Once the activity coefficient model’s parameters are known for a given system, the K-ratio can be calculated as a function of composition using Eqn. 11.1. For the one-parameter Margules equation, the activity coefficients are given by Eqn. 11.6. Then the bubble, dew, and flash routines can be executed from Table 10.1 on page 373. Because the activity coefficients depend on xi, the algorithms where xi is unknown require an initial guess to calculate a value for γi, and an iterative procedure to converge. Raoult’s law is often used for the initial guess for xi. Flow sheets for the methods are summarized in Appendix C, Section C.1. The method for bubble pressure does not require iteration because the activity coefficient depends on temperature and liquid composition and both are specified as inputs, as shown by Eqn. 11.2. This simple method is shown in Fig. 11.4.
Figure 11.4. Bubble-pressure method for modified Raoult’s law.
Let us use the algorithm for bubble pressure to determine the pressure and vapor phase compositions predicted by the one-parameter Margules equation at new compositions based on the fit of GE at the composition from Example 11.1. In fact, we can generate the entire diagram by repeating the bubble-pressure calculation across the composition range.
Example 11.2. VLE predictions from the Margules equation
Use the fit of Example 11.1 to predict the P-x-y diagram for isopropanol + water at 30°C. The data used for Fig. 9.5 from Udovenko et al. for 2-propanol(1) + water(2) at 30°C show x1 = 0.1168 and y1 = 0.5316 at P = 60.3 mmHg.
Solution
This is a Stage III problem, since the first two stages have been completed earlier. Let us start by generating activity coefficients at the same composition where experimental data are provided, x1 = 0.1168; we find
Note that these activity coefficients differ substantially from those calculated in Example 11.1 because the liquid composition is different. We always recalculate the activity coefficients when new values of liquid composition are encountered.
Substituting into modified Raoult’s law to perform a bubble-pressure calculation:
The total pressure is found by summing the partial pressures,
P = y1P + y2P = 50.4 mmHg
We manipulate modified Raoult’s law as shown in step 3 of Fig. 11.4:
y1 = y1P/P = 21.48/50.4 = 0.426
Therefore, compared to the experimental data, the model underestimates the pressure and the vapor composition of y1 is too low, but the use of one measurement and one parameter is a great improvement over Raoult’s law. The estimation can be compared with the data by repeating the bubble-pressure calculation at selected xi values across the composition range; the results are shown in Fig. 11.5. Recall that in Fig. 11.3 we noted that the excess Gibbs energy model using A12 = 1.42 fails to capture the skewness of the excess Gibbs energy curve. The deficiency is evident in the P-x-y diagram also. Fig. 11.5 includes a two-parameter fit that will be discussed later.
Figure 11.5. (a) One-parameter and two-parameter Margules equation fitted to a single measurement in Examples 11.2 and 11.5 compared with the experimental data points from Fig. 10.8 on page 395. Data are tabulated in Example 11.8. (b) Activity coefficients predicted from the parameters fitted in Example 11.5 compared with points calculated from the data.
This example has demonstrated that a single experiment can be leveraged to generate an entire P-x-y diagram with a greatly improved representation of the system. There is an even better method to use a single experiment with a two-parameter model, but we can explain that later. Let us look at one more example using the one-parameter model, but let us integrate the fitting of the excess Gibbs energy (Stage II) simultaneously with the bubble-pressure calculation (Stage III).
Example 11.3. Gibbs excess characterization by matching the bubble point
The 2-propanol (1) + water (2) system is known to form an azeotrope at 760 mmHg and 80.37°C (x1 = 0.6854). Estimate the Margules parameter by fitting the bubble pressure at this composition. Then compare your result to the Raoult’s law approximation and to the data in Fig. 10.8(c) (at 30°C), where P = 66.9 mmHg at x1 = 0.6369 as used in Example 11.1.
Solution
The Antoine coefficients for 2-propanol and water are given in Appendix E. At T = 80.37°C, 
, and 
. We seek P = 760 mmHg. Let us use trial and error at the azeotropic composition to fit A12 to match the bubble pressure.
