns. 13.57 and 13.61 is that the formulas are derived entirely differently. The matrix of a_jk is related to balancing electrostatic charges, as given by:

where α(j,k) and β(j,k) characterize the hydrogen bonding between the j^th and k^th segments, similar to the α and β parameters of the SSCED model. We assume that hydrogen bonding occurs regardless of which σ value (j or k) surpasses the threshold value because the energetic reward is sufficient for them to find each other regardless of where the segments are. Mathematically, this becomes,

Similar to UNIFAC, this approach leads to a nonunity value for the activity coefficient of a pure fluid. So,

where is computed using the same formula as for UNIFAC or UNIQUAC; A_eff = 7.5 is the normalization for the area, and Γ_k⁽ⁱ⁾ is the segment activity in pure component i, computed by applying Eqn. 13.62 and so forth, with the appropriate composition.

Although we can conceive of COSMO-RS as being similar to UNIFAC, it is really much more general. The UNIFAC method requires experimental data to characterize the a_ik matrix. COSMORS, on the other hand, computes this matrix based on σ-profiles that have been computed with no experimental data except the empirical constants in Eqn. 13.65. Usually, more precise results are obtained with UNIFAC if all the groups have been characterized, but the COSMO-RS approach can be used to supplement the UNIFAC method when no experimental data exist.

a. The UNIQUAC model (right figure) was fit to the data of Marinichev A.N., Susarev M.P. 1965. Zhur. Prikl. Khim, 38(2):371.

b. These components are available in the educational version of COSMOtherm. Estimates are more accurate with COSMOtherm, but the mirror-image effect is consistent.

13.7. The Molecular Basis of Solution Models

As discussed during the development of quadratic mixing rules, there comes a point at which the assumption of random mixing cannot completely explain the nonidealities of the solution. Local compositions are examples of nonrandomness. The popularity of local composition models like Wilson’s equation or UNIFAC means that we need to develop some appreciation of the strength of the underlying theory and its limitations. Similar to the situation for the random mixing models, there are limitations to the local composition models. At this time, however, we are not exactly sure what all the limitations are. This is still a question for active research. Nevertheless, we can provide an understanding of the assumptions in these models, because the assumptions are the sources of errors that impose limitations.

Extending the Energy Equation to Mixtures

We begin the discussion with the energy equation, not the pressure equation as we did for pure fluids. This is because we are presently concerned with the Gibbs energy of mixing and its excess change relative to ideal solution behavior. It turns out that the excess Gibbs energy is dominated by the excess internal energy in most cases. In other words, the entropy of mixing is given with reasonable accuracy by the mixing rule on “b” given in Eqn. 12.1. Therefore, we focus our attention on extending the energy equation to mixtures. This requires revisiting our development of the energy equation for pure fluids and applying the same principles to extend it to mixtures. With two small modifications, the energy equation we developed for pure fluids becomes:

The small modifications are: (1) We have put the equation on an atomic basis instead of a molar basis by noting n_iN_A = N_i and Nk = nR; (2) We recognize that these are the contributions to the energy departure that arise from atoms of type “1.” For the pure fluid, it so happens that we only have atoms of type “1.”

In developing the energy equation for pure fluids, we recognized that the average internal energy departure per atom [i.e., (U₁ – U₁^ig)/N₁] was equal to the energy per pair per unit volume times the local density in that volume integrated over the total volume. To make the extension to a mixture, we must simply recognize that there are now atoms of “type 2” around those atoms of “type 1.” To illustrate, consider a parking lot full of blue cars and green cars. If one parking lot had only green cars, then the average energy per green car would involve the average number of green cars at each distance around one green car times the energy associated with green cars being that distance from each other. If you pack them too close, you will have to work hard to pack them, and so forth. Now consider the next parking lot, where blue cars are mixed with green cars. The average energy per green car will now involve contributions from green-green interactions and blue-green interactions. In equation form, this becomes

We can check this equation by noting that it approaches the pure green car equation when all the blue cars leave the parking lot (i.e., as N_b → 0). We may next write the average energy per blue car by symmetry and the total energy by addition.

Finally, making the substitutions in terms of the mole fractions, recognizing that u_bg=u_gb and g_bg=g_gb, and converting back to a molar basis, we see that for multicomponent mixtures,

Comparing to the van der Waals equation for pure fluids,

where and it is understood that a_ij = a_ji.

