Chapter 12. Van Der Waals Activity Models

Nothing is more practical than a good theory.1

Clausius

Empirical models like the Redlich-Kister expansion provide a significant improvement over the ideal solution approximation, but they lack the kind of connection with the molecular perspective that we have developed in Chapters 1 and 7. The empirical models of Chapter 11 are useful for determining the activity coefficients from a given expression for GE, but they suggest little about the form that GE should assume. As we develop models that incorporate physical insight more closely, we also obtain greater predictive power.

Section 12.1 introduces concepts common to many models based on the van der Waals approach including random mixing rules and the use of the “regular solution” assumption. Section 12.2 introduces the van Laar equation which has a meaningful functional form, but, like the Redlich-Kister and Margules equations, is fitted to experimental data. Section 12.3 introduces the Scatchard-Hildebrand theory, which represents historically the first widely-accepted predictive model, primarily useful for mixtures of nonpolar molecules. Though the original theory has limited direct use today, it is the basis of several other models, and solubility parameters developed therein are currently widely-used to characterize solvents. Section 12.4 develops the Flory equation and the Flory-Huggins model that combines the Flory representation of entropy of mixing with the Scatchard-Hildebrand theory for energy of mixing. The Flory-Huggins approach is widely used in polymer thermodynamics. Section 12.5 extends the Scatchard-Hildebrand theory by including acidity and basicity corrections in a manner that is very successful in estimating infinite dilution activity coefficients. That section also develops the SSCED and MAB models, which are pedagogical simplifications that apply the concepts with approximate methods. The final sections document the relation of the theories to the original van der Waals equation, and extend the models to multicomponent systems.

Chapter Objectives: You Should Be Able to...

1. Compute VLE phase diagrams using modified Raoult’s law with the van Laar, Scatchard-Hildebrand, SSCED, MOSCED, or Flory-Huggins models.

2. Compute the relative volatility of key components in a multicomponent mixture.

3. Explain the relationship between molecular properties like energy density, acidity, and basicity and macroscopic behavior like activity coefficients and azeotropes, enabling predictions and formulation design.

12.1. The van der Waals Perspective for Mixtures

We have seen that the van der Waals EOS in Chapter 7 provides a simple basis for understanding the interplay between entropy, energy, repulsion, and attraction of pure fluids. Even the embellishments of the Peng-Robinson equation add little to the qualitative physical picture envisioned by van der Waals. Therefore, the van der Waals model provides a reasonable starting point for conceiving the physics of mixtures. The key quantities to be considered are van der Waals’ a and b. If only we knew how to compute a = a(x) and b = b(x), then we could solve for Z for a mixture. Through Z, we can integrate our density-dependent formulas to obtain G, then differentiate with respect to composition to obtain partial molar Gibbs energies, chemical potentials, and component fugacities. This is the general strategy. The formulas for a = a(x) and b = b(x) are called mixing rules.

A Simple Model for Mixing Rules

Recognizing the significance of the Gibbs excess function, it should not be surprising that many researchers have studied its behavior and developed equations that can represent its various shapes. In essence, these efforts attempt to apply the same reasoning for mixtures that was so successful for pure fluids in the form of the van der Waals equation. The resultant expressions contain parameters that are intended to characterize the molecular interactions within the context of the theory. The utility of the theory is judged by how precisely the experimental data are correlated and by how accurately predictions can be made. Given that molecules in solution must actually interact according to some single set of laws of nature, one might wonder why there are so many different theories. The challenge with mixtures is that there are many different kinds of interactions occurring simultaneously, for example, disperse attractions, hydrogen bonding, size asymmetries, branching, rings, and various rotations and aspect ratios. As a result, many specific terms must be invoked to describe these many specific interactions. Because incorporation of all these effects makes the resultant model unwieldy, many researchers have made different approximations in their models. Each model has its proponents. It is difficult even to describe the various models without expressing personal prejudices. For practical applications, the perspective we adopt is that these equations usually fit the data, and that extrapolations beyond the experimental data must be performed at some risk.

The models considered in the remainder of this chapter are based on extension of the van der Waals equation of state (Eqn. 7.12) to the energy departure for mixtures. When we extend equations of state to mixtures, the basic form of the equation of state does not change. The fluid properties of the mixture are written in terms of the same equation of state parameters as for the pure fluids; however, equation of state parameters like a and b are functions of composition. The equations we use to incorporate compositional dependence into the mixture constants are termed mixing rules.


Image The composition dependence is introduced into an equation of state by mixing rules for the parameters. The basic equation form does not change.


The parameter b represents the finite size of the molecules. For many mixtures of roughly equal-sized molecules, the dense volumes mix ideally. Therefore, it is reasonable to assume2

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As for a, we must carefully consider how this term relates to the internal energy of mixing, because the Gibbs energy of mixing is closely related. The departure function can be quickly found using Eqn. 7.12 with the departure formula Eqn. 8.22, resulting in

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The parameter a should represent the average attraction resulting from the many varied molecular interactions in the mixture.

In a binary mixture there are three types of interactions for molecules (1) and (2). First, a molecule can interact with itself (1+1 or 2+2 interactions), or it can interact with a molecule of the other type (a 1+2 interaction). Assuming a random fluid,3 the probability of finding a (1) molecule is the fraction of (1) atoms, x1. The probability of a 1+1 interaction is a conditional probability. A conditional probability is the probability of finding a second interacting molecule of a certain type given the first is a certain type. For independent events, a conditional probability is calculated by the product of the individual probabilities. Therefore, the probability of a 1+1 interaction is Image. By similar arguments, the probability of a 2+2 interaction is Image. The probability of a 1+2 interaction is x1x2 and the probability of a 2+1 interaction is also x1x2.4 If the attractive interactions are characterized by a11, a22, and a12, the mixing rule for a is given by:

Image

where the pure component a parameters are indicated by identical subscripts on a. In other words, a11 represents the contribution of 1+1 interactions, a22 represents 2+2 interactions, a12 represents the contribution of 1+2 interactions, and the mixing rule provides the mathematical method to sum up the contributions of the interactions. a12 is called the cross coefficient, indicating that it represents two-body interactions of unlike molecules. In the above sum, it is understood that a12 is equivalent to a21. Note that a12 is not the a for the mixture. This type of mixing rule is called a quadratic mixing rule, because all cross-products of the compositions are included. It represents a fairly obvious approximation to the way mixing should be represented.

The quantity a12 plays a major role in the solution behavior. Predicting solution behavior is largely the same task as predicting a12. This prediction takes the form of

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where k12 is an adjustable parameter called the binary interaction parameter. The default is k12 = 0, giving a preliminary estimate of a12. Going beyond this estimate requires contemplating whether we expect positive or negative deviations from Raoult’s law. Eqns. 12.3 and 12.2 show that larger values of a12 result in more negative (exothermic) energies of mixing. More negative values of k12 lead to more negative energies of mixing, which is favorable to mixing.