At A12 = 1, γ1 = exp[1(1 – 0.6854)2] = 1.104; γ2 = exp[1(1 – 0.3146)2] = 1.600; the bubble pressure is by Eqn. 11.2
P = 0.6854(694.)1.104 + 0.3146(359.9)1.600 = 706.3 mmHg
The pressure is too low. We need larger activity coefficients, so A12 must be increased. Typing the bubble-pressure formula into Excel or MATLAB (see file Ex11_03.m), we can adjust A12 until P = 760 mmHg.
Ex11_03.m.at A12 = 1.368, γ1 = exp[1.368(1 – 0.6854)2] = 1.145; γ2 = exp[1.368(1 – 0.3146)2] = 1.902; the bubble pressure is
P = 0.6854(694.)1.145 + 0.3146(359.9)1.902 = 760.0 mmHg
Now, for the second part of the problem, to apply this at T = 30°C, 
, 
. When x1 = 0.6369 the ideal solution gives,
P = 0.6369(58.28) + 0.3631(31.74) = 48.64 mmHg
At A12 = 1.368, γ1 = exp[1.368(1 – 0.6369)2] = 1.1976;
γ2 = exp[1.368(1 – 0.3631)2] = 1.7418; the bubble pressure is
P = 0.6369(58.28)1.1976 + 0.3631(31.74)1.7418 = 64.53 mmHg
Comparing, we see that the Raoult’s Law approximation, P = 48.6 mmHg, deviates by 27% whereas the Margules model deviates by only 3.5%. Furthermore, the Margules model indicates an azeotrope because 
means that there is a pressure maximum. Hence the Margules model “predicts” an azeotrope at this lower temperature, qualitatively consistent with Fig. 10.8(c), whereas the ideal solution model completely misses this important behavior.
x-y Plots
The “x-y” plot introduced in Fig. 10.4 can be prepared for the azeotropic system of Fig. 11.5 by plotting the pairs of y-x data/calculations at each pressure or temperature. Such a plot is shown in Fig. 11.6. The curve represents the two-parameter fit that is shown in Fig. 11.5. Note when an azeotrope exists that the y-x curve crosses the diagonal at the azeotropic composition.
Figure 11.6. Data and the two-parameter fit of Fig. 11.5 plotted as pairs of x and y. Both T-x-y and P-x-y data can be plotted in this way.
Looking Ahead
Careful readers may notice that A12 = 1.42 from Example 11.1 and A12 = 1.37 from Example 11.3. The compositions were slightly different. We also noted in Example 11.1 when we peeked at additional data that the single parameter model was insufficient to represent the system all the way across the composition range, so this was also a factor in the difference. There is also a another possibility for fitting the activity model that we did not consider. After determining the activity coefficients from Eqn. 11.3, we could have used the values directly in the model Eqn. 11.6. This method was not used because the solution is overspecified with two equations and one A12 parameter value that would have been different for each. We could have calculated the two values and averaged them, but we chose instead to use the excess Gibbs energy or the bubble pressure directly—methods that used thermodynamic properties directly. We can see that improved models are desirable.
Clearly, the one-parameter Margules model has limitations, but it sets us on a path of continuing improvement that is fundamental to engineering: Observe, predict, test, evaluate, and improve. Observations for ideal solutions suggested a crude model in Raoult’s law. When predictions with Raoult’s law were tested for a broader range of mixtures, however, we observed deficiencies. Evaluating the model, a correction factor was suggested that conformed to physical constraints like γi(xi) = 1 when xi = 1. Then a slightly more sophisticated model equation was suggested, the one-parameter Margules model. We followed through on several of the implications of this model (e.g., Eqns. 11.4 and 11.6) and arrived at new predictions. We tested those predictions and found improvements, but still deficiencies. Now we are ready to begin a new round of evaluation. Each successive round of evaluation requires deeper insight into the physical constraints, ultimately leading to careful consideration of the interactions at the molecular scale.
The activity coefficient models that we discuss in upcoming sections enable a broad range of engineering analyses. For example, we may wish to design a distillation column that operates at constant pressure and requires T-x-y data. However, the available VLE data may exist only as constant temperature P-x-y data. We may use the activity coefficient models to convert isothermal P-x-y data to isobaric T-x-y data, and vice versa. Furthermore, parameters from binary data can be combined and extended to multicomponent systems, even if no multicomponent data are available. Elementary techniques for fitting GE (or activity coefficient) models are presented to fit single data points. Advanced techniques for fitting GE or VLE data across the composition range are presented in specific examples and Section 11.9. In the next few pages, we fill in some of the theoretical development that we have skipped in our overview.
Preliminary Predictions Based on a Molecular Perspective
Ultimately, we would like to make predictions that go beyond fitting a single data point for a single binary mixture. We would like to design formulations to solve practical problems. For example, suppose somebody had sprayed graffiti on the Mona Lisa. Could we formulate a solvent that would remove the spray paint while leaving the original painting intact? What about an oil spill in the Gulf of Mexico? What kind of treatment could disperse it best? What kind of molecule could be added to break the azeotrope in ethanol + water? What formulation could promote the permeation of insulin through the walls of the small intestine? These may sound like very different problems, but they are all very similar to a thermodynamicist. To formulate a compatible solvent, we simply need to minimize the activity coefficient. For example, we should seek a solvent that has a low activity coefficient with polymethylmethacrylate (PMMA, a likely graffiti paint) and a high activity coefficient with linseed oil (the base of oil paint). We could imagine randomly testing many solvents, but then we might hope to observe patterns that would lead to predictions. These would be predictions of a higher order than simply extrapolating to a different temperature or composition, but they would enable us to contemplate the solutions to much bigger problems. You already possess sufficient molecular insight to begin this process. Elucidating that will simultaneously make these problems seem less daunting and help us on the way to more sophisticated model evaluation.
You know that acids and bases interact favorably. An obvious example would be mixing baking soda and vinegar which react. You could also mix acid into water. These interactions are “favorable” because they release energy, meaning they are exothermic. They release energy because their interaction together is stronger than their self-interactions with their own species. A subtler exothermic example is hydrogen bonding, familiar perhaps from discussions of DNA, where the molecules do not react, but form exothermic hydrogen bonds. Unlike a covalent bond, the hydrogen sits in a minimum energy position between the donor and acceptor sites. The proton of a hydroxyl (-OH) group is acidic while an amide or carbonyl group acts as a base. We can extend this concept and assign qualitative numerical values characterizing the acidity and basicity of many molecules as suggested by Kamlet et al.3 These are the acidity parameter, α, and basicity parameter, β, values listed on the back flap. For example, this simple perspective suggests that chloroform (α > 0) might make a good solvent for PMMA (a polymer with a molecular structure similar to methyl ethyl ketone, β > 0) because the α and β values should lead to favorable interactions. This is the perspective suggested by Fig. 11.7(a).
Figure 11.7. Observations about complexation. (a) A mixture of acid with base suggests favorable interactions, as in acetone + chloroform. (b) Hydrogen bonding leads to unfavorable interactions when one component associates strongly and the other is inert, as in isooctane + water. (c) Hydrogen bonding solutions can also be ideal solutions if both components have similar acidity and basicity, as in methanol + ethanol.
Hydrogen bonding may sound familiar, but there are subtleties that lead to complex behavior. These subtleties are largely related to the simultaneously acidic and basic behavior of hydroxyl species. We know that water contains -OH functionality, but its strong interaction with acids also indicates a basic character. It is both acidic and basic. The subtlety arises when we consider that its acidic and basic interactions link together when it exists as a pure fluid. Then a question arises about how water might interact with an “inert” molecule that is neither acidic nor basic, as illustrated in Fig. 11.7(b). Clearly, the water would squeeze the inert molecule out, so it could maximize its acid-base interactions. Referring to the back flap, α = β = 0 for molecules like octane-hexade-cane, and these components have molecular structures similar to oil. Thus, we see that this acid-base perspective correlates with the old guideline that oil and water do not mix. Oils are said to be “water-fearing,” or hydrophobic. Finally, Fig. 11.7(c) illustrates what happens when two molecules have similar acidity and basicity, like methanol and ethanol. Then they can substitute for each other in the hydrogen bonding network and result in a solution that is nearly ideal. Molecules like alcohols are called hydrophilic (“water-liking”).
This perspective is not a large leap from familiar concepts of acids, bases, and hydrogen bonding, but it does provide more insight than guidelines such as “like dissolves like” or “polarity leads to nonideality.” Acids are not exactly “like” bases, but they do interact favorably. Methanol and ethanol are both polar, but they can form ideal solutions with each other.
We can go a step further by formulating numerical predictions using what we refer to as the Margules acid-base (MAB) model. The model provides first-order approximations. The model is:
Margules acid-base (MAB) model.
where 
is the liquid molar volume at 298.15K in cm3/mol. The MAB model is introduced here for pedagogical purposes. MAB is a simplification of SSCED4 which is in turn a simplified adaptation of MOSCED5, both of which are covered in Chapter 12. Typical values of V, α, and β are presented in Table 11.1. For example, with chloroform + acetone at 60°C, this formula gives
Table 11.1. Acidity (α) and Basicity (β) Parameters in (J/cm3)1/2 and Molar Volumes (cm3/mol) for Various Substances as liquids at 298 Ka
a. Additional parameters are on the back flap.
Note how the order of subtraction results in a negative value for A12 when one of the components is acidic and the other is basic. If you switched the subscript assignments, then Δα would be negative and Δβ would be positive, but A12 would still be negative. This negative value makes the value of γi smaller, and that is basically what happens when hydrogen bonding is favorable. Something else happens when one compound forms hydrogen bonds but the other is inert. Taking isooctane(1) as representative of oil (or gasoline) and mixing it with water(2) at 25°C,
This large positive value results in γ1 > 7.5 for the isooctane. We can use γi > 7.5 to suggest a liquid phase split, as we should expect from the familiar guideline that oil and water do not mix. Furthermore, we can quantify the solubilities of the components in each other (aka. mutual solubilities) by noting that xi ≈ 1/γi when γi >100. Knowing the saturation limit of water contaminants can be useful in environmental applications. As a final example, note that we recover an ideal solution when both components hydrogen bond similarly, as in the case of ethanol + methanol at 70°C.
In this case, we see that hydrogen bonding by itself is not the cause of solution non-ideality. A mismatch of hydrogen bonding is required to create non-idealities.
Example 11.4. Predicting the Margules parameter with the MAB model
Predict the A12 value of the 2-propanol (1) + water (2) system using the MAB model at 30°C. Then compare your result to those of Examples 11.1 and 11.3.
Solution
From Eqn. 11.9, A12 = (50.13 – 9.23)(15.06 – 11.86)(76.8 + 18.0)/[4(8.314)303] = 1.08. This compares to the value A12/RT = 1.42 from Example 11.1 and A12/RT = 1.37 from Example 11.3 at 30°C. The MAB model does not provide a precise prediction, but qualitatively indicates a positive deviation of the right magnitude.
With this perspective we can begin to contemplate formulations of very broad problems, but this is only the beginning. We will see in Section that the MAB model overlooks an important contribution to the activity coefficient, even in the context of the relatively simple van der Waals perspective. In Section 13.1 we show limitations of the van der Waals perspective. Finally, Chapter 19 shows that accounting for hydrogen bonding as a chemical reaction results in a description of the Gibbs excess energy that is quite different from the perspectives in Chapters 11 to 13. All of these presentations focus primarily on relatively small molecules with single functionalities like alkyl, hydroxyl, or amide. Modern materials (including biomembranes and proteins) are composed of large molecules with deliberate arrangements of the functionalities resulting in self-assembly to perform remarkably diverse macroscopic functions.
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