In this form we can recognize that the radial distribution functions may be dependent on composition as well as temperature and density. Therefore, assuming the quadratic mixing rule simply neglects the composition dependence of the a parameter, as well as the temperature and density dependence. We found in Unit II that the assumptions about temperature and density in the van der Waals equation were flawed and that is what motivated the Peng-Robinson equation. Similarly, neglecting the composition dependence of the radial distribution functions leads to some limitations that give rise to local composition theory.

Local Compositions in Terms of Radial Distribution Functions

Recalling the energy equation for mixtures, Eqn. 13.71 multiplied by RT,

We may now define the local compositions in terms of the radial distribution functions.

where r_ij = r / σ_ij

          R_ij = “neighborhood”

          N_c,i = total # of atoms around sites of type “i,” that is, the coordination number.

Rearrangement gives the molecular definition of the local composition parameter Ω_ij,

and we note the similarity between the integral in the energy equation and the integral in the definition of local composition.

For a square-well fluid, ε_ij = constant, so we can factor it out of the integral,

Substituting N_c,j, and x_ij into the energy equation for mixtures (multiply Eqn. 13.75 by N_c,j and substitute into Eqn. 13.78),

This is the equation previously applied as the starting point for development of the Gibbs excess energy model from a local composition perspective. The rest of the derivation proceeds as before.

Assumptions in Local Composition Models

In the previous section, we discussed some of the currently popular expressions for activity coefficients. We listed the assumptions involved in developing the expressions but we did not take time to discuss those assumptions. Instead, we directly applied the expressions as a practical necessity and moved on. In this section, we recall those assumptions and attempt to put them in perspective. After developing this perspective, we conclude with a word of caution; the reliability of the predictions depends largely on the accuracy of the assumptions.

The local composition theory, upon which UNIQUAC and others are based, has the general intent of correcting regular solution theory for asymmetries in solution behavior due to fluid structure near a central species. Relaxing the assumption that S^E = 0 also leads to the necessity of considering the entropy of mixing, and differences between the UNIQUAC model and Wilson’s model are primarily due to differences in treatment of S^E. In review, four assumptions are shared by the local composition theories when considered with respect to spheres:

1. The average energy of an i-j interaction is independent of temperature, density, and other species present.

2. (A – A^is) = (G – G^is).

3. The “coordination number” of a specie in a mixture is the same as that of the pure species.

4. The temperature-dependent part of the energy of mixing is given by Ω_ij = (σ_ij/σ_ij)³ exp[z(ε_ij – ε_jj)/2kT] where z is a “coordination number.”

Wilson’s equation makes the following assumptions:

5w. N_c,j = z = 2 for all j at all densities.

6w. .

7w. (σ_ij/σ_jj)³ = V_i/V_j for all i, j, and , Λ_ji = Ω_ij.²⁰

UNIFAC(QUAC) makes the following assumptions:

7u. (σ_ij/σ_jj)³ = N_c,i/N_c,j = q_i/q_j for all i, j, .

Assumption 1 involves factoring some average energy out of the energy integral such that the local composition integral is obtained. As noted, this assumption would be correct for a square-well potential, so we can probably trust that it would be reasonable for other similar potentials like the Lennard-Jones. The doubt which arises, however, involves the application of this approximation to highly nonideal mixtures. The square-well and Lennard-Jones potentials rarely give rise to highly nonideal mixtures when realistic values for their parameters are chosen. There is very little evidence to judge whether the ε_ij factored out in this way is really independent of temperature and density for nonideal mixtures. In fact, in Chapter 1, we showed that dipole interactions are temperature dependent.

Assumption 2 has to do with neglecting ln(Z/Z^is). For liquids, this may seem dangerous until one realizes that it amounts to neglecting ln(1 + ρ^E/ρ) ≈ ρ^E/ρ. Relative to the density of a liquid, the excess density is generally small (but easy to measure with a high degree of accuracy) and this assumption is acceptable.

Assumption 3 has to do with convenience. If the coordination number of each species was assumed to change with mixing, the theory could become very complicated. That is not a very good physical reason, of course. Physically speaking, this assumption could become quite poor if the sizes of the molecules (or segments in the case of UNIFAC) were very different.