We can relate a12 to the molecular properties by considering the square-well model. Fig. 1.2 shows the square-well model for a binary mixture. We are now in a better position to interpret its significance. When k12 is positive (case (c)), the depth of the attraction is shallower. The most positive value would be k12 = 1, in which case there would be no attraction at all. This is an easy way to remember that positive values of k12 lead to repulsion. When k12 is negative (case (b)), the attraction can be very deep. In fact, k12 < –1 is a possibility for very strong acid-base interactions. Recalling our discussion of the MAB model, we should begin to understand how to predict k12. We further develop this understanding through the remaining discussion in this chapter.

Activity Models from the van der Waals Perspective

Several models derive from the van der Waals equation of state with the assumption of a constant packing fraction (bρ). They are distinguished by different approximations of the terms in the resultant equations. For example, the entropic contributions to GE are neglected by “regular” solution models (van Laar, Scatchard-Hildebrand, SSCED). The van Laar model is distinguished by treating the ratio of V2/V1 as an adjustable parameter, whereas Scatchard-Hildebrand and SSCED estimate that ratio from the liquid molar volume. SSCED differs from Scatchard-Hildebrand by accounting for the effects of hydrogen bonding (like the MAB model) as well as the influences of attractive dispersion forces (aρ).

Many of these distinctions may seem superficial, but they are part of the historical development that forms the lexicon of activity models. Furthermore, they provide a convenient shorthand for referencing the various contributions to overall solution behavior. It is important to remember, however, that the assumed activity model in no way alters the procedures for computing VLE developed in Chapters 10 and 11. You should be able to adapt any solution model to solving VLE problems by simply substituting the appropriate expression for γ.

“Regular” Solutions

The Gibbs energy, GE = UE + PVETSE, is composed of three contributions. For liquids, the PV term is small to begin with, and PVE is even smaller, so it makes sense to ignore the excess volume contribution and assume that the solution volume follows ideal mixing rules. Recalling how entropy is related to volume suggests the hypothesis that entropy might mix ideally also. These simple suppositions lead to the theories of van Laar or Scatchard and Hildebrand. We refer to “regular solution theories” when applying the assumptions of VE = SE = 0, leading to GEUE.

The energetics of a of the mixture are given by Eqn. 12.3 and 12.4. For the volume, assuming zero excess volume, Image according to regular-solution theory. Combining into Eqn. 12.2,

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For the pure fluid, taking the limit as xi → 1,

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For a binary mixture, subtracting the ideal solution result, UE = (UUig) – (UUig)is to get the excess energy gives,

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Collecting terms over a common denominator,

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Image
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12.2. The van Laar Model

Johannes van Laar found that the parameters from the van der Waals equation of state were not accurate in predicting excess energy of mixing, and empirical fitting was required. He simplified the equation for the excess internal energy by arbitrarily defining a single symbol, “Q,” to represent the final term in the equation:

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J. van Laar was a student of Johannes van der Waals and Jacobus Henricus van’t Hoff.


It would appear that this equation contains three parameters (V1, V2, and Q), but van Laar recognized that it could be rearranged such that only two adjustable parameters need to be determined.

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with the final result:5

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Differentiating Eqn. 12.12 gives expressions for the activity coefficients. To show this for γ1,

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Applying Eqn. 11.28 for n1 and differentiating the ratio using the product rule on (n2A12A21)(n1)(1/(n1A12+n2A21)),

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Obtaining a common denominator, rearranging, and applying symmetry for γ2,

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Image Van Laar model.


Note: The parameters A12 and A21 for the van Laar and Margules equations have different values for the same data. Do not interchange them.

When applied to binary systems, it is useful to note that these equations can be rearranged to obtain A12 and A21 from γ1 and γ2 given any one VLE point. This is the simple manner of estimating the parameters that we generally apply in this chapter. Similar to the two-parameter Margules, a single experimental point can be used as described in Section 11.6:

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Care must be used before accepting the values of Eqns. 12.16 applied to a single experiment, because experimental errors can occasionally result in questionable parameter values. Eqn. 12.16 applied to the activity coefficients from Example 11.1 results in A12 = 2.38, A21 = 1.15, and GE is plotted in Fig. 11.3. Methods of fitting the parameters in optimal fashion for many data are covered in Section 11.9.


Example 12.1. Infinite dilution activity coefficients from the van Laar theory

n-Propyl alcohol (1) forms an azeotrope with toluene (2) at x1 = 0.6, 92.6°C, and 760 mmHg. Use the van Laar model to estimate the infinite dilution activity coefficients of these two species at this temperature.

Solution

The vapor pressures using parameters from Antoine.xlsx are Image, Image.

Applying the azeotropic data as explained in Section 11.7 gives: Image = 1.191; Image. Eqn. 12.16 gives: A12 = 1.643; A21 = 1.193.

Taking the limits of Eqn. 12.15 as the respective components approach zero composition results in Image and Image; Similarly Image.


Linear Fitting of the van Laar Model

Like the Margules models, the van Laar model can be linearized. Eqn. 12.12 can be rearranged:

Image

Therefore, if numerical values for the left-hand side are determined using GE from experimental data as illustrated in Example 11.1 on page 414 and plotted versus x1, the slope yields (1/A21 – 1/A12), and the intercept yields 1/A12. The value of 1/A21 can also be determined by the value at x1 = 1. Sometimes plots of the data are non-linear when fitting is attempted. This does not necessarily imply that the data are in error. It implies that an alternative model may fit the data better. By plotting the data in both the Margules and van Laar linearized forms, the better model can be identified as the one that is most linear. Fitting the data for 2-propanol + water presented in Example 11.8 (ignoring the first mixture point), results in an intercept = 1/A12 = 0.4993, slope = 0.3486, resulting in A12 = 2.00, A21 = 1.18.

12.3. Scatchard-Hildebrand Theory

Returning to Eqn. 12.10, G. Scatchard in Europe and Joel H. Hildebrand in the United States both made similar adjustments to match the van der Waals equation to experiment and provide a model capable of predictions for nonpolar fluids. They made an assumption that is equivalent to assuming k12 = 0 in Eqn. 12.4. Setting Image, and collecting terms,

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J.H. Hildebrand is credited with suggesting that helium be mixed with breathing air in deep sea diving to minimize “the bends.” He was awarded the ACS Priestly award in 1962. He continued to maintain an active professional life until he was 100.


Scatchard and Hildebrand recognized the unknown parameters in terms of volume fractions and disperse attraction energies that could be related to the pure component values. Defining a term called the “solubility parameter,”

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Image
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To estimate the value of δi, Scatchard and Hildebrand suggested that experimental data be used such that

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(Note the units on the “a” parameter from Eqn. 12.2, and from comparing Eqns. 12.21 and 12.22, and note the way Vi moves inside the root in Eqn. 12.22.)

In other words, δi is assumed to provide a standard measure of the “energy density” for each component. Because it represents the energy departure divided by volume, it is called the cohesive energy density. As long as a standard reference condition is used, any convenient set of ΔUvap and Vi may be applied. A convenient set of conditions that has become customary is the saturated liquid at 298 K. On this basis, we can tabulate a fair number of solubility parameters for ready reference, as shown in Table 12.1. Note that many similar compounds have similar values for their solubility parameters. Since similar solubility parameters yield small excess energies, solutions of similar components are predicted to be nearly ideal, as intuitively expected. By scanning the tables for the values of solubility parameters, we can quickly estimate whether the ideal solution model should be accurate or not. This approach gives a quantitative flavor to the old adage “like dissolves like.”

Table 12.1. Solubility Parameters in (J/cm3)1/2 and Molar Volumes (cm3/mol) for Various Substances as liquids at 298 K

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Turning to the Gibbs energy, the regular solution assumptions give,

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And the resultant activity coefficients are

Image

Image Scatchard-Hildebrand theory.


Image

Image Actcoeff.xlsx, worksheet REGULAR.



Image Bubble-temperature calculation.



Example 12.2. VLE predictions using the Scatchard-Hildebrand theory

Benzene and cyclohexane are to be separated by distillation at 1 bar. Use the Scatchard-Hildebrand theory to predict whether an azeotrope should be expected for this mixture.

Solution

We will implement the algorithm to test for an azeotrope from Section 11.7 on page 432. Given xB and P, we should perform bubble-temperature calculations.

Using parameters from Table 12.1, at xB = 0.99, guess T = 350 K:

ΦB = 0.99(88)/[0.99(88) + 0.01(107)] = 0.9879

Calculating vapor pressures:

Image

Applying Eqns. 12.24 and 12.25:

lnγB = 88(1 – 0.9879)2(18.74 – 16.75)2/(8.314·350) = 1.8E-5 γB = 1.00002
lnγC = 107(0.98789)2 (18.74 – 16.75)2/(8.314·350) = 0.1443 γC = 1.1552

Calculating the pressure and vapor mole fractions:

Image = 0.99(1.00)(686.9) + 0.01(1.1552)(680.0) = 687.9 mmHg

yB = 0.99(1.00)(686.9)/687 = 0.895, yC = 0.01(1.1552)(680.0)/687 = 0.010

Since Image, we must guess a higher temperature.

Guess T = 354 K Image; Image; γB = 1.00; γC = 1.1533

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Interpolating between the first guesses:

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At xB = 0.01, guess T = 353.5 K ΦB = 0.01(88)/[0.01(88) + 0.99(107)] = 0.0082

lnγC = 107(0.0082)2(18.74 – 16.75)2/(8.314·350)≈ 0 γC = 1.00
lnγB = 88(1 – 0.0082)2(18.74 – 16.75)2/(8.314·350) = 0.1232 γB = 1.126

Image

converged with yB = 0.011 > xB = 0.01

Therefore, (yBxB) changes sign between 0.01 and 0.99, so the system has an azeotrope.



Image Actcoeff.xlsx, worksheet REGULAR.


When the Scatchard-Hildebrand solution theory is used, the {δi} and {Vi} are available directly from pure component data, and in principle, there are no adjustable parameters. The theory is entirely predictive. The van Laar theory, on the other hand, treats both A12 and A21 as adjustable parameters. We can also obtain a compromise by assuming

Image

The activity coefficient expressions for binary solutions become:

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Image Scatchard-Hildebrand model with an adjustable parameter.


In mixtures of compounds that deviate moderately from ideal-solution behavior, the Scatchard-Hildebrand solution theory with binary interaction parameters can be extremely helpful. The binary interaction parameter in those cases serves to adjust the magnitude of the excess Gibbs energy without addressing the skewness directly. Large deviations, however, are generally accompanied by non-ideal mixing in the volume and entropy. In those cases, the van Laar equations can often be useful in correlation, but the physical meaning behind the parameters is generally lost.


Paul J. Flory was awarded the Nobel Prize in chemistry in 1974. The original derivation used statistical mechanics, but is consistent with this alternative derivation.


12.4. The Flory-Huggins Model

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 = RT2 – 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 = RT2 – 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 = RT2 – 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:

Image
Image

Substituting and noting that (x1V1 + x2V2) cancels between numerator and denominator,

RTlnγ1 = RT2 – δ1)2 [V1Φ1Φ2 + (Φ2 V1(1 – Φ1) – Φ1V12)]

The first and last terms in the brackets cancel. Also, for a binary mixture, Φ2 = (1 – Φ1). So,

RTlnγ1 = RT2 – 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.

Image

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.



Image Actcoeff.xlsx, sheet REGULAR, Ex12_04.m.


The free volume available to any pure component is

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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

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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:

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Note that Eqn. 12.32 reduces to the ideal solution result, Eqn. 10.63, when V1 = V2. The excess entropy is

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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.

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Image 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,

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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,

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Image 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 ≡ V11 – 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.

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Figure 12.2. Illustration of excess entropy according to Flory’s equation for various pure component volume ratios.


Example 12.5. Polymer mixing

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.

Image

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.

12.5. MOSCED and SSCED Theories

The Scatchard-Hildebrand theory provides reasonable results for hydrocarbon mixtures, but the results can be highly unreliable if one of the components hydrogen bonds, especially if one of the components is water. The MOSCED, SSCED, and HSP models remedy this problem by accounting for hydrogen bonding as a separate contribution to the solubility parameter. MOSCED (pronounced moss-ked) stands for MOdified Separation of Cohesive Energy Density. SSCED (pronounced sked) stands for Simplified Separation of Cohesive Energy Density. Recall that the cohesive energy density is the term for δ2. The HSP model (Hansen Solubility Parameters) is similar in concept, but does not distinguish between acidity and basicity.8 Therefore, it cannot predict negative deviations from ideality in the manner of MOSCED or SSCED and is omitted from detailed discussion. It has been broadly applied, however, and provides impressive demonstrations of what can be achieved with these kinds of theories. The concept behind these models is that hydrogen bonding should be counted separately from the physical interactions envisioned by Scatchard and Hildebrand.

The MOSCED Model

The MOSCED model is given by,9

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where λi is the dispersion factor (e.g., equal to δ for n-alkanes), τi is the polarity factor (e.g., 3.95 for benzene), qi is a factor ranging from 0.9 to 1 (for our purposes, qi = 1 for all i, cf. reference 9 for details) and α and β are acidity and basicity parameters. Values of λ and τ are given in Table 12.2. Values of α and β are given on the back flap. We recognize d12 as the Flory-Huggins contribution. aa, ψ1 and ξ1 are parameters characterizing solvent properties.

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Table 12.2. Dispersion (λ) and Polarity (τ) Parameters in (J/cm3)1/2 liquids at 293 K

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Note that MOSCED is not intended to describe the entire solution behavior directly. Instead, it provides estimates for the infinite dilution activity coefficients. At that point, another activity coefficient model can be applied by fitting its parameters at infinite dilution to the MOSCED predictions.

The essential feature of this model is the explicit representation of acidity and basicity. Consider, for example, the acetone + chloroform system of Fig. 9.6(c). MOSCED model predicts a negative deviation from ideality for this system, as is the experimental behavior:

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For 2-propanol + water, a positive deviation is properly indicated:

Image

These measures of acidity and basicity provide useful insights into the chemical nature of compounds. It is notable that they can be characterized spectroscopically by mixing a range of compounds with standard bases and acids. To measure acidity, for example, one might individually mix pyridine with acetone, benzene, chloroform, and acetic acid. More acidic molecules would bind more strongly to the pyridine nitrogen and shift the ultraviolet absorption more strongly. So the magnitude of the shift would provide a relative measure of the acidity.10 By verifying the trend with other standard bases, an average indicator could be developed for the acidity of each compound (and its variance). The spectroscopic measurement is independent from the VLE measurement, so the two observations strongly reinforce each other. This kind of chemical insight combined with spectroscopic evidence can be useful in a wide variety of settings. In catalyst design, for example, one might devise acidic adsorption sites to attract a reactant with high basicity, then measure spectroscopically whether the device was working. Students should look for creative opportunities to relate concepts like these across the curriculum.

The SSCED Model

Despite its attractions, the MOSCED model is relatively cumbersome. It has many terms, and in the end, another model (e.g., Redlich-Kister, van Laar, or a model from the next chapter) must be used to compute the phase behavior. As an pedagogical introduction to MOSCED, we would like to estimate the phase behavior with sufficient accuracy to predict whether an azeotrope or liquid-liquid separation may occur, but to make only approximate estimates with a single self-consistent theory. These motivations suggest a need for a simplified version of MOSCED.

Similar to MOSCED, the SSCED model retains the simple form of the Scatchard-Hildebrand model while correcting its gross misrepresentation of polar mixtures by taking advantage of the acidity and basicity measures of MOSCED. For a binary mixture, the SSCED model is

Image
Image

Image SSCED model.


Image
Image

Example 12.6. Predicting VLE with the SSCED model

Amines often function as bases that can moderate interactions with acidic compounds. In the case of triethylamine, however, the high hydrocarbon content competes with the basicity and it is difficult to intuitively assess how the solution ideality may turn out.

a. Predict the bubble pressure and vapor composition of triethylamine (1) + ethanol (2) at 308 K and x1 = 0.59 using the SSCED model.

b. Compute the relative volatility, αLH, at x1 = 0.01 and 0.99, where triethylamine is the light key component. Is an azeotrope indicated? Copp and Everett (1953) report an azeotrope at x1 = 0.59 and P = 119 mmHg.

SSCED and Antoine constants for triethylamine are:

Image

Solution

a. From the Antoine equation, P1sat = 109; P2sat = 102 mmHg (cf. Appendix E for ethanol);
δ1′ = 15.17 – 0 = 15.17; δ2′ = (26.132 – 2·12.58·13.29)½ = 18.67; V1= 101/0.72= 140; V2=58.5.

From Eqn. 12.51, k12 = (12.58 – 0)(13.29 – 7.70)/(4·15.17·18.67) = 0.062;
[(δ2′ – δ1′)2 + 2k12δ1′δ2′]/RT = [(18.67 – 15.17)2 + 2·0.062·18.67·15.17]/(8.314·308) = 0.0185.
At x1 = 0.59, Φ1 = 0.59·140/(0.59·140 + 0.41·58.5) = 0.774.

γ1 = exp(140(1 – 0.774)20.0185) = 1.141; γ2 = exp(58.5(0.774)20.0185) = 1.916;
P = 0.59·1.141·109 + 0.41·1.01·102 = 154; y1 = 0.59·1.141·109/154 = 0.477.

b. At x1 = 0.01, Φ1 = 0.01·140/(0.01·140+0.99·58.5) = 0.024;
γ1 = exp(140(1 – 0.024)20.0185) = 11.9; γ2 = exp(58.5(0.024)20.0185) = 1.001;
P = 0.01·11.9·109 + 0.41·1.001·102 = 114; y1 = 0.99·1.001·109/114 = 0.113.
αLH(0.01) = 11.9·109/(1.001·102) = 12.6

At x1 = 0.99, Φ1 = 0.99·140/(0.99·140 + 0.01·58.5) = 0.996;
γ1 = exp(140(1 – 0.996)20.0185) = 1.0001; γ2 = exp(58.5(0.996)20.0185) = 2.93;
P = 0.99·1.0001·109 + 0.01·2.93·102 = 111; y1 = 0.99·1.0001·109/111 = 0.973.

αLH(0.99) = 1.0001·109/(2.93·102) = 0.363

Therefore, an azeotrope is suspected since αLH – 1 changes sign as discussed in Section 11.11. The system should be evaluated experimentally or with a literature search.


To follow up, Fig. 12.3 shows through comparison to experiment that the SSCED model overestimates the nonideality of the solution, but the prediction of an azeotrope is valid. For a broader perspective, we can go beyond Example 12.6 and compare to the Scatchard-Hildebrand model, but the Scatchard-Hildebrand (ScHil) model is not even close. In fact, the Scatchard-Hildebrand model indicates VLLE where none exists. This is a common problem with Scatchard-Hildebrand theory in the presence of hydrogen bonding. It undermines the viability of the Scatchard-Hildebrand model for most applications, but the SSCED model retains its simplicity while providing a reasonable basis for conceiving formulations predictively. As a final note, the MAB model performs slightly better than the SSCED model for this mixture with P = 148 mmHg.

Image

Figure 12.3. VLE of triethylamine(1)+ethanol at 308 K. The basicity of the amine moderates the non-ideality of the solution by solvating the hydroxyl interaction. Data of Copp J.L., Everett D.H. 1953. Disc. Faraday Soc.15:174.

12.6. Molecular Perspective and VLE Predictions

Can we just forget about some of these models? After all, the van der Waals perspective inherently accounts for the molecular properties through a and b. Why should we worry with so many variations? Is the SSCED model really so different from the Scatchard-Hildebrand model? What about the MAB model? The van Laar model? Which model is “best?” In every case, there is a factor mitigating against entirely eliminating any one of the models from consideration.

To begin, it is possible to derive the MAB model as a special case of the SSCED model. When V1 = V2 and δ1′ = δ2′, we obtain Φi = xi, and we obtain the Margules one-parameter model,

Image

Technically, we could simply substitute V1 or V2 instead of (V1 + V2)/2, but writing it this way provides a small compensation for the observation that it is extremely unlikely that V1 = V2. So, if the MAB model is such an oversimplification, why not forget about it and just use the SSCED model? That would be a good argument, except that MAB is such a simple model. It does not require converting from mole fraction to volume fraction and you can anticipate the predicted sign on GE without using a calculator.

On the other hand, the Δδ′ term of SSCED is quite significant in some cases. For example, n-hexane + methylethylketone has a significant non-ideality that is overlooked by MAB. A similar argument could be made about polyethylene + polypropylene. Furthermore, the distinction between V1 and V2 in SSCED correctly indicates that it is much less favorable to squeeze a large molecule into a fluid of small molecules than vice versa, as in the n-butanol + water system.

Another argument could be made about the difference between the SSCED and Scatchard-Hildebrand models. Both models have the same skewness when k12 is fit to experimental data. The Scatchard-Hildebrand model takes precedence historically. So maybe we should forget the SSCED model. On the other hand, SSCED provides better a priori predictions of phase behavior when an associating component is involved. Eqn. 12.50 shows that the solubility parameter is unaltered if α = 0 or β = 0, but it is substantially diminished for associating compounds like alcohols and water. With this change alone, the estimated nonideality is substantially diminished. This is a step in the right direction because overestimating the nonideality (Scatchard-Hildebrand) may cause more confusion than treating the solution as ideal (Fig. 12.4(a)). Remember to “First, do no harm.” A similar observation is illustrated in Fig. 12.4(b). The peculiar lines of the Scatchard-Hildebrand (ScHil) model show what happens if a VLE model is applied when the activity model indicates VLLE. The experimental data indicate no LLE for either system, showing the qualitative deficiency of the Scatchard-Hildebrand model for associating mixtures, as well as its quantitative deficiency. The SSCED model, on the other hand, is qualitatively correct, and semi-quantitative in its predictions.

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Figure 12.4. Predictions of SSCED and Scatchard-Hildebrand theories at 1 bar. For Scatchard-Hildebrand theory, kij = 0 is applied. For SSCED, Eqn. 12.51 applies. (a) Acetone+ethanol (data of Amer, H.H., Paxton, R.R. van Winkle, M. 1956. I&EC, 48:142); (b) ethanol+water (data of Bloom, C.H., Clump, C.W., Koeckert, A.H. 1961. Ind. Eng. Chem. 53:829).

Another distinction between SSCED and the Scatchard-Hildebrand model is the guideline for k12 given by Eqn. 12.51. We can compute the VLE with this guideline and compare to the VLE at k12 = 0 to get a range of estimates that suggests the direction of the nonideality and its magnitude. Having multiple VLE estimates may seem like a bad idea, but computing a single crude prediction ignoring alternative estimates would be a much worse idea. Contradictory estimates should remind us of the value of using experimental data to correlate k12 whenever possible. The procedure for correlating k12 of SSCED based on experimental data is not different from the procedure for the Scatchard-Hildebrand model, nor from the procedure for the Margules or van Laar models.

One could also argue that the van Laar model provides the best fit of the VLE data, so it should be preferred. On the other hand, it offers no predictive capability, and we really would like to conceive designs for formulations. Furthermore, the van Laar model cannot be extended to multicomponent mixtures in a manner that is consistent with the van der Waals perspective, as discussed in the next section.

Finally, one might consider the MOSCED model to be the best model. First, it includes the Flory-Huggins correction for polymer-solvent interactions. Second, when combined with the van Laar or two-parameter Margules model, it provides predictive capability that is superior to the SSCED model. In a particular case study involving organic nitrates, the SSCED model gave roughly 10% deviation in predicted bubble pressure while the MOSCED model gave only 5%. Furthermore, the predictive insight of MOSCED is at least as good as that of SSCED because it uses the same values of α, β, δ, and V. On the other hand, it requires several more intermediate calculations than the SSCED model, including the translation of the infinite dilution activity coefficients into a different activity model and subsequent computation of the activity coefficient at the concentration of interest. The requirement of so many intermediate computations generally necessitates the use of a computer, in which case the methods of Chapter 13 are generally preferred. Overall, the argument in favor of MOSCED over SSCED is probably the best, however.

In summary, the “best” overall activity model should account for simple molecular properties in addition to providing a basis for fitting the data of a specific binary system, because our overall goal includes conceiving of formulations. Formulations generally involve more than two components and designing them requires simple intuitive insights like energy density (reflected in δi’) and hydrogen bonding (reflected in α and β). A small molecule with a high energy density disfavors molecules with less energy density just as an associating compound squeezes out an inert one. The interplay between these types of molecular interactions is significant and it is best to contemplate both influences when predicting phase behavior. Through the SSCED model, several observations can be cited about this interplay: (1) Ignoring hydrogen bonding leads to overestimates of solution nonideality, as observed for the Scatchard-Hildebrand model; (2) accounting for hydrogen bonding reduces the differences in disperse interactions (i.e. Δδi′ < Δδi); (3) competition between the effects of complexation and energy density can make it difficult to predict whether deviations from Raoult’s law are positive or negative; and (4) large nonidealities result when both association and energy density indicate unfavorable mixing, typified by the hydrophobic effect.

In this context the SSCED and MOSCED models would appear to provide the best overall models within the van der Waals perspective because they both account for the interplay between energy density and hydrogen bonding. The trade-off between them is one of precision versus simplicity. Resolving this trade-off requires the context of a particular application. If you are always working with similar solvents and species, the greater detail of the MOSCED model may provide opportunities for refining your predictions.

12.7. Multicomponent Extensions of van der Waals’ Models

Most systems encountered in chemical processes and formulations are multicomponent. If the application requires bypassing an azeotrope, a third component (called an entrainer) might be added. If a biomembrane is to be penetrated by a pharmaceutical treatment, the formulation must at least account for water, the pharmaceutical, the biomembrane, and any additive. The output of a simple reaction like A + B → C would mean that at least three components must be separated if conversion was less than 100%. These examples and more lead to the conclusion that a multicomponent solution model is a necessity.

Unfortunately, the van Laar model makes the assumption that V2/V1 can be treated as an adjustable parameter. While this is fine for a binary mixture, it becomes problematic for a ternary mixture because knowing V2/V1 and V3/V2 means that V3/V1 must be implied. The normal procedure for the van Laar model would fit the binaries for 2-1, 3-2, and 3-1 independently, however. The likelihood of achieving a consistent value of V3/V1 from such a fit is practically zero.

On the other hand, the simple relation of the Scatchard-Hildebrand theory to the van der Waals equation permits a simple extension to multicomponent systems. The derivation of this extension is given by practice problem 12.4. The result is

Image
Image

where <δ> = ΣΦjδj; <<kmm>> = ΣΦiδi<kim>; and <kim> = ΣΦjδj<kij>.

The SSCED model is similar to the Scatchard-Hildebrand model by design. This similarity guarantees that the algebra extending to multicomponent mixtures is identical, giving

Image

where <δ.′> = ΣΦjδj’<<kmm>> = ΣΦiδi’<kim>; and <kim> = ΣΦjδj’<kij>.

At first sight, the notation may seem confusing. The following example clarifies the necessary summations.


Example 12.7. Multicomponent VLE using the SSCED model

A partial condenser is generally indicated when distilling a stream that has a light gas impurity, even if the gas is dilute. A total condenser is impractical because of the “noncondensable” gas. Of course, we know that it must condense at some sufficiently low temperature, but it is wasteful to cool the stream that far or reflux the light gas component. A partial condenser condenses just as much liquid as necessary to keep the column functioning then sends the remainder downstream as a vapor distillate product. Downstream, a second partial condenser will provide liquid and the remaining gas can be purged to a flare tower.

Fermentation of corn to form acetone (A), n-butanol (B), and ethanol (E) in water (W) followed by a drying step results in the stream below to be distilled.a Dissolved carbon dioxide (CO2) is a prevalent by-product of the fermentation process. Use the SSCED model and assume 2 bar pressure with splits of 99% ethanol and 2.8% water. Assume that acetone moderates the ethanol+water interaction to achieve this tops composition without an azeotrope interfering. You may estimate vapor pressures using the shortcut equation.

Image

where δ’ has been computed from δ in accordance with Eqn. 12.50.

a. Estimate the flow rate and composition of the distillate based on the key component splits.

b. Estimate the dew temperature of the distillate stream as a preliminary estimate of the operating temperature in the partial condenser.

Solution

We will indicate the vapor outlet of the partial condenser stream as D.

a. E will be the light key, and W the heavy key. Everything lighter than the light key is assumed to go out the top, and everything heavier than the heavy key is assumed to go out the bottom. We estimate the boiling temperatures (K) at 2 bar in the first row. Technically, CO2 does not have a boiling temperature at 2 bar. Nevertheless, we apply the shortcut model here (at 351 K, where CO2 is supercritical) merely as an estimate, as suggested in the problem statement. Components are sorted in decreasing volatility to show that a split between E and W sends CO2 and A completely to distillate and B completely to the bottoms. The third row shows the consequent distillate (D) flows and fourth row shows the mole fractions.

Image

The tops composition of ethanol to water on a solvent-free basis is 4.95/(4.95+0.42) = 92%, which exceeds the azeotropic composition only slightly.

b. For the dew-temperature calculation, we use option (b) from Appendix C and refer to the step numbers. As an initial estimate, Image and assuming Raoult’s law (step 1), Image gives Image:

Image

Image Dew - temperature calculation.


In the last row, we divide xi(0) by Σxi(0) (the process is known as normalization), such that the mole fractions xi(0’) sum to one. To prepare for step 3, the SSCED model gives the values below for kij. For example, the value for ethanol + water is

kEW = (12.58 – 50.13)(13.29 – 15.06)/(4(18.67)27.94) = 0.0319

Similarly, kAE = 0.0184; kAW = 0.0896.

Finally, we are ready to apply the SSCED model. We drop butanol and CO2 from further calculations because their liquid compositions are zero. xi(1) is the result of the first iteration in which the γi are applied. xi(2) results from varying the temperature while assuming the γi are constant (even though the composition is changing slightly too). We are approximating to force xT = 1 to get a good T guess (step 5) for the next iteration.

Image

where γA, γE, γW, <δ.′>, <kim>, and <<kmm>> are computed by Eqn. 12.55. For example,

.′> = 0.6094(19.6) + 0.3710(18.67) + 0.0196(27.94)0.0319 = 19.4

<kEm> = 0.6094(19.6)0.0184 + 0.3710(18.67)0.0 + 0.0196(27.94)0.0319 = 0.239

<<kmm>> = 0.6094(19.6)0.177 + 0.3710(18.67)0.239 + 0.0196(27.94)1.294 = 4.478

γE = exp(58.5((18.67 – 19.4)2 + 2(0.239)18.67 – 4.478)/(8.314 . 351)) = 1.106

In going from xi(1’) to xi(2), we have varied the temperature assuming that the γi are unchanged. This must be validated with one final iteration at T = 354.88 K.

Image

Note

1. The liquid composition of less volatile components is enhanced in a partial condenser.

2. For this particular condenser, KW >KE means there is a limit to splitting W from E.


a. This problem is based on the AIChE 2009 National Student Design Competition, Richard L. Long Coordinator, AIChE New York (2008).

Noting that MAB is a special case of the SSCED model, the expression for the multicomponent SSCED model suggests a similar form for the MAB model.

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The relation between the Margules form and the Redlich-Kister form suggests a similar relation.

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In this manner, we can envision extending many activity models to multicomponent applications.

The power to design formulations that solve practical problems generally involves multicomponent systems. Design requires synthesis of many fundamental principles into a working toolbox as well as creative thinking. If we want to disperse an oil spill, we should consider the fate of the additive as well as the oil. After the oil is dispersed, where will the additive go? What is its toxicity? If we want to circumvent an azeotrope, where will the entrainer go? Should it be more volatile or less volatile than the key components? How should its molecular properties relate to those of the key components? What about the volatility of a solvent to remove paint? What about the solvent’s water solubility? The multicomponent SSCED perspective empowers us to think creatively about these kinds of problems.


Example 12.8. Entrainer selection for gasohol production

The ethanol + water system has a well-known azeotrope at 89.4 mol% ethanol and 353 K, frustrating efforts to distill fuel grade ethanol cheaply and easily. Industrially, the azeotrope is broken using adsorption. Another strategy is to add a third component (an “entrainer”) that reduces the activity coefficient of the water. In this way, the relative volatility of ethanol to water can remain greater than one. Noting that activity coefficients pertain to the liquid phase, our entrainer should stay in the liquid phase. In other words, it should be less volatile than either key component. We can envision pouring it into the top of the distillation column and letting it trickle down. This process is called extractive distillation. We would like to use as little entrainer as possible and it should not form another azeotrope with water. One suggested entrainer is 2-pyrrolidone, for which the key properties are given below. You can assume the shortcut VP model for 2-pyrrolidone and SSCED predictions for kij interactions with 2-pyrrolidone. How much 2-pyrrolidone would be needed to keep the relative volatility greater than 1.1 at 99mol% ethanol? Can you suggest any other prospective entrainers based on their molecular structure?

Molecular properties for 2-pyrrolidone are

Image

Solution

Noting that azeotropes are sensitive to vapor pressure, we should use the Antoine constants from Appendix E for ethanol and water. The vapor pressure of the entrainer is less important because it must be substantially lower than that of water to prevent another azeotrope. The SSCED predicted value of k12 = 0.0319 fails to reproduce the experimentally observed azeotrope, but a value of k12 = 0.058 gives an azeotrope with xE = 0.894 and 353 K. Checking the relative volatility at xE = 0.99 and 353 K we find αLH = 0.94. We should add entrainer until αLH = 1.1.

We can assume T = 353 K for now, so the vapor pressure ratio stays constant. We can check our result at the bubble temperature of the final formulated composition. Recalling the definition of relative volatility from Section 10.6 and substituting modified Raoult’s law,

Image

From the Antoine equation, PLsat/PHsat = 2.03. Therefore, we seek γLH > 1.1/2.03 = 0.542.

Taking a basis of 1 mole of ethanol (and 0.0101 water), increasing the ratio of 2-pyrrolidone (P) to ethanol to 0.05 gives a final composition of {0.9433, 0.0095, 0.0472} for {xE, xW, xP}. This gives a value of αLH = 2.03(1.001/2.075) = 1.11. Setting the pressure to 0.101 MPa and finding the bubble temperature gives T = 352.6, so the temperature is altered very little.

Another candidate for entrainer might be ethylene glycol. From its molecular structure, it is similar to half ethanol and half water. Following the same procedure, we find that a final composition of {0.8532, 0.0086, 0.1382} is required to achieve the same αLH. This is roughly a factor of 3 for entrainer on a mole basis or a factor of 2 on a weight basis. 2-pyrrolidone is predicted to be superior because of the large value for its basicity, larger than water’s. This causes the kij parameters with 2-pyrrolidone to be negative, so less is needed. Glycol is less effective.


12.8. Flory-Huggins and van der Waals Theories

We have shown that the contribution to the excess internal energy in the Flory-Huggins theory is identical to that in the Scatchard-Hildebrand theory. We derived the Scatchard-Hildebrand theory from the excess internal energy function of the van der Waals equation on page 468 and 12.3 on page 471. Therefore, any potential difference between the Flory-Huggins theory and the van der Waals equation must pertain to the entropy. Reviewing briefly, the van der Waals equation of state gives

Image
Image

Recall that the van der Waals equation gives Image.

Therefore, Image. Comparing Image to the result for regular solutions, we see that,

UE = Φ1Φ21 – δ2)2(x1V1 + x2V2)

which is the same. We may also note that Image is a very small number because: 1) These are liquid compressibility factors, so all Z’s are small numerically; and 2) the excess volume is usually a small percentage of the total volume, Image. Thus, we may neglect Image.

Turning to the differences between the entropy terms, the van der Waals equation gives

Image
Image

Note: (1 – biρi) = (Vibi)/Vi ≡ (Viϖ)/Vi = ϖ. This means that bi ρi = bρ for all i.

If we assume that ϖ is a universal constant for all fluids, including the mixture, then

Image

This expression is identical to Flory’s equation (and note the importance of the ln(Z) term as the second term on the right-hand side of Eqn. 12.62, which derived from the ideal gas reference state). Therefore, the only difference between van der Waals’ and Flory’s theories is the assumption that ϖ is a universal constant. This is equivalent to saying that the packing fraction () is a constant (the packing fraction is one minus the void fraction). In other words, the Flory-Huggins theory is simply the van der Waals theory with the assumption that bρ = constant. The difference between the Flory-Huggins theory and the Scatchard-Hildebrand theory is accounting for mixing at constant pressure instead of mixing at constant packing fraction. This is related to the argument about free volume being larger for larger molecules because fitting a polymer in the same volume as a solvent must lead to a deviation from the ideal gas law at some degree of polymerization. Therefore, the V in PV/RT must be proportional to the volume of the molecule.

The suggestion that = constant is actually quite consistent with another observation that should seem more familiar. That is, the mass density of a polyatomic species is only weakly dependent on its molecular weight. For example, the mass density of decane is 0.73 g/cm3 and the density of n-hexadecane is 0.77 g/cm3. Since the molar density decreases inversely as molecular weight increases but the b-parameter increases proportionally as molecular weight increases, a constant value for the mass density implies a constant value for . When you consider that the mass density for almost all hydrocarbons, alcohols, amides, amines, and their polymers lies between 0.7 and 1.3 g/cm3, you begin to get an idea of how broadly applicable this approximation is.

Nevertheless, there are some obvious limitations to the assumption of a constant packing fraction. A little calculation would make it clear that the ϖ for liquid propane at Tr = 0.99 is significantly larger that ϖ for toluene at Tr = 0.619. Thus, a mixture of propane and toluene at 366 K would not be very accurately represented by the Flory-Huggins theory. Note that deviations of ϖ from each other are related to differences in the compressibilities of the components. Thus, it is common to refer to the Flory-Huggins theory as an “incompressible” theory and to develop alternative theories to represent “compressible” polymer mixtures. Not surprisingly, these alternative theories closely resemble the van der Waals equation (with a slightly modified temperature dependence of the a parameter). This observation lends added significance to Rayleigh’s statement: “I am more than ever an admirer of van der Waals.”

12.9. Summary

This has been a somewhat theoretical chapter. We have gone through iterations of observation, prediction, testing, and evaluation with several theories (e.g., van Laar, Scatchard-Hildebrand, Flory-Huggins, SSCED, and MOSCED). With each iteration, we have achieved increasing precision and insight. Sulfuric acid and water may react very favorably toward each other (GE << 0), while 2-propanol and water have enhanced escaping tendencies because the energy required for forcing them to mix is counteracting the entropic driving force (GE > 0). These facts are due in large part to the interaction energies characterized by molecular quantities like the solubility parameter, acidity, and basicity. We showed that two major effects influence the mixing: energy density and complexation. This molecular perspective provides clarification of the familiar guideline that “like dissolves like.” With this perspective, we begin to acquire the capability to intuitively reason about chemical formulations. We illustrated the value of this reasoning with several practical examples. You should be able to distinguish favorable molecular interactions from unfavorable.

Carrying forward the molecular perspective, we can characterize these tendencies in terms of the cross-interaction energy (a12). If the cross-interaction energy is weaker than the geometric mean (k12 > 0), then each component prefers its own company. If the cross-interaction energy is stronger than the geometric mean (i.e., k12 < 0), then the components are strongly attracted, releasing energy as they fall into the well of their mutual attraction. The energy to break the favorable interactions must be added to separate such a mixture and this may show up as a maximum boiling azeotrope. Conversely, mixtures with k12 > 0 tend to exhibit minimum boiling azeotropes, or VLLE if γi > 10.

Molecular insight can help a lot when conceiving of formulations, but these conceptions must ultimately be tested and validated experimentally.

Important Equations

This chapter has built the connection between the molecular perspective offered by the van der Waals model and semi-empirical estimates of activity coefficients for binary and multicomponent mixtures. The key equations in this extension are

Image
Image

where k12 is an adjustable parameter called the binary interaction parameter.

This extension results in several closely related activity models of varying complexity. The choice of the “best” model depends on the application and the individual’s comfort level with a particular degree of complexity. As a summary, the SSCED model provides a reasonable compromise.

Image
Image
Image

With this equation, the important roles of energy density (δ2a/V2), molecular size (Vk), and hydrogen bonding are all evident. Understanding these roles enables you to go beyond fitting data and make predictions about the behavior to be expected when various chemicals are combined. With practice, these predictions evolve to provide intuitive insight into formulations that achieve specific engineering objectives.

12.10. Practice Problems

P12.1. Acrolein + water exhibits an atmospheric (1 bar) azeotrope at 97.4 wt% acrolein and 52.4°C.

a. Determine the values of Aij for the van Laar equation that match this bubble-point pressure at the same liquid and vapor compositions and temperature. (ANS. 2.97, 2.21)

(You may use the shortcut vapor pressure equation for acrolein: Tc = 506 K; Pc = 51.6 bar; and ω = 0.330; MW = 56.)

b. Tabulate P at 326.55 K and x = {0.1,0.3,0.5} via the van Laar equation using the A12 and A21 determined above. (ANS. P = {1.11, 1.15, 1.03}

P12.2. The system α-epichlorohydrin(1) + n-propanol(2) exhibits an azeotrope at 760 mmHg and 96°C containing 16 mol% epichlorohydrin. Use the van Laar theory to estimate the composition of the vapor in equilibrium with a 90 mol% epichlorohydrin liquid solution at 96°C. (α-epichlorohydrin has the formula C3H5ClO, and IUPAC name 1-chloro-2,3-epoxypropane. Its vapor pressure can be approximated by: log10Psat = 8.0270–2007/T, where Psat is in mmHg and T is in Kelvin. You can use the shortcut vapor pressure equation for n-propanol.) (ANS. 0.72, 0.63 bar)

P12.3. The following free energy model has been suggested for a particularly unusual binary liquid-liquid mixture. Derive the expression for the activity coefficient of component 1,

Image

where, Image and Image.

(ANS. Image)

P12.4. The Scatchard-Hildebrand model can be extended to multicomponent mixtures in the following manner. Setting aij = (aiiajj)½ and aii/Vi = Viδ2 Eqn. 12.7 can be rewritten as

Image

Recognizing that the quadratic term is separable and simplifying the square-root of the square:

Image

where <δ> ≡ ΣxiViδi/V ≡ ΣΦiδi and V ≡ ΣxiVi.

This result can be made even simpler by adding and subtracting V<δ>2 and rearranging to obtain:

Image

where we have substituted the definition of <δ> in the second term and the definition of V in the third.

Collecting all terms in a common summation, we obtain:

Image

This is a remarkably simple result. Derive an equally simple expression for the activity coefficient of a component in a multicomponent mixture. (Hint: It is easier to start with Eqn. 12.65.) (ANS. Eqn. 12.54)

12.11. Homework Problems

12.1. The compositions of coexisting phases of ethanol(1) + toluene(2) at 55°C are x1 = 0.7186, and y1 = 0.7431 at P = 307.81 mmHg, as reported by Kretschmer and Wiebe, 1949. J. Amer. Chem. Soc., 71:1793. Estimate the bubble pressure at 55°C and x1 = 0.1, using

a. The Scatchard-Hildebrand model with k12 = 0

b. The SSCED model with a default value of k12

c. The SSCED model with k12 matched to the data

d. The van Laar equation

12.2. A vapor/liquid experiment for the carbon disulfide(1) + chloroform(2) system has provided the following data at 298 K: Image, Image, x1 = 0.2, y1 = 0.363, and P = 34.98 kPa. Estimate the dew pressure at 298 K and y1 = 0.6, using

a. The Scatchard-Hildebrand model with k12 = 0

b. The SSCED model with a default value of k12

c. The SSCED model with k12 matched to the data

d. The van Laar equation

12.3. The (1) + (2) system forms an azeotrope at x1 = 0.75 and 80°C. At 80°C, Image, Image. The liquid phase can be modeled by the van Laar model.

a. Estimate the activity coefficient of component 1 at x1 = 0.75 and 80°C. [Hint: The relative volatility (given in Eqn 10.32) is unity at the azeotropic condition.]

b. Qualitatively sketch the P-x-y and T-x-y diagrams that you expect.

12.4. Ethanol(1) + benzene(2) form azeotropic mixtures. Compare the specified model to the experimental data of Brown and Smith cited in problem 10.2.

a. Prepare a y-x and P-x-y diagram for the system at 45°C assuming the van Laar model and using the experimental pressure at xE = 0.415 to estimate A12 and A21.

b. Prepare a y-x and P-x-y diagram for the system at 45°C with the predictions of the Scatchard-Hildebrand theory with k12 = 0.

c. Prepare a y-x and P-x-y diagram for the system at 45°C assuming the SSCED model and using the standard guideline to estimate k12.

d. Prepare a y-x and P-x-y diagram for the system at 45°C assuming the SSCED model and using the experimental pressure at xE = 0.415 to estimate k12.

12.5. The CRC Handbook lists the azeotrope for the acetone + chloroform system as 64.7°C and 20 wt% acetone.

a. Use the van Laar model to estimate the T-x-y diagram at 1 bar.

b. Use the SSCED model to estimate the T-x-y diagram at 1 bar with predicted k12.

c. What value of V2/V1 is implied by the van Laar parameters?

12.6. Using the van Laar model and the data from problem 11.3, estimate the total pressure and composition of the vapor in equilibrium with a 20 mol% ethanol(1) solution in water(2) at 78.15°C.

12.7. A liquid mixture of 50 mol% chloroform(1) and 50% 1,4-dioxane(2) at 0.1013 MPa is metered into a flash drum through a valve. The mixture flashes into two phases inside the drum where the pressure and temperature are maintained at 24.95 kPa and 50°C. The compositions of the exiting phases are x1 = 0.36 and y1 = 0.62.

Your supervisor asks you to adjust the flash drum pressure so that the liquid phase is x1 = 0.4 at 50°C. He doesn’t provide any VLE data, and you are standing in the middle of the plant with only a calculator and pencil and paper, so you must estimate the new flash drum pressure. Fortunately, your supervisor has a phenomenal recall for numbers and tells you that the vapor pressures for the pure components at 50°C are Image and Image. What is your best estimate of the pressure adjustment that is necessary without using any additional information?

12.8. 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

a. The Scatchard-Hildebrand model with k12 = 0

b. The SSCED model with a default value of k12

c. The SSCED model with k12 matched to the data

d. The van Laar equation

e. Plot the P-x-y diagram at 80°C, based on the fits specified by your instructor.

12.9. Fit the data from problem 11.10 to the following model by regression over all points, and compare with the experimental data on the same plot, using

(a) – (d) as in problem 12.8.
(e) Plot the T-x-y diagram at 1 bar, based on the fits specified by your instructor.

12.10. Fit the data from problem 11.26 to the following model by regression over all points, and compare with the experimental data on the same plot, using

(a) – (e) as in problem 12.8.

12.11. Fit the data from problem 11.27 to the following model by regression over all points, and compare with the experimental data on the same plot, using

(a) – (e) as in problem 12.8.

12.12. Starting from the excess Gibbs energy formula for Flory’s equation, derive the formula for the activity coefficient of component 1 in a binary mixture.

12.13. Crime scene investigators have determined that an acrylic spray paint (polymethylmethacrylate, PMMA) was used to deface the Mona Lisa. Leonardo used linseed oil. We would like a solvent that interacts more strongly with acrylic than with linseed oil. Based on their chemical structures, we can approximate the SSCED parameters of linseed oil as n-hexadecane and acrylic paint as methylethylketone. Do you recommend CHCl3, toluene, or acetone as the solvent? Explain.

12.14. R410a is a replacement for R22 in air conditioners and heat pumps. Air conditioners require a different refrigerant because they operate in a different temperature range. R410a avoids the problems with the ozone layer caused by chlorofluorocarbons, but its longevity may be limited because it has a relatively high global warming factor (1725 times the effect of CO2). Roughly, it is a 50wt% mixture of difluoromethane (D) and pentafluoroethane (P) (i.e., 70mol% D). Kobayashi and Nishiumi (1998) report a pressure of 1.098 MPa at 283.05K.11 You may assume the SCVP equation

Image

a. Assuming kij = 0 for the binary interaction parameter of the SSCED equation, predict whether an azeotrope should be expected in this system at 283.05 K. Tabulate the relative volatilities at xD = 0.01 and 0.99.

b. Solve for the value of k12 that matches the reported pressure.

c. What acidity value for pentafluoroethane matches the value of kij determined in part (b)?

12.15. As part of a biorefining effort, butanediols are being produced by fermentation. The problem is that the isomers are all mixed up. Furthermore, 1,3-propanediol comprises roughly 30mol% of the mixture on a dry basis (i.e., water has been removed). The problem is to assess the prospects for azeotrope formation and avoidance. The following steps should shed some light on the problem.

a. Plot log10(Psat) versus 1000/T(K) in the range of T°C = [100,400] for 1,3-propanediol, 1,3-butanediol, and 1,4-butanediol on the same axes. Are there Bancroft points?

b. Compile a table of Image for each component in each solvent based on the SSCED model. Which combinations show the greatest tendency to form azeotropes?

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