Chapter 11. An Introduction to Activity Models

First, do no harm.


The subject of non-ideal solutions includes just about everything from aqueous acids to polymers to semiconductors. Not surprisingly, there is no completely general model for non-ideal solutions. But there are several popular approaches for specific situations like VLE of alcohols or LLE of organic solvents with water. We discuss typical models and briefly explain their forms and history. Moreover, the challenge of developing accurate descriptions of non-ideal solution behavior means that model development is still an active research area. The presentation here should provide enough background to understand the rationales behind new developments as well as the old.

Chapters 11 through 13 are concerned with correction factors to Raoult’s law known as “activity coefficients.” The difference is that Chapter 11 is concerned with purely empirical models of binary mixtures while Chapter 12 focuses on model equations that can be derived from the van der Waals equation of state. These models are not as complex as those in Chapter 13, but they convey the key concepts and serve as an introduction. The models in Chapter 13 reformulate the analysis in terms of the radial distribution function for mixtures. Students need to recognize that Raoult’s law can lead to gross errors, failing to represent an azeotrope or liquid-liquid equilibrium (LLE), for example. With this in mind, we take a hierarchical approach. We begin with a simple illustration using the Margules 1-parameter model. This suffices to account crudely for mixtures of components that “like” or “dislike” each other. Later, we demonstrate that this simple model is related to a generalized empirical form called the Redlich-Kister expansion. To proceed beyond empirical fitting and begin understanding the molecular driving forces for nonideality, Chapter 12 returns to the van der Waals model and extends it to multicomponent mixtures. With molecular understanding, we can develop more accurate and general models, permitting us to formulate solutions that accomplish broad design goals, in addition to fitting and extrapolating specific data for specific systems.

Chapter Objectives: You Should Be Able to...

1. Compute VLE phase diagrams using modified Raoult’s law. Perform bubble, dew, and flash calculations using modified Raoult’s law.

2. Characterize adjustable parameters in activity models using experimental data.

3. Derive an expression for an activity coefficient given an arbitrary expression for the Gibbs excess energy.

4. Assess the degree of non-ideality of a given mixture based on the molecular properties of the components.

5. Comment critically on the merits and limitations of the following solution models: Henry’s law, Margules models, and the Redlich-Kister expansion, including the ability to identify the most appropriate model for a given application.

6. Apply Henry’s law to estimate fugacities of dilute components.

7. Explain osmotic pressure and compute its value and implications for biological systems.

11.1. Modified Raoult’s Law and Excess Gibbs Energy

In Section 10.7 we demonstrated that deviations from Raoult’s law were manifested by changes in the bubble line and thus characterized positive and negative deviations. With a purely mathematical perspective for modeling the behavior, we could develop a “correction” to Raoult’s law as illustrated in Fig 11.1.


Figure 11.1. P-x-y diagram for isopropanol water at 30°C illustrating the rationale for activity coefficients using modified Raoult’s law. Data from Udovenko, V.V., and Mazanko, T.F. 1967. Zh. Fiz. Khim. 41:1615.

Note that the bubble line for Raoult’s law misses the shape completely. The bubble pressure formula for Raoult’s law is linear in x1 because x2 = 1 – x1 and the Psat values are constants with respect to x1. Suppose that we were to apply a “correction” to VLE K-ratio for each component and attribute the deviations to the liquid phase, where the molecules are closely packed and molecular geometry and intermolecular potentials are much more important than in the vapor phase. The resultant method provides


The relation to the bubble line is not yet obvious. Cross-multiplying and summing gives,


Image Bubble pressure calculation.

Now we can begin to see how this approach adjusts the model to the bubble pressure. When a system has positive deviations from Raoult’s law, the bubble line lies above the Raoult’s law bubble line Image, therefore γi > 1. When a system has negative deviations, γi < 1. This correction factor γi, is referred to as the activity coefficient. Therefore, P-x-y data are related to the deviations of the activity coefficients from unity, or it may be helpful to consider the sign of ln γi.1 Look back at Fig. 11.1 and the figures in Section 10.7 and note that the deviations from Raoult’s law disappear as pure compositions are approached. This means that the deviations depend on composition, and that the γi(x) that we have introduced in Eqn. 11.1 must go to 1 as the solution becomes pure in the ith component.

As you might imagine, there are rules that we should follow to develop feasible functions. We discuss those theoretical aspects in the upcoming sections. The accepted method of modeling the system is to build a model for γi(x) based in the excess Gibbs energy. The excess Gibbs energy is also discussed in upcoming sections, but briefly it is the Gibbs energy in “excess” of an ideal solution. It is generally positive when γi > 1 and negative when γi < 1. The model for excess Gibbs energy can be developed from a theoretical model of mixing behavior, and usually contains adjustable parameters to adjust the magnitude and skewness of the “excess” and thus fit the experiment.

Activity coefficients may be determined from experimental measurements by rearranging modified Raoult’s law as implied by Eqn. 11.1:


From the values of the activity coefficients, a value for excess Gibbs energy can be calculated:


These values may be tabulated to provide GE/RT vs. x1 and the resultant curve can be regressed to fit a reasonable analytical expression for convenient interpolation at all compositions. Empirically, it is often preferable to fit the γi’s directly. We demonstrate both approaches.

The three major stages of working with activity coefficients are shown in Fig. 11.2. In Stage I, the activity coefficients are determined at various compositions from the experiments. In Stage II, a model is selected and various techniques can be used to fit the model to the experimental data. Finally, in Stage III, the model is utilized to solve many different types of phase equilibria problems. Initially, we work some examples where Stage II is separate from Stage III. In advanced fitting, Stages II and III are combined. In addition to development of these skills, much of the chapter is devoted to demonstrating the theoretical foundations for the models so that you are familiar with some of the popular models available in process design software. We integrate examples of bubble, dew, and flash calculations throughout the discussions.


Figure 11.2. Strategy for using excess Gibbs energy models for activity coefficients. Experiments are used to determine γi in Stage I. A model is selected and the model is fitted in Stage II. The model is utilized to extend and extrapolate the experimental results in Stage III.

The One-Parameter Margules Equation

The simplest expression for the Gibbs excess function is the one-parameter Margules equation (also known as the two-suffix Margules equation). For a binary system,


Image The one-parameter Margules equation is the simplest excess Gibbs expression.

Note: The parameter A12 is a constant which is not associated with the other uses of the variable (equation of state parameters, Helmholtz energy, Antoine coefficients) in the text. The parameter A12 is typically used in discussions of the Margules equation, so we use it here.2

The one-parameter Margules equation is symmetrical with composition. It has an extremum at x1 = 0.5 in a binary system and becomes zero at purity of either component. The activity coefficients in a binary system for the one-parameter Margules equation are (derived later as Eqns. 11.2911.31),


Image Margules one-parameter model for a binary mixture.

Let us fit the model to experimental data to demonstrate Stage I and Stage II procedures.

Image gammaModels/Marg1P.m.

Example 11.1. Gibbs excess energy for system 2-propanol + water

Using data from the 2-propanol(1) + water(2) system presented in Fig. 10.8 calculate the excess Gibbs energy at x1 = 0.6369 and fit the one-parameter Margules equation. Data from the original citation provide T = 30°C, Image, Image, and y1 = 0.6462 when x1 = 0.6369 at P = 66.9 mmHg.


The approach is to determine the activity coefficients and then relate them to the excess Gibbs energy. The Stage I step is


If we were given more experimental data, we could repeat the calculation for each data point, thus creating a plot of GE versus x1 like the points shown in Fig. 11.3.


Figure 11.3. Illustration of calculation of GE from experiment and fitting of Margules models to a single point as discussed in Examples 11.1 and 11.5, for 2-propanol + water, with the experimental data points from Fig. 10.8 on page 395. Data are tabulated in Example 11.8. The van Laar model fit to a single point is explained in Section 12.2.

Then, we have been instructed to use the one-parameter Margules model for Stage II. Let us fit the model as given by Eqn. 11.5 and using the value from Eqn. 11.7.


The curve of GE versus x1 is shown in Fig. 11.3 along with the two-parameter models to be discussed in Section 11.6 and Example 11.5.

Note from the example that the single point fit gives only an approximate representation of the excess Gibbs energy (because we compared to some additional data that were not included in the problem statement). However, let us proceed to see how this one data point can be leveraged to study the phase behavior.

11.2. Calculations Using Activity Coefficients

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.


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.

Image Bubble-pressure calculation.

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.

Image Bubble-pressure calculation.