Assumption 4 is the primary assumption behind all of the current local composition approaches, but it is not required by the concept of local compositions. It is simply computationally expedient in the equations that develop. The crucial aspect of the assumption is the simple form of the temperature dependence of Ω_ij. The main motivation for this assumption appears to be obtaining an expression which can be integrated analytically. But how accurate is this assumption on a physical basis? Moreover, how can we determine the physical behavior for the behavior of Ω_ij in an unequivocal manner? Merely fitting experimental data for the Gibbs excess function is equivocal because some set of adjustable parameters will provide a good fit even if the model has no physical basis. An alternative available to us that was not available to van der Waals is to apply computer simulation of square-well mixtures over a specific range of densities and temperatures and test the validity of Wilson’s approximation directly through the simulated local compositions. This approach was undertaken by Sandler and Lee.²¹

Sandler and Lee have developed a correlation for what amounts to Ω_ij of a square-well potential.

This expression reproduces the local compositions of a substantial set of molecular simulation data which Sandler and coworkers have generated. We can therefore use this expression along with the molecular simulation data as a guide to the accuracy of Wilson’s assumption.

The most important consideration is the temperature dependence of this parameter because it is the integration with respect to temperature that allows us to get from energy to free energy. As for density, it could be argued that the density of all liquids is roughly the same, so it is not unreasonable to pick a specific density and just study the temperature effects. Suppose

If the natural logarithm term on the right-hand side in the above equation is small or gives rise to a contribution which is linear in 1/kT, then Wilson’s assumption is basically correct and his definition of ε_ij would just be a little different from Eqn. 13.18. If ln Ω_ij versus 1/kT is not linear, however, then Wilson’s assumption is very questionable. Fig. 13.4 shows ln Ω_ij is fairly linear over certain ranges of temperature. This suggests that the primary assumption of Wilson and UNIFAC(QUAC) is not unreasonable. Does this mean that the problem of nonideal solutions is solved? Maybe, but maybe not. Unfortunately, we must look closely at the range of temperatures that are applicable. This range is limited by the tendency for the mixture to phase-separate. That is, dropping the temperature at constant overall density eventually places the conditions inside the binodal curve. For the composition and density listed above, the binodal occurs when N_A(ε₁₂ – ε₁₂)/RT < –0.4. This corresponds to a maximum value of A₁₂ in Wilson’s equation on the order of ~300 cal/mol at temperatures near 400 K. To use larger values for A₁₂ would be unsupported, but larger values are often used, as illustrated in Example 13.3.

Figure 13.4. Temperature-dependence of local composition parameters. Points are molecular simulation data and the curve is the correlation of Sandler and Lee.²¹ The approximate linearity of the plot lends support to the assumption applied in integrating the internal energy to obtain the free energy.

Another indication of the potential for error with the local composition approach is given by experimental data for excess enthalpies of mixing. Relations from classical thermodynamics make it possible to estimate the enthalpy of mixing by taking the derivative of the Gibbs energy of mixing with respect to temperature. Larsen et al., have developed a modified form of UNIFAC to address this problem.²² Not surprisingly, the modification involves the introduction of a substantial number of additional adjustable parameters. Even though the modified form does improve the accuracy of all the thermodynamic properties for a large number of systems there are many systems for which the predicted heats of mixing are in error by 100%–700%. More importantly, there is no way of knowing in advance when the predictions will be in error or when they will be accurate.

So why do these approximations fit the activity coefficient data? Because they have enough adjustable parameters to fit the data. Even the Margules one-parameter equation is good enough for that in many cases, but we suffer few delusions about its physical accuracy. In conclusion, we must say that local composition theory has much to recommend it. It does fit a great wealth of experimental data and there is some justification for its form via the theoretical physics which can be applied. But it is often extrapolated too far and that can lead to miscalculations by unwary users. In the end, we must never underestimate the value of experimental data for nonideal mixtures and apply the currently available theory with a careful and mildly critical view.

Assumptions 5 through 7 have to do with the entropy of mixing. The inclusion of the Staverman-type modifications to address the differences between surface fraction and volume fraction are generally recognized to be reasonable based on polymer lattice computer simulations. This modification and the estimate of molecular volumes by group contributions instead of liquid molar volumes comprise the primary differences between the Wilson and UNIQUAC models.

The UNIFAC theory is distinguished from UNIQUAC in that the solution is assumed to be a mixture of functional groups, not molecules. The UNIQUAC theory is then applied to each type of group interaction. The values for the group interactions are then regressed from a data base that includes phase equilibrium data for many, many systems. In one sense, the UNIFAC method is more like a massive regression than a truly predictive method. Thus, it lies somewhere between the purely correlative method of fitting van Laar constants and the purely predictive method of the Scatchard-Hildebrand theory (or any equation of state with k_ij = 0). Like any regressed equation, it can be unreliable if extrapolated far beyond the originally applied data. If you are ever in the position of designing truly novel chemical systems, you should be especially sensitive to the need for specific experimental data.

13.8. Summary

The theories developed in this chapter are based on the local composition concept. Similar to models developed in the previous chapter, accurate representation of highly nonideal solutions requires the introduction of at least two adjustable parameters. These adjustable parameters permit us to compensate for our ignorance in a systematic fashion. By determining reasonable values for the parameters from experimental data, we can interpolate between several measurements, and in some cases extrapolate to systems where we have no measurements. Learning how to determine reasonable values for the parameters and apply the final equations is an important part of this chapter. We also introduce the UNIFAC model, which is useful for predicting behavior when no experimental measurements are available. Similar to UNIFAC, COSMO-RS methods are predictive, but are based on quantum mechanical simulations that can be applied when experimental data are entirely lacking for a particular functional group. UNIFAC would require at least a small quantity of experimental data to characterize the group contributions.

We should also note, however, that using an equation of state is similarly simplified by using a computer, so the basic motivation for developing solution models specific to liquids is simultaneously undermined by requiring computers for implementation. From this perspective, what we should be doing is analyzing the mixing rules and models of interaction energies in equations of state if we intend to use a computer anyway. We return to this point in Unit IV, when we discuss hydrogen-bonding equations of state for nonideal solutions.

13.9. Important Equations

The UNIFAC method receives broad application throughout thermodynamic modeling. In fact, occasional applications of UNIFAC may be too broad in the sense that experimental data specific to a particular binary system are ignored and UNIFAC predictions are not validated with actual measurements. A literature search should be conducted for experimental data pertaining to every molecular interaction in a mixture and compared to the UNIFAC predictions. If significant deviations are observed, then UNIQUAC (or Wilson or NRTL) should be applied as the general activity model with system-specific parameters whenever possible and parameters inferred from UNIFAC predictions only in cases where no experimental data are available. In that context, an important equation for this chapter is best characterized by the UNIQUAC model.

Fundamentally, the energy equation for mixtures summarizes all local composition models:

13.10. Practice Problems

P13.1. The following lattice contains x’s, o’s, and void spaces. The coordination number of each cell is 8. Estimate the local composition (X_xo) and the parameter Ω_ox based on rows and columns away from the edges. (ANS. 0.68,1.47)

13.11. Homework Problems

13.1. Show that Wilson’s equation reduces to Flory’s equation when A_ij = A_ji = 0. Further, show that it reduces to an ideal solution if the energy parameters are zero, and the molecules are the same size.

13.2. The actone(1) + chloroform(2) system has an azeotrope at x₁ = 0.38, 248 mmHg, and 35.17°C. Fit the Wilson equation, and predict the P-x-y diagram.

13.3. Model the behavior of ethanol(1) + toluene(2) at 55°C using the UNIQUAC equation and the parameters r₁ = 2.1055, r₂ = 3.9228, q₁ = 1.972, q₂ = 2.968, a₁₂ = –76.1573 K, and a₂₁ = 438.005 K.

13.4. The UNIFAC and UNIQUAC equations use surface fraction and volume fractions. This problem explores the differences.

a. Calculate the surface area and volume for a cylinder of diameter d = 1.0 and length L = 5 where the units are arbitrary. Calculate the surface area for a sphere of the same volume. Which object has a higher surface area to volume ratio?

b. Calculate the volume fractions and surface area fractions for an equimolar mixture of the cylinders and spheres from part (a). Use subscript s to denote spheres and subscript c to denote cylinders.

c. For this equimolar mixture, calculate the local composition ratios x_cs/x_ss and x_sc/x_cc for the UNIQUAC equation if the energy variables τ_cs and τ_sc are unity. For the equimolar mixture, substitute the values of volume fraction and surface fraction into the expression for UNIQUAC activity coefficients, and simplify as much as possible, leaving the q’s as unknowns.

d. Consider n-pentane and 2,2-dimethyl propane (also known as neopentane). Calculate the UNIQUAC r and q values for each molecule using group contribution methods. Compare the results with part (a). [Hint: You might want to think about the -C-C-C-bond angles.]

13.5. Consider a mixture of isobutene(1) + butane(2). Consider a portion of the calculations that would need to be performed by UNIFAC or UNIQUAC.

a. Calculate the surface area and volume parameters for each molecule.

b. Provide reasoning to identify which component has a larger liquid molar volume. Which compound has a larger surface area?

c. Calculate the volume fractions for an equimolar mixture.

13.6. Solve problem 10.14 using UNIFAC to model the liquid phase.

13.7. The flash point of liquid mixtures is discussed in Section 10.5. For the following mixtures, estimate the flash point temperature of the following components and their equimolar mixtures using UNIFAC:

a. methanol (LFL = 7.3%) + 2-butanone (LFL = 1.8%)

b. ethanol (LFL = 4.3%) + 2-butanone (LFL = 1.8%).

13.8. Use the UNIFAC model to predict the VLE behavior of the n-pentane(1) + acetone(2) system at 1 bar and compare to the experimental data in problem 11.11.

13.9. According to Gmehling et al. (1994),²³ the system acetone + water shows azeotropes at: (1) 2793 mmHg, 95.1 mol% acetone, and 100°C; and (2) 5155 mmHg, 88.4 mol% acetone and 124°C. What azeotropic pressures and compositions does UNIFAC indicate at 100°C and 124°C? Othmer et al. (1946) (cf. Gmehling²⁴) have studied this system at 2570 mmHg. Prepare T-x-y or P-x-y plots comparing the UNIFAC predictions to the experimental data.

13.10. Consider the experimental data of Brown and Smith (1954) cited in problem 10.2. Prepare a P-x-y plot and a plot of experimental activity coefficients vs. composition. Then use UNIFAC to predict the activity coefficients across the composition range and add the calculations to the plots.

13.11. Flash separations are fundamental to any process separation train. A full steady-state process simulation consists largely of many consecutive flash calculations. Use UNIFAC to determine the temperature at which 20 mol% will be vaporized at 760 mmHg of an equimolar mixture liquid feed of n-pentane and acetone.

13.12. A preliminary evaluation of a new process concept has produced a waste stream of the composition given below. It is desired to reduce the waste stream to 10% of its original mass while recovering essentially pure water from the other stream. Since the solution is very dilute, we can use a simple equation known as Henry’s law to represent the system. According to Henry’s law, . Use UNIFAC to estimate the Henry’s law constants when UNIFAC parameters are available. Use the Scatchard-Hildebrand theory when UNIFAC parameters are not available. Estimate the relative volatilities (relative to water) of each component. Relative volatilities are defined in problem 11.2.

Compositions in mg/liter are:

13.13. Derive the form of the excess enthalpy predicted by Wilson’s equation assuming that A_ij’s and ratios of molar volumes are temperature-independent.

13.14. Orbey and Sandler (1995. Ind. Eng. Chem. Res. 34:4351.) have proposed a correction term to be added to the excess Gibbs energy of mixing given by UNIQUAC. To a reasonable degree of accuracy the new term can be written

where

Derive an expression for the correction to the activity coefficient. [Hint: Do you remember how to differentiate implicitly?]

13.15. The energy equation for mixtures can be written for polymers in the form:

By analogy to the development of the Scatchard-Hildebrand theory, this can be rearranged to:

where

N_d,i = degree of polymerization for the ith component

ρ = molar density

x_i = mole fraction of the ith component

N_A = Avogadro’s number

U = molar internal energy.

a_ii* = 3 + 2/Nd_i

a_ij* = (a_ii* a_jj*)^1/2

ε_ij = (ε_ii ε_jj)^1/2

Derive an expression for lnγ₁ for the activity coefficient model presented above.

13.16. Use the UNIFAC model to predict P-x-y data at 90°C and x₁, = {0, 0.1, 0.3, 0.5, 0.7, 0.9, 1.0} for propanoic acid + water. Fit the UNIQUAC model to the predicted P-x-y data and report your UNIQUAC a₁₂ and a₂₁ parameters in kJ/mole.

13.17.

a. Rearrange Eqn. 13.22 to obtain Eqn. 13.23.

b. Use Eqns. 13.16 and 13.18 in Eqn. 13.17 and perform the integration to obtain Eqn. 13.19.

c. Use Eqns. 13.16 and 13.31 in Eqn. 13.17 and perform the integration to obtain Eqn. 13.40.

13.18. Fit the data from problem 11.11 to the following model by regression over all points, and compare with the experimental data on the same plot, using the

a. Wilson equation

b. NRTL equation

c. UNIQUAC equation

13.19. Work problem 11.25 using the

a. Wilson equation

b. NRTL equation

c. UNIQUAC equation

13.20. Work problem 11.26 using the

a. Wilson equation

b. NRTL equation

c. UNIQUAC equation

13.21. Using the data from problem 11.27, fit the specified model equation and then plot the P-x-y diagram at 80°C using the

a. Wilson equation

b. NRTL equation

c. UNIQUAC equation