The Antoine coefficients for 2-propanol and water are given in Appendix E. At T = 80.37°C, Image, and Image. 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.

Image 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, Image, Image. 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 Image 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:


Image Margules acid-base (MAB) model.

where Image 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.


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.

11.3. Deriving Modified Raoult’s Law

In Chapter 10 we demonstrated that Raoult’s law requires an ideal solution model for the vapor and liquid phases as well as conditions where the fugacity coefficients can be ignored. In Section 11.1 we have shown that a relatively simple function is able to capture a major correction to Raoult’s law, but we have superficially made the connections to fundamental properties and we must now develop that understanding of how this function is related to component fugacity and Gibbs energy of the mixture.

To perform VLE calculations, we begin with the fundamental criterion Image. The fugacity in non-ideal systems is modeled in terms of deviations from either the ideal gas model or the ideal solution model. The Venn diagram in Fig. 11.8 may be helpful in visualizing these relations. Ideal gas behavior is the simplest type of mixture behavior because the particles are non-interacting. This is shown in the center of Fig. 11.8. Clearly, ideal gas behavior is not followed by all mixtures, and therefore ideal gases are a subset of real mixtures. Strictly, ideal gas molecules cannot condense because they have no attractive forces; if fluids were ideal gases, there would be no liquids, and VLE would not occur. However, at low densities, gas molecules are frequently separated far enough that the effective intermolecular potentials are insignificant, and we can frequently model the gas phase as if it is an ideal gas. The fugacity of a component in an ideal-gas mixture is particularly simple; it is equal to yiP, the partial pressure. The ideal gas model is acceptable for most small molecular weight vapors near atmospheric pressure;6 and the primary failure of Raoult’s law is due to the liquid phase where the molecules are closely packed. When the molecules are closely packed in a liquid phase the intermolecular potential is significant for the same molecules whose potential energies could be disregarded in the vapor phase. The size and shape of molecules and differences in chemical functionality at close distances often violate the assumptions of ideal solution behavior. We introduce the concept of the activity and activity coefficient, γi, to correct the ideal mixture model as summarized in Fig. 11.8.


Figure 11.8. Schematic of the relations between different fluid models. Ideal gases are a subset of ideal mixtures, which in turn are a subset of real mixtures. Departure functions (fugacity coefficients) characterize deviations from ideal-gas behavior, and excess properties (activity coefficients) characterize deviations from ideal-solution behavior.

The activity coefficient is defined as a ratio of the component fugacity to the ideal solution fugacity at the same mole fraction:


Image Activity coefficients are commonly used for highly non-ideal solutions.

A value of γi = 1 will denote an ideal solution; Image is the value of the fugacity at standard state. A standard state is slightly different from a reference state. The standard state is a specified temperature, pressure, and composition. The most common standard state in solution thermodynamics is the pure component at the same temperature pressure as the system. You can see that this is clearly different from a reference state that would stay fixed throughout a series of calculations. For this standard state, standard state fugacity is simply the pure component fugacity which we introduced in Chapter 9, fi. In Fig. 11.8 we also introduce the fugacity coefficient Image for a component in a mixture to characterize the component fugacity relative to the ideal gas partial pressure. We defer most discussion of the component fugacity coefficients to Chapters 15 and 19.

Now, let us look rigorously at the development of modified Raoult’s law. For the vapor phase, we begin with the rigorous expression from Fig. 11.8 including deviations from the ideal gas model, Image. For the liquid phase, we use an activity coefficient, γi, giving7


Typically the Poynting method (Section 9.8) is used to calculate the pure-component liquid phase fugacities, Image. Combining these expressions,


When used in this full form, Eqn. 11.15 is called the gamma-phi method. This may be written in terms of the K-ratio, Ki = yi/xi,


At the low pressures of many chemical engineering processes the Poynting corrections and the ratios of Image for the components approach unity. Recalling Image, we find


We usually write

Image Modified Raoult’s law.


We can see that modified Raoult’s law depends on the fugacity coefficient ratio being close to unity, not necessarily on the ideal gas law being exact. Next, let us demonstrate how the activity coefficient is related to the excess Gibbs energy.

Image Modified Raoult’s law.

11.4. Excess Properties

The deviation of a property from its ideal-solution value is called the excess property. For a generic property M, the excess property is given the symbol ME, and ME is the value of the property for the mixture relative to the property for an ideal mixture, ME = M – Mis. Ideal solutions were discussed in Section 10.10. The molar volume of an ideal solution is just the weighted sum of the molar volumes of the components, Image. The excess volume is then,

Image Excess volume.


Although the excess volumes of liquids are typically a very small percentage of the volume, the concepts of excess properties are easily grasped by first studying the excess volume and then exploring the more abstract quantities of excess enthalpy, entropy, or Gibbs energy.

The excess volume of the system 3-pentanone (1) + 1-chlorooctane (2) at 298.15 K has been measured by Lorenzana, et al.,8 and is shown in Fig. 11.9. The molar volumes of the pure components are V1 = 106.44 cm3/mol and V2 = 171.15 cm3/mol. At the equimolar concentration, the excess volume is 0.204 cm3/mol. Therefore, the molar volume is V = VE + Vis = 0.204 + 0.5 · 106.44 + 0.5 · 171.15 = 139.00 cm3/mol. The excess volume is only 0.15% of the total volume. The partial molar excess volume is calculated in a manner analogous to the partial molar volume, Image. If an algebraic expression is available for the excess volume, it may be differentiated by this relation to yield formulas for the excess volumes. Graphically, the partial molar volumes at any point may be found by drawing the tangent line to the excess volume curve and reading the intercepts. At the composition shown at the tangent point in Fig. 11.9, the intercepts give Image and Image. The partial molar volumes depend on composition.


Figure 11.9. Excess volume for the 3-pentanone (1) + 1-chlorooctane (2) system at 298.15 K.

The excess enthalpy is very similar to the excess volume,


A solution with HE > 0 has an endothermic heat of mixing, and when HE < 0, the heat of mixing is exothermic. In an adiabatic mixing process an endothermic mixing process will cool and an exothermic mixing process will heat.

In directly analogous fashion, the excess Gibbs energy can be defined as the difference between the Gibbs energy of the mixture and the Gibbs energy of an ideal solution, GE = G – Gis. Then, instead of speaking of partial molar volumes, we speak of partial molar Gibbs energies. But you should recognize the partial molar Gibbs energy as the chemical potential as introduced in Section 10.8, and the significance of the chemical potential to phase equilibrium calculations should resonate strongly after reading Chapter 10. What remains is to rearrange the mathematics into the final relations to show how this property is related to the activity coefficients.

11.5. Modified Raoult’s Law and Excess Gibbs Energy

The excess Gibbs energy is


Image Excess Gibbs energy.

where we have added and subtracted the sum of the component Gibbs energies in the second line, and used Eqns. 10.65 and 10.66 in the last line. Let us further examine Image.

Recall that Image, (Eqn. 10.42). Previously in Eqn. 10.48 we expressed μi relative to a pure component value using a ratio of fugacities. Generalization of this equation will provide the link between Gibbs energies and component fugacities that will lead to the activity coefficient models. If we calculate the chemical potential relative to a standard state, we find by generalizing 10.48


This ratio of fugacities will appear often, and it is convenient to define the ratio as the activity, and it can be related to composition using the activity coefficient,


Image Activity.

Using a standard state at the T and P of the system (cf. page 425), Image, we can develop an expression for ΔGmix,


Substituting into Eqn. 11.21:


Image Excess Gibbs energy is zero for an ideal solution, and activity coefficients are unity.

Note that the activity coefficients and excess Gibbs energy are coupled—when the activity coefficients of all components are unity, the excess Gibbs energy goes to zero. The excess Gibbs energy is zero for an ideal solution.

Activity Coefficients as Derivatives

Activity coefficients are related to derivatives of the excess Gibbs energy, specifically the partial molar excess Gibbs energy. We have a very simple relation between partial molar quantities and molar quantities developed in Eqn. 10.41,


Applying this relation to excess Gibbs energy,


Comparing this with Eqn. 11.27, we see that


Image Activity coefficients are related to the partial molar excess Gibbs energy.

where we have recognized that the partial molar excess Gibbs energy of a component is also the component excess chemical potential. So, for any expression of GE (T,P,x), we can derive γ’s.

For the one-parameter Margules,


Applying Eqn. 11.28 for i = 1, and using the product rule on n1/n = n1(1/n),


11.6. Redlich-Kister and the Two-Parameter Margules Models

We noted in Example 11.1 a shortcoming in the one-parameter model’s representation of the skewness of experimental excess Gibbs energy. In principle, adjusting both the magnitude and skewness of GE is possible with a two-parameter model equation. The mathematical relations in Sections 11.311.5 liberate us to conjecture freely about forms of GE that may fit any given set of VLE data. Based on any model, developing working equations for activity coefficients is a straightforward matter of taking derivatives. These are the considerations behind many empirical models like the Redlich-Kister and Margules two-parameter models.

Many models can be rewritten in the Redlich-Kister form given as9


Two-Parameter Margules Model

The two-parameter Margules model is a simplification of the Redlich-Kister,


where we relate the constants to the Redlich-Kister via A21 = B12 + C12, A21 = B12C12, and D12 = 0. The constants A21 and A12 are fitted to experiment as we show below. Note that if A21 = A12, the expression reduces to the one-parameter model. The expression for the activity coefficient of the first component can be derived as


Applying Eqn. 11.28 for i = 1, and using the product rule on n1/n = n1(1/n) and n2/n = n2(1/n),


Image Actcoeff.xlsx, sheet Margules; gammaModels/Marg2P.m.

The two parameters can be fitted to a single VLE measurement using


where the activity coefficients are calculated from the VLE data. Care must be used before accepting the values from Eqn. 11.38 applied to a single measurement because experimental errors can occasionally result in questionable parameter values.

Example 11.5. Fitting one measurement with the two-parameter Margules equation

We mentioned following Example 11.2 that a single experiment could be used more effectively with the two-parameter model. Apply Eqn. 11.38 to the two activity coefficients values calculated in Example 11.1 and estimate the two parameters. This is an example of a Stage II calculation.


From Example 11.1, x1 = 0.6369, x2 = 0.3631, and γ1 = 1.118, γ2 = 2.031. From Eqn. 11.38,


The parameters from Example 11.5 provide the representation of GE shown in Fig. 11.3. Using the concepts from earlier examples along with Eqn. 11.37 for the activity coefficients, bubble-pressure calculations across the composition range (Stage III calculations) result in the curve of Fig. 11.5 designated as the two-parameter model.

Example 11.6. Dew pressure using the two-parameter Margules equation

Use the parameters of Example 11.5 to predict the dew-point pressure and liquid composition for the 2-propanol(1) + water(2) system at T = 30°C, y1 = 0.4, and compare with Fig. 11.5. Use the vapor pressures, Image, Image.


We will apply the procedure in Appendix C and refer to step numbers there.

Image Dew-pressure calculation.

Step 1. Refer to Chapter 10, P = 1/(0.4/60.7 + 0.6/32.1) = 39.55 mmHg, x1 = 0.4(39.55)/60.7 = 0.26. We skip Step 2 the first time.

Step 3. Using parameters from Example 11.5 in Eqn. 11.37, γ1 = exp(0.742(1.99 – 1.8(0.26)))=2.30; γ2 = exp(0.262(1.09 + 1.8(0.74)))=1.18.

Step 4. P = 1/(0.4/(2.3·60.7) + 0.6/(1.18·32.1)) = 53.46 mmHg.

Note the jump in P compared to Step 1 for the first loop.

Step 5. x1 = 0.4(53.46)/(2.3·60.7) = 0.153. Continuing the loop:


Continuing for several more iterations with four digits, P = 50.63 mmHg, and x1 = 0.0649. The calculations agree favorably with Fig. 11.5. The dew calculations are consistent with a bubble calculation at x1 = 0.0649.

Because this is a long chapter, we summarize the relations between the activity coefficient models developed thus far in Table 11.2.

Table 11.2. Summary of Empirical Activity Models and Simplifications Relative to the Redlich-Kister


11.7. Activity Coefficients at Special Compositions

Two parameter models provide sufficient flexibility with a balance of relative simplicity to provide successful VLE modeling. Determination of activity for each component permits two parameters to be fitted, and special compositions can be used.


The location of an azeotrope is very important for distillation design because it represents a point at which further purification in a single distillation column is impossible. Look back at Fig. 11.1 on page 412. Looking at dilute isopropanol concentrations, note x2-propanol = 0.01 < y2-propanol, but near purity, x2-propanol = 0.99 > y2-propanol. The relative magnitudes have crossed and thus we expect y2-propanol = x2-propanol (i.e., there is an azeotrope) somewhere in between. If the relative sizes are the same at both ends of the composition range, then we expect that an azeotrope does not exist.11 Certainly, the best way to identify an azeotrope is to plot T-x-y or P-x-y, but a quick calculation at each end of the diagram is usually sufficient.

Image A simple algorithm to decide if an azeotrope exists.

Note that the relative volatility introduced in Section 10.6 on page 390 also changes significantly in an azeotropic system. For the reasons above, αij > 1 on one side of the azeotrope, αij = 1 at the azeotrope, and αij < 1 on the other side. Because shortcut distillation calculations fail at αij = 1, they must not be used if (αij – 1) changes sign between column ends. This means that screening systems for azeotropes such as using the algorithm above is important before blindly plugging numbers into shortcut distillation calculations.

Image Relative volatility crosses 1 at an azeotrope.

Image Any deviation from ideality will create an azeotrope at a Bancroft point.

We noted in Section 10.7 on page 393 that azeotropic behavior was dependent on the magnitude of deviations from ideality and the vapor pressure ratio. Look back at Fig. 11.1 on page 412 and recall that deviations from Raoult’s law create the curve in the bubble line. When the pure component vapor pressures are nearly the same then a slight curve due to non-ideality can cause an azeotrope. The same size deviation in a system with widely different vapor pressure may not have an azeotrope. A plot of logPsat versus 1/T with both components may show a point where the two curves cross when the heats of vaporization are different. This point is called a Bancroft point. Since the vapor pressures are exactly equal at the Bancroft point, any small non-ideality generates an azeotrope. This might be avoidable if the system pressure is raised or lowered to circumvent the Bancroft point in the temperature range of a distillation column.

Many tables of known azeotropes are commonly available.12 For systems with an azeotrope, the azeotropic pressure and composition provide a useful data point for fitting activity coefficient models because x1 = y1. Then Image; Image. Then the typical single point fitting formulas are used with the azeotrope composition to find the model parameters.

Image The azeotrope is a useful point to fit parameters.

Example 11.7. Azeotrope fitting with bubble-temperature calculations

Consider the benzene(1) + ethanol(2) system which exhibits an azeotrope at 760 mmHg and 68.24°C containing 44.8 mole% ethanol. Using the two-parameter Margules model, calculate the composition of the vapor in equilibrium with an equimolar liquid solution at 760 mmHg given the following Antoine constants:



At T = 68.24°C, Image; Image, and the azeotrope composition is known, x1 = 0.552; x2 = 0.448. At this composition, the activity coefficients can be calculated.


Image Positive deviations from Raoult’s law, γi > 1.

Using Eqn. 11.38 with the composition and γ’s just tabulated, A12 = 1.2947, A21 = 1.8373. New activity coefficient values must be found at the composition, x1 = x2 = 0.5. Using Eqn. 11.37, γ1 = 1.583; γ2 = 1.382. The problem statement requires a bubble-temperature calculation. Using the method of Table 10.1 (a flow sheet is available in Appendix C, option (a); a MATLAB example is provided in Ex11_07.m),

Image Bubble-temperature calculation.

Guess Image; Image. For this model, the activity coefficients do not change with temperature. The K-ratio depends on the activity coefficients:


Image Ex11_07.m.

Checking the sum of yi, Image is too low. Try a higher T.

After a few trials, at T = 68.262°C, Image; Image


Note: The bubble temperatures at x1 = 0.55 and 0.5 are almost the same. The T-x diagram is quite flat near an azeotrope. This has an important effect on temperature profiles in distillation columns.

Purity and Infinite Dilution

A component is said to be infinitely dilute when only a trace is present. Thus, when a binary mixture is nearly pure in component 1, it is infinitely dilute in component 2. The activity coefficients take on special values at purity and infinite dilution.


Find these limiting values in Fig. 11.5. As an example, consider the infinite dilution composition limits of Eqn. 11.37, Image, Image. Infinite dilution activity coefficients are sometimes available in the literature and can be useful for fitting if no data are available near the composition range of interest, but it should be recalled that extrapolations are less reliable than interpolations. In other words, one might experience significant errors in predictions of bubble pressures near equimolar compositions when basing parameters on infinite dilution activity coefficients. The same principle can be used with other activity coefficient models. Infinite dilution activity coefficients are especially important in applications requiring high purity. In those cases, several stages may be required in going from 99% to 99.999% purity.

11.8. Preliminary Indications of VLLE

Recall from Section 10.7 that azeotropes occur at x=y, where a maximum or minimum appears in all the plots. Also note that the bubble and dew lines do not cross, but they touch at the azeotrope composition. Occasionally when a P-x-y or T-x-y diagram is generated in a Stage III calculation, the diagram can look very odd. The two-parameter fit in Fig. 11.5 was generated using A12 = 1.99, A21 = 1.09 as fitted in Example 11.5. Suppose, due to a slight calculation error or programming typo, we generated a diagram using parameters A12 = 2.99, A21 = 1.09. The predicted phase diagram and y-x diagram would look like those shown in Fig. 11.10.


Figure 11.10. Phase diagram calculations for the 2-propanol + water system at 30°C compared with data cited in Example 11.2. The parameters where selected as described in the text to illustrate how a numerical error can result in thermodynamically unstable loops. Note the dew line has a has a loop and the maximum in the bubble line is not at the azeotropic condition. Note in the y-x plot that the coexistence curve has maxima and minima. These calculated conditions are indicative of LLE as discussed in the text, though the experiments do not show LLE.

The behavior of the lines using these parameters actually predicts that two liquid phases exist. However, the diagram requires additional modification before coexisting compositions and the vapor-liquid-liquid equilibria (VLLE) can be read from the diagram. It is important to understand that the diagram has been generated assuming that only one liquid phase exists. Though we started the discussion by assuming that a parameter calculation error resulted in predictions, all systems that exhibit VLLE will have similarly odd diagrams when only one liquid phase is assumed to exist. This assumption is the default in common process simulators such as ASPEN Plus and ChemCAD because the calculations are faster when the simulator can avoid checking for two phases. When working with simulators, you should check the phase diagrams to see if liquid-liquid phase behavior exists and you should understand where to change the simulator settings to calculate liquid-liquid behavior when it exists. Within this chapter, you should be ready to recognize that such diagrams are indicative of two liquid phases. Also recall that a T-x-y diagram qualitatively resembles an inverted P-x-y, so peculiar loops appear on a T-x-y diagram if a similar situation exists. When models incorrectly predict VLLE behavior that we know to be incorrect, we need to check our calculations. We learn how to rigorously characterize VLLE phase diagrams and how to eliminate the loops in Chapter 14.

11.9. Fitting Activity Models to Multiple Data

Fitting of the Margules equations to limited data has been discussed in Examples 11.2 and 11.5. Fits to multiple points are preferred, which requires regression of the parameters to optimize the fit. In a few cases, the Gibbs excess function can be rearranged to form the basis for a linear regression. In general, a non-linear regression may be required. Modern computers facilitate either method.

Linear Fitting of the Margules Equation

Eqn. 11.33 can be linearized:.


Image Margules models can be linearized for fitting of parameters.

Therefore, plotting GE/(x1x2RT) versus x1 gives a slope of (A21A12) and an intercept of A12. The value of A21 can also be determined by the value at x1 = 1. Using the data for 2-propanol + water listed in Example 11.8 (ignoring the first mixture point) results in a slope = –0.9289, intercept = 2.001. Thus, A12 = 2.001, A21 = 1.072, slightly different from the single point fit of Example 11.5.

Image Nonlinear parameter fitting is possible in Excel and MATLAB.

Nonlinear fitting techniques

In general, parameters for excess Gibbs models are nonlinearly related to GE or γ. Even in the cases of the Redlich-Kister and Margules equations, it may be more convenient to simply apply a nonlinear fitting procedure. The parameters can be fitted to the experimental data to optimize the fit to the experimental bubble pressure (P-x-y data) or bubble temperature (T-x-y data). In the case of fitting bubble-pressure data, the parameters can be used to generate a bubble-pressure calculation at each experimental xi as demonstrated with the one-parameter Margules in Example 11.3. The Excel Solver tool or the MATLAB fminsearch or lsqnonlin can provide rapid fits. The spreadsheet Gammafit.xlsx or MATLAB m-file GammaFit.m permit nonlinear fitting of activity coefficient parameters for the Margules equation by fitting total pressure. Either can be easily modified to find parameters for any activity coefficient model. The strategy implemented here is to calculate the activity coefficients using assumed values for the parameters and generate a bubble-pressure calculation using Eqn. 11.2 for each of the experimental points. Then each bubble pressure is compared to the experimental bubble pressure, and Image is calculated. The optimizer is invoked to keep adjusting the parameters until OBJ is minimized.

Example 11.8. Fitting parameters using nonlinear least squares

Measurements for the 2-propanol + water system at 30oC have been published by Udovenko, et al. (1967).a Use the pressure and liquid composition to fit the two-parameter Margules equation to the bubble pressure. Plot the resultant P-x-y diagram.

Image Gammfit.xlsx, Gammafit.m.


In the experimental data, the researchers report experimental vapor pressures. It is best to use experimental values from the same publication to reduce the effect of systematic errors which may be present in the data due to impurities or calibration errors. The solution will be obtained by minimizing the sum of squares of error for the bubble pressures across the composition range.

MATLAB (condensed to show the major steps):

function GammaFit()
% statements omitted to load experiments into matrix 'Data'
x1 = Data(:,1); %data have been entered into columns of 'Data'
y1expt = Data(:,2); Pexpt = Data(:,3);
Ps1Calc = 60.7; Ps2Calc = 32.1; %experimental values used for Psat
x2 = 1-x1; % calculate x2
x  = [x1  x2];  % create a 2 column matrix of x1 & x2
A = [0 0];  % initial guess for A12 and A21
A = lsqnonlin(@calcError,A); %optimize, calling 'calcError' as needed
   function obj = calcError(A)
        A12 = A(1); %extract coeffs so eqns look like text
        A21 = A(2);
        Gamma1Calc = exp((x2.^2).*(A12 + 2* (A21 - A12).*x1));
        Gamma2Calc = exp((x1.^2).*(A21 + 2* (A12 - A21).*x2));
        Pcalc = (x1.*Gamma1Calc)*(Ps1Calc) + ...
        obj = Pcalc - Pexpt;

The resultant parameters are A12 = 2.173, A21 = 0.9429. The distributed file includes statements to plot the final figure similar to that shown below. Note that fminsearch can be used if lsqnonlin is not available due to the toolboxes on your MATLAB installation. See the fit in Fig. 11.11.


Figure 11.11. Comparison of experimental data with regressed model as explained in Example 11.8.

Excel: The spreadsheet “P-x-y fit P” in the workbook Gammafit.xlsx is used to fit the parameters as shown below. Antoine coefficients are entered in the table for the components shown at the top of the spreadsheet. The flag in the box in the center right determines whether experimental vapor pressures are used in the calculations or values calculated from the Antoine equation.


Experimental data for x1 and Pexpt are entered in columns A and I. Initial guesses for the constants A12 and A21 are entered in the labeled cells in the top table. Solver is then called to minimize the error in the objective function by adjusting the two parameters. Calculated pressures are determined by bubble-pressure calculations.

The results of the fit are shown by the plot on spreadsheet “P-x-y Plot.” See the fit in Fig. 11.11.

Note that the system is the same used in Example 11.2 on page 417 and Example 11.5 on page 430. The fit in this example using all data is superior. The parameters are also slightly different from the linear fit discussed above because the objective function is different.

a. Udovenko, V.V., Mazanko, T.F. 1967. Zh. Fiz. Khim. 41:1615.

Alternative Objective Functions

An alternative choice of objective function for a given set of data usually results in a slightly different set of parameters. While the total pressure is often measured accurately, it may be desired to include vapor compositions in the objective function. For example, it is not uncommon for the pure component vapor pressures measured by investigators to differ from literature data; this is indicative of an impurity or a systematic error. One method of incorporating additional considerations into the fitting procedure is to use weighted objective functions, where recognition is made of probable errors in measurements. One of the most rigorous methods uses the maximum likelihood principles, which asserts that all measurements are subject to random errors and therefore have some uncertainty associated with them. Such techniques are discussed by Anderson, et al.,13 and Prausnitz, et al.14 The objective function for such an approach takes the form


where σ represents the variance for each type of measurement. The “true” values are calculated as part of the procedure. Typical values for variances are: σP = 2 mmHg, σT = 0.2 K, σx = 0.005, σy = 0.01, and therefore the weight of a measurement depends on the probable experimental error in the value.

11.10. Relations for Partial Molar Properties

Gibbs-Duhem Equation

A useful expression known as the Gibbs-Duhem equation results when we analyze Eqn. 10.40 together with Eqn. 10.42. Consider the differential of Eqn. 10.42 using the product rule,


Substituting for dG in Eqn. 10.40 results in


Simplifying, we obtain the Gibbs-Duhem equation,


Therefore, we conclude at constant T and P:


The relation is typically applied in the context of activity coefficients, as described below.

Gibbs-Duhem Relation for Activity Coefficients

To extend the Gibbs-Duhem equation to excess properties, the excess Gibbs energy can be manipulated in an manner analogous to the derivation above. Therefore,


resulting in


Inserting the relation between excess chemical potential and activity coefficients gives


Technically it is not possible to vary composition for two coexisting phases in a binary without either T or P changing. However, experimental analysis of isothermal P-x-y data or isobaric T-x-y data shows that the SE and VE terms are almost always very small compared to the last term and Eqn. 11.48 is generally an excellent approximation. If we apply the expansion rule, Eqn. 6.17, using one of the mole fractions, and divide by n, in a binary


This equation means that the activity coefficients for a binary system, when plotted versus composition, must have slopes with opposite signs, and the slopes are related in magnitude by Eqn. 11.49. A further deduction is that if one of the activity coefficients in a binary system exhibits a maximum, the other must exhibit a minimum at the same composition. We find this relation useful in: 1) testing data for experimental errors (grossly inconsistent data); 2) generating the activity coefficients in a binary for a second component based on the behavior of the first component in experimental techniques where only one activity coefficient is measured; 3) for development of theories for the Gibbs energy of a mixture, since our model must follow this relation. The Gibbs-Duhem equation is also useful for checking thermodynamic consistency of data; however, the applications are subject to uncertainties themselves because the activity coefficient is itself derived from assumptions, (e.g., modified Raoult’s law).15 Fortunately, developers of activity coefficient models are generally careful to ensure the models satisfy the Gibbs-Duhem equation. An understanding of the restrictions of the Gibbs-Duhem equation is helpful when studying alternative standard states such as Henry’s law or electrolyte models. A particularly useful application of the Gibbs-Duhem equation in a binary mixture is the use of the activity coefficient of one component to calculate the activity coefficient of the other component.16 Often this can be done by fitting a model to the activity of the first component, but the Gibbs-Duhem equation provides a method that does not require the application of a particular mixture model with its associated assumptions.

Relations between Various Excess Properties

For the development of accurate process calculations, a thermodynamic model should accurately represent the temperature and pressure dependence of deviations from ideal solution behavior. The excess functions follow the same relations as the total functions, HE = UE + PVE, GE = HETSE, and AE = UETSE. The derivative relations are also followed,


resulting in


The Gibbs-Helmholtz relation applies:


Particularly useful is Eqn. 11.53 using the relation with activity coefficients:


Therefore, excess enthalpy data from calorimetry may be used to check the temperature dependence of the activity coefficient models for thermodynamic consistency. Typically, activity coefficient parameters need to be temperature-dependent for representing data accurately, which implies an excess enthalpy. Likewise, any system with a heat of mixing will have temperature-dependent activity coefficients. A simple model modification is to replace the parameters with functions, for example, Aij = aij + bij/T, where aij and bij are constants and T is in K. This sets the stage for computing heats of mixing of any activity model, as shown in Example 11.9.

Example 11.9. Heats of mixing with the Margules two-parameter model

Fitting the VLE of methanol + benzenea in the range of 308–328 K with the Margules two-parameter model and then fitting the parameters to Aij = aij + bij/T gives A12 = 0.1671 + 714/T and A21 = 2.3360 – 247/T. Estimate the heat of mixing at 318 K and 50 mol% benzene.


The Margules two-parameter model is,


The relation between GE and HE is given by Eqn. 11.53. Noting the right side Margules parameters for the problem statement are simple functions of (1/T), we can manipulate the derivative for this function. Since d(1/T) = –T–2dT,




At 318 K and x1 = x2 = 0.5, HE = 8.314(0.5)(0.5(0.5·714 – 0.5·247) = 485 J/mol. Note that direct measurement of excess enthalpy is recommended when possible. Phase equilibria data must be very precise to provide an accurate enthalpy of mixing.

a. Gmehling J., et al., 1977-. VLE Data Collection. Frankfurt/Main: DECHEMA; Flushing, N.Y.: Distributed by Scholium International.

This example illustrates what it means for the “activity coefficient parameters to be temperature-dependent” and the manner of taking the derivative of GE. Though we have calculated the excess Gibbs energy in the previous example, the parameters may also be used to calculate temperature dependence of activity coefficients. Activity coefficients are strong functions of composition but weak functions with respect to temperature. This becomes apparent as you study more systems.

Recalling that heat of mixing for an ideal solution is zero, we note that the heat of mixing and the excess enthalpy are one and the same. The heat capacity for liquid methanol is about 80 J/mol-K and about 130 J/mol-K for benzene. In an adiabatic mixing process, we would thus expect this equimolar mixture to be colder after mixing by roughly 5°C. (Note: The excess heat capacity, CpE = CpCpis, is the temperature derivative of the excess enthalpy; can you determine its value for this mixture?) In general, we could write that the enthalpy of any given composition is given by,


In this way, we can represent the enthalpy of any stream to perform energy balances.

11.11. Distillation and Relative Volatility of Nonideal Solutions

To illustrate the impact of activity coefficients on practical applications, it is helpful to revisit our discussion of distillation. The relative volatility of the light to heavy key, αLH, is important to distillation, as discussed in Section 10.6. Since αLH may not be constant over an entire distillation column, it is common to estimate the average value by the geometric mean of the bottom and top.


Recalling the definition of αLH from Eqn. 10.32, substituting Eqn. 11.18, and canceling pressures,


Suppose in a binary mixture that we specify splits so that the top is xLKtop = 0.99, and xLKbot = 0.01. Then recognizing that the activity coefficients go to unity near purity,


Note: It is required that αLH > 1 at both ends of the column in order to avoid an azeotrope. In other words, Eqns. 10.35 and 11.57 CANNOT be applied unless αLH > 1 at both ends of the column.

Example 11.10. Suspecting an azeotrope

Make a preliminary estimate of whether we should suspect an azeotrope in the system benzene (B) + 2-propanol (I) at 80°C. Assume the MAB model. A convenient feature of Margules one-parameter models (including the MAB model) is that the infinite dilution activity coefficients are equal. (Note that “convenient” may not equate to “accurate.”)


Note that this problem is isothermal rather than a distillation column design, but we can evaluate the relative volatility at either end of the composition range. Antoine.xlsx gives vapor pressures of PBsat = 757 mmHg and PIsat = 683 mmHg at 80°C, so benzene is the LK. For the MAB model,

A12 = (9.23 – 0.63)(11.86 – 2.24)(89.8 + 76.8)/[4(8.314)353] = 1.174;

γi = exp(1.174) = 3.235

Using the component key assignments, PLKsat/PHKsat = 757/683 = 1.108. Therefore, at the end rich in LK, αLH = (PLKsat)/(γHKPHKsat) = 1.108/3.235 = 0.343, and the end rich in HK, αLH = (γLKPLKsat)/(PHKsat) = 3.235·1.108 = 3.58. MAB predicts an azeotrope since (αLH–1) changes sign. The prediction should be validated with experimental data and/or more accurate models because of the approximations in the MAB model.

11.12. Lewis-Randall Rule and Henry’s Law

In this chapter, we have thus far introduced the standard state using the pure component properties at the state of the system (e.g., same T, P, and liquid state). What if the pure liquid substance does not exist at these conditions? For example, in liquid-phase hydrogenation reactions, H2 is far above its critical temperature, yet exists in liquid solution at small concentrations. A pure standard state of liquid H2 is impractical. Similarly, salts dissolve as ions in aqueous solution, but the ions cannot exist as pure liquids. A model for dilute liquid solutions would be convenient, particularly if it is possible to model the liquid as some type of ideal solution. To develop models for this behavior, we first consider the general compositional behavior of the component fugacities. Based on these observations, we introduce Henry’s law to model the solution behavior relative to an ideal solution at dilute concentrations.

The Henry’s Law Standard State

Consider the shape of the component-2 fugacity versus composition that results when Eqn. 11.13 is used along with an activity coefficient model developed in this chapter. If the liquid-phase model parameters provide positive deviations from Raoult’s law, the shape of the component fugacity curve is represented by the curve in Fig. 11.12. In the figure, we follow the widely used convention for dilute binary solutions, where the solvent is designated as component-1, and the dilute solute is component-2. The use of a pure component property as a standard state creates an ideal solution line (Eqn. 10.68) known as the Lewis-Randall rule ideal solution line. Raoult’s law is a special case of the Lewis-Randall ideal solution where we use the vapor pressure to approximate the standard state fugacity.17 Thus, the activity coefficient models that we have developed previously are relative to the Lewis-Randall ideal solution. For a Lewis-Randall ideal solution, the activity coefficients approach one as the concentration approaches purity for that component, and the activity coefficients are usually farthest from unity at infinite dilution.


Figure 11.12. Schematic representation of the fugacity of component 2 in a binary mixture.

Consider that the fugacity curve in Fig. 11.12 is nearly linear at low concentrations. Thus, we could express the component fugacity as proportional to concentration using a tangent line near infinite dilution,


which is the behavior of an ideal solution given by Henry’s law.

Image Ideal Henry’s law component.

The Henry’s law constant, hi, is usually determined experimentally, and depends on temperature, pressure, and solvent. The fact that it depends on solvent makes it very different from a pseudo-vapor pressure because a vapor pressure would be independent of solvent.

Looking at Fig. 11.12, note that Henry’s law fails at high concentrations of a component unless an activity coefficient method is developed. Introducing a Henry’s law activity coefficient to represent non-idealities,


Image Henry’s law for non-ideal solution.

where γi* is the Henry’s law activity coefficient. For the fugacity curve shown in Fig. 11.12 the Henry’s law activity coefficient needs to be less than one at high concentrations, and the Henry’s law activity coefficient goes to one at infinite dilution (compare relative to the Henry’s law ideal solution line). In contrast, Fig. 11.12 shows that the activity coefficient relative to the Lewis-Randall ideal solution would be greater than one. This can be confusing if you have grown accustomed to activity coefficients less than one meaning that the components “like” each other. The component will have negative deviations from Henry’s law and positive deviations from the Lewis-Randall rule.

Although the Henry’s law activity coefficient goes to one at infinite dilution, it is inaccurate to designate the Henry’s law standard state at that composition. In fact, the correct standard state designation is a hypothetical pure component fugacity (often not experimentally accessible) selected in a manner such that the infinite dilution activity coefficient goes to one. Applying the definition of activity, Eqn. 11.23, we see that Image. Comparing with Eqn. 11.61, we see that the standard state is Image. So the important activity coefficient value is at infinite dilution, but the standard state composition is a hypothetical pure state. This perspective is especially useful for electrolyte solutions.

Relating γi for Henry’s Law and Lewis-Randall Rule

Note in Fig. 11.12 that both Henry’s approach and the Lewis-Randall approach must represent the same fugacity. Equating the two approaches,


Taking the limit at infinite dilution where γ2* approaches one, and we see


resulting in the relation between the Henry’s and the Lewis-Randall fugacity and activity coefficient,


Image Relation between Henry’s Law constant and Lewis-Randall fugacity.

In Fig. 11.13, look at the right side where lnγ2 approaches lnγ2 and lnγ2* approaches zero. The difference in the intercept at x2 = 0 is ln(h2/f2). To model the Henry’s law activity coefficient, the restriction that the activity coefficients must follow the Gibbs-Duhem Eqn. 11.49 remains; thus, the slope of the logarithm of the Henry’s law activity coefficient must be the same as the slope of the logarithm of the Lewis-Randall activity coefficient—the shift is independent of composition. The shift is illustrated in Fig. 11.13. We may adapt any activity model developed for the Lewis-Randall rule to Henry’s law by shifting the intercept values for the components modeled by Henry’s law. Thus,


Figure 11.13. Schematic illustration of the relation of the Henry’s law activity coefficient compared to the Lewis-Randall rule activity coefficient.

Image Formula to shift a Lewis-Randall activity model to a Henry’s law activity model.

where any Lewis/Randall model can be used for γi and the same model is used for γi. Usually the activity coefficient model is manipulated to obtain the infinite dilution activity coefficient expressed in terms of the activity model parameters, and the difference is expressed analytically. For the one-parameter Margules equation, Image. If applied to component 2 (dilute solute) of a binary, with component 1 (rich solvent) represented by the Lewis-Randall rule,


Image Unsymmetric activity coefficients for Henry’s law based on the one-parameter Margules equation.

Readers should recognize that we have been careful to distinguish between the two standard states in the presentation here, including distinct symbols for the different activity coefficients. When one component is represented with Henry’s law and the other represented by the Lewis-Randall rule, the overall model is described as using the unsymmetrical normalization convention for the activity coefficients.

Henry’s Law on Molal Activity Scale

Eqn. 11.61 suggests that the units for the Henry’s law constant should be pressure, but other conventions also exist. For example, a common way of presenting Henry’s constants for gases is to express the liquid phase concentration in molality and provide a constant inverted relative to hi. The result is


where the change in units for concentration requires a change in the activity coefficient and a change in units of the Henry’s law constant.18 Details on the molal activity coefficient Image are deferred until Chapter 18, but like Image it goes to one at infinite dilution. For a gas phase component, we have seen that Image, and we may use the vapor phase fugacity in the Henry’s law calculation. Many Henry’s law constants in the NIST Chemistry WebBook follow the KH convention for molality concentration units. The relation between molality and mole fraction in water is xi = mixw0.001Mw, w as we derive later (cf. Eqn. 18.147) where Mw,w is the molecular weight of water, and xw is the mole fraction of water.

Image Molal activity coefficients and Henry’s law.

Example 11.11. Solubility of CO2 by Henry’s Law

Carbon dioxide solubility in water plays a critical role in biological physiology and environmental ocean chemistry, affects the accuracy of acid-base titrations in analytical chemistry, and makes many beverages fizzy. The Henry’s law constant for CO2 in water is listed on the NIST Chemistry WebBooka as KH = 0.035 mol/kg-bar at 298.15 K. Estimate the mole fraction of CO2 in water at 0.7 MPa total pressure and 298.15 K. Treat the vapor phase as an ideal gas and the liquid as an ideal solution with the Henry’s law standard state. Neglect formation of ionic carbonate species.


KHyCO2 P = mCO2 where mCO2 is mol/(kgH2O). First find yCO2 by using Raoult’s law for water. Taking Pwsat from the steam tables, Raoult’s law for water gives

yw = Pwsat/P = 0.00317/0.7 = 0.0045 => yCO2 = 0.9955.

The solubility of CO2 is thus,


The ionic species ignored here in this binary system are sufficient to lower the pH, and though essential for comprehensive understanding, the concentrations are small relative to the molecular CO2 modeled here. In physiology or ocean chemistry, many other salts are involved which make the equilibrium more complicated. Chapter 18 addresses several issues of ionization.


Dilute Solution Calculations Using Hypothetical Lewis-Randall Fugacities

Dissolved gas solubilities can be modeled by treating the liquid phase and vapor phase both with direct use of an equation of state (to be discussed in Chapter 15). However, Eqn. 11.63 suggests that we can model dilute solutions relative to the Lewis-Randall rule. Looking at Eqn. 11.63, you can appreciate why the Henry’s law constant depends on solvent—the Lewis-Randall γi will be different for every solvent. The activity coefficient models we have developed can take γi into account. What we need is a manner to correlate the fugacity of hypothetical liquids above the critical point. A prevalent model for light gases in petrochemicals is the Grayson-Streed model (and the closely related Chao-Seader and Prausnitz-Shair models).19

Note that the shortcut vapor pressure equation yields finite numerical results for the Lewis-Randall fugacity even when T > Tc. Close to the critical temperature, it is sensible to simply extrapolate the shortcut equation. In that case, the shortcut result is more properly referred to as an estimate of fL since the liquid vapor pressure is not rigorously defined at T > Tc. Common experience is that increasing temperature decreases gas solubility. At temperatures where T > 2Tc, however, a surprising thing happens. The solubility of the gas increases with increasing temperature. In other words, fugacity of the hypothetical gaseous component decreases with increasing temperature. This behavior is predicted by equations of state (Chapter 15), but for simple correlations the previously cited researchers developed correlations of the hypothetical Lewis-Randall fugacities that matched experiments together with simple mixture models (predominantly the regular solution models discussed in the next chapter). We introduce the concepts here using the MAB model.

Fig. 11.14 shows several generalized estimates for fL as a function of reduced temperature. The Grayson-Streed estimates vary substantially depending on whether the general correlation is applied (GS-0) or specific correlations as for methane (CH4) or hydrogen (H2). The Grayson-Streed estimates have limits in their range of temperature that are reflected by their ranges in Fig. 11.14. The Prausnitz-Shair correlation gives a single curve for all compounds regardless of their acentric factor. For our purposes, we would like to have a generalized correlation that improves qualitatively on the shortcut equation at high temperatures. This goal can be achieved with the following correlation for the Lewis-Randall fugacity:


Figure 11.14. Comparison of correlations for liquid fugacity at high temperature as described in the text. GS = Grayson/Streed, PS = Prausnitz/Shair, SCVP = shortcut vapor pressure equation., SCVP+ = extended shortcut vapor pressure equation. (ω = 0.21 for SCVP and SCVP+)

Image SCVP+ model for extrapolating Lewis-Randall fugacity above Tc.


This correlation is designed to match the shortcut vapor pressure (SCVP) equation at T < Tc. It provides a reasonable match of the Grayson-Streed estimates for CH4 at T > Tc and a fairly accurate match to the Prausnitz-Shair correlation when a value of ω = 0.21 is applied to Eqn. 11.68. It also provides reasonable results for all temperatures. We refer to this as the SCVP+ equation.

Example 11.12. Henry’s constant for CO2 with the MAB/SCVP+ model

The solubility for CO2 in water at 298 K and 7 bar can be estimated as xCO2 = 0.0044. Treating the gas phase as an ideal gas and neglecting any aqueous ionic species, (a) fit γi using the Lewis-Randall rule and the SCVP+ equation for pure CO2 and determine the one-parameter Margules parameter; (b) estimate A12 of the MAB model for CO2 in water and γi and compare to part (a); (c) predict Henry’s constant at 311 K using the MAB and the SCVP+ equation.


a. First use the SCVP+ equation to predict the hypothetical liquid fugacity, log10(fL / 73.82) = 7(1 + 0.228)(1 – 304.2/298)/3 – 3exp(–3·304.2/298) fCO2L = 73.82·10–0.200 = 47 bar.

(FYI: The Lewis-Randall standard state by the SCVP model would be 64 bar instead of 47 bar.) Referring to Example 11.11, the ideal gas vapor fugacity has been calculated there, and we can equate it with Henry’s law and use the fugacity just calculated with the experimental xCO2,

yCO2P = 7·0.9955 = xCO2γCO2fCO2L γi = 7·0.9955/(0.0044·47) = 34.

lnγCO2 = A12 = ln(34) = 3.52

b. For MAB the default estimate is A12 = (α2 – α1)(β2 – β1)(V1 + V2)/(4RT)

A12 = (1.87 – 50.13)(0 – 15.06)(44/1.18 + 18/1)/(4(8.314)298) = 4.05

γi = exp(4.05) = 57.4

The MAB prediction for A12 is approximately (100%)(4.05 – 3.52)/3.52 = 15% too high.

c. At 311 K, the fitted MAB model suggests that A12 = 3.52(298/311) = 3.37 = lnγCO2. So, γCO2 = 29.

By Eqn. 11.68, log10(fL/73.82) = 7(1 + 0.228)(1 – 304.2/311)/3 – 3exp(–3·304.2/311) = –0.0968

By Eqn. 11.64, hCO2 = γCO2 fCO2L = 29(10–0.0968) = 23 bar

Note that γi > 10 for CO2 in H2O with the SCVP+ model of Henry’s law, suggesting that CO2 and water are not very compatible. In fact, the CO2+H2O system does exhibit VLLE, affirming that this approach to Henry’s law maintains consistency with the Lewis-Randall perspective. In a similar manner, other activity models, compounds, and conditions can be characterized.

11.13. Osmotic Pressure

Semi-permeable membranes exhibit the remarkable ability to sort molecules at the nanoscale. Semi-permeable membranes are used in reverse-osmosis water purification where water can permeate but salts cannot and in dialysis membranes where blood is purified. Cell walls and cell membranes in biological systems also have selective permeability to many species. Consider the membrane shown in Fig. 11.15(a) where pure W is on the left and a mixture of W + C is on the right. (Often the solvent is water but in polymer chemistry organic solvents can be used.) The membrane is permeable to W but not to C. If the solutions are at the same pressure, P, then component W spontaneously flows from the left chamber (higher chemical potential because higher mole fraction) to the right chamber (lower chemical potential because lower mole fraction) in the condition of osmosis. If the pressure on the right side is increased, the degree of flow can be decreased. When the pressure has been increased by the osmotic pressure, Π, the sides achieve phase equilibrium and flow stops. If the pressure on the right side is increased by more than the osmotic pressure, a condition of reverse osmosis exists and component W flows from the right to the left. Reverse osmosis is on the verge of becoming the largest scale chemical engineering unit operation in the world as populations grow and water becomes scarce.


Figure 11.15. (a) Illustration of a semipermeable membrane. The membrane is permeable to component W, but not component C. The label of W is convenient because water is a typical solvent. (b) Illustration of the path used in calculation of the chemical potential in the mixture at the osmotic pressure.

At the pressure (P + Π) on the right side, inward flow of W stops and the chemical potential is balanced. Let us create a convenient pathway to relate the chemical potential for the pure fluid at P to the mixture at P + Π. We can consider pressurizing the pure fluid and then mixing, or we can consider mixing the fluid and then pressurizing at fixed composition. Following historical derivation, it is common to use P as the standard state pressure for the mixing. The mixing process can be represented by the activity, aW = xWγW (cf. Eqn. 11.23)


For W in a mixture, the pressure effect on chemical potential at constant T is Image. Because the liquid is nearly incompressible, for the pressure step Image and overall,


The calculation path is illustrated in Fig. 11.16(b). The initial state represents the left side of the membrane and the final state represents the right side. Equating the chemical potential expressions for the two sides of the membrane results in


Figure 11.16. (a) Osmotic pressure in mmHg for solutions of bovine serum albumin (BSA) in water at 25°C at different pH values. Data from Vilker, V.L.; Colton, C.K.; Smith, K.A. J. Colloid Int. Sci. 1981. 79:548. (b) Fits of osmotic pressure at pH 5.4 as explained in Example 11.13.

Leading to the relation between osmotic pressure and activity of the permeable species,


Image Relation between osmotic pressure and activity of the permeable species.

The activity can be calculated from any activity coefficient model. Note that because the solution is very nearly pure W on a molar basis, we calculate activity relative to the Lewis-Randall rule for W, and it is common to replace the partial molar volume with the volume of pure W. A method known as the McMillan-Mayer framework20 is used frequently in biology to express lnaW, writing the logarithm of the activity as an expansion in terms of the molar concentration of the solute, [C]. Using the molar volume of W to normalize the expression per mole of W, the McMillan-Mayer framework results in


where B2(T) and B3(T) are functions of temperature known as the osmotic virial coefficients. Combining the two expressions, eliminating the molar volume, and rewriting the expression using solute generic subscript i, Ci the solute mass density (in units of grams/(volume of solution)), and Mw,i molecular weight, results in the form which is common in presenting data:


The osmotic virial coefficients are explicitly given temperature dependence, though they also depend on pH for biological molecules that change charge as a function of pH. Note that a plot of Π/(RTCi) will have an intercept related to the reciprocal of molecular weight and the plot can be used to determine molecular weight of solutes. Experimental data for osmotic pressure for the pig blood protein bovine serum albumin (BSA) in water at various pH values are shown in Fig. 11.16. The pH effects on charge are explained when we discuss electrolytes in Chapter 18.

Example 11.13. Osmotic pressure of BSA

Bovine serum albumin (BSA) has a molecular weight of 66399 g/mol. The osmotic pressure of an aqueous solution at 25°C and pH 5.4 is 74 mmHg when the concentration is 130 g/L and 260 mmHg at 234 g/L.a Using only these data, determine the second and third osmotic virial coefficients and estimate the pressure needed to concentrate a solution to 450 g/L across a membrane with pure water on the other side.


Since two points are given, let us linearize the equation for osmotic pressure to relate the coefficients to the slope and intercept. Defining a variable s to hold the rearranged variables,


Converting the osmotic pressure to MPa, Image is


Then at 234 g/L, si = 1.269×10-2L/g. The third coefficient is given by the slope of si versus Ci,


The second coefficient is found using the third coefficient with either of the original data points. From the point at 130 g/L:


Now at 450 g/L,


Therefore, we must apply a minimum estimated pressure of 1400 mmHg to concentrate the BSA to 450 g/L. The original paper cited gives a value of approximately 1500 mmHg. The estimate is within 10%. The prediction is sensitive to noise in the data points selected. A better method is to collect a few more data and regress a best fit.

Fig. 11.16(b) shows three fits of the data. For the “Linear Fit”, the data are linearized following the procedure in this example, and then linear regression is used over all points. For the ‘“Non-Linear Fit”, the error in the osmotic pressure prediction of Eqn. 11.73 is minimized using nonlinear regression. The “Example 11.13” curve uses the coefficients fitted in this example. The second osmotic coefficient for this data set is sensitive to the regression method. For the linear fit [B2 B3] = [1.93E-4 5.352E-5], for the nonlinear fit, [–3.57E-3 6.360E-5]. Careful analysis of the regression statistics shows that the uncertainty in the value of B2 is larger than the value—the uncertainties for the 95% confidence limit of the nonlinear fit are ±[5.25E-3 1.25E-5].

a. Vilker, V.L., Colton, C.K., Smith, K.A. 1981. J. Colloid Int. Sci. 79:548. Note the original paper uses a molecular weight of 69000 g/mol.

Cell membranes are excellent examples of semipermeable membranes, especially when considering water permeability. One implication of this property is that altering the osmotic pressure in the cellular environment can make the cells “uncomfortable.” Specifically, a higher salt concentration outside the cell might cause dehydration. On the other hand, zero salt concentration outside the cell might cause the cell to swell or rupture. This property extends to cell aggregates like skin, or the epithelium of the eye. For example, one requirement to minimize discomfort caused by eye drops is to make the solution isotonic, meaning that the osmotic pressure of water in the solution is the same as that of the reference cellular material, the eyes in this case.21

A common situation in pharmaceutical preparation is that the drug concentration is determined by the treatment protocol. However, the delivery solution should be isotonic with bodily fluids. Therefore, the solution must be supplemented with sodium chloride to make it isotonic. Fortunately, solute concentrations in cells are so low that B2 and B3 of Eqn. 11.73 can be neglected. This results in the interesting observation that osmotic pressure is independent of the nature of the compound as long as the molar concentration is the same. In other words, the concentrations of all constituents can simply be added up until the isotonic concentration is achieved. A property that follows this rule of adding up the constituents regardless of chemical nature is called a colligative property, of which osmotic pressure is an example (as long as the concentration is sufficiently low that B2 and B3 may be neglected).22 As a point of reference, human blood is in the concentration range where colligative properties can be assumed and isotonic with any solution of 0.308 mol/L solute.

Example 11.14. Osmotic pressure and electroporation of E. coli

E. coli are bacteria commonly used to express desired proteins through genetic modification because they replicate and express whatever intracellular DNA they find. Introducing foreign DNA requires weakening the cell membrane by washing twice briefly (~10 min.) with pure water at 4°C, followed by a wash with 10wt% glycerol solution, centrifuging to isolate the cell pellet from the medium before washes. After the cells are rendered “electro-competent” through washing, all but 1 ml of the glycerol solution is removed and the aliquots are frozen for storage until the “electroporation” step (electrically shocking the cells) is conducted. What concentration of glycerol (wt%) is necessary to make a solution that is isotonic with human blood? Describe what happens to the water in the cells and the glycerol outside the cells when the medium is replaced with 10 wt% glycerol.


The molecular weight of glycerol can be found from the NIST Chemistry WebBook as 92.1. This means that a 0.308 mol/L solution has 0.308·92.1 g/L of glycerol. Assuming 1000g/L as the density (the same as water since the concentration is low), this gives a weight fraction of 0.308·92.1/1000 = 0.0284 = 2.84 wt%. Therefore, the 10 wt% is hypertonic. The activity of water is too low to be isotonic. The driving force is for water to come out of the cells, diluting the glycerol outside the cells. The cells will shrink and shrivel.

11.14. Summary

The strategies for problem solving remain much the same as the strategy set forth at the end of Chapter 10 and a review of that strategy is suggested. Use Table 10.1 on page 373 and the information in Sections 10.110.8 to identify known variables and the correct routine to use. Then apply the valid approximations.

The introduction of activity coefficients is new in this chapter. First, we showed that there are three different stages in working with activity coefficients: obtaining them from experiments; fitting a model to the experiments; and using the models to extrapolate to new compositions or different temperatures and pressures.

We provided several methods of fitting activity coefficient models to experiments, and we demonstrated bubble-pressure and bubble-temperature calculations. We presented the strategy for relating the non-idealities to the excess Gibbs energy. We hypothesized models to fit the correct shape of the excess Gibbs energy and we differentiated the models to obtain expressions for the activity coefficients. We related the nature of the non-idealities to the chemical structures in the mixture through the concepts of acidity and basicity.

We introduced the concept of activity. The foundation is laid here for relating the fugacity of a component in a mixture to its pure component fugacity. Subtle details pertain to the characterization of standard state, as discussed in the introductions to Henry’s law and osmotic pressure and we superficially introduced the molality scale for Henry’s law. We will refer back to this discussion in the context of electrolytes.

Recognize that the primary difference between this chapter and Chapter 10 is the γi used to calculate Ki. We also pointed out the need for a greater understanding of solution behavior to assist in developing theory-based activity models, which will lead us into the next two chapters.

Important Equations

The starting point for many phase equilibrium problems is Eqn. 11.13 on page 425:


The various activity models alter the method of computing γi, but do not alter this basic equation. Eqn. 11.13 will appear in the simplified form for modified Raoult’s law:


Another significant equation can be summarized as the Redlich-Kister expansion (Eqn. 11.32 on page 429), in that this implicitly represents all the Margules models. When a GE model is combined with Eqn. 11.28, the activity coefficients can be derived at any composition and substituted into modified Raoult’s law to solve a wide variety of problems. The relations between GE and activity coefficient are given by


The simplest binary phase equilibrium equation to keep in mind is the bubble pressure,


Through this equation, it is very easy to compute the implications of non-ideality and assess qualitatively whether process complications like azeotropes or LLE are likely. A simple equation to guide your assessment is the MAB estimate of A12 in the Margules one-parameter model.


When considering distillation applications you must first check that αLH > 1 at top and bottom:


We also developed Henry’s law,


We showed how to relate Henry’s law to the Lewis-Randall rule used for modified Raoult’s law and how to predict the solubilities of supercritical gases in liquid solvents with the SCVP+ model.

11.15. Practice Problems

P11.1. Ninov et al. (J. Chem. Eng. Data, 40:199, 1995) have shown that the system diethylamine(1) + chloroform(2) forms an azeotrope at 1 bar, 341.55 K and x1 = 0.4475. Is this a maximum boiling or minimum boiling azeotrope? Determine the bubble temperature and vapor composition at x1 = 0.80 and 1 bar. (ANS. 331 K, 0.97)

P11.2. Derive the expression for the activity coefficient of the Redlich-Kister expansion.

11.16. Homework Problems

11.1. The volume change on mixing for the liquid methyl formate(1) + liquid ethanol(2) system at 298.15 K may be approximately represented by J. Polack, Lu, B.C.-Y. 1972. J. Chem Thermodynamics, 4:469:

ΔVmix = 0.8x1x2 cm3/mol

a. Using this correlation, and the data V1 = 67.28 cm3/mol, V2 = 58.68 cm3/mol, determine the molar volume of mixtures at x1 = 0, 0.2, 0.4, 0.6, 0.8, 1.0 in cm3/mol.

b. Analytically differentiate the above expression and show that


and plot these partial molar excess volumes as a function of x1.

11.2. In vapor-liquid equilibria the relative volatility αij is defined by Eqn. 10.32.

a. Provide a simple proof that the relative volatility is independent of liquid and vapor composition if a system follows Raoult’s law.

b. In approximation to a distillation calculation for a nonideal system, calculate the relative volatility α12 and α21 as a function of composition for the n-pentane(1) + acetone(2) system at 1 bar using experimental data in problem 11.11.

c. In approximation to a distillation calculation for a non-ideal system, calculate the relative volatility α12 and α21 as a function of composition for the data provided in problem 10.2.

d. Provide conclusions from your analysis.

11.3. After fitting the two-parameter Margules equation to the data below, generate a P-x-y diagram at 78.15°C.


11.4. A stream containing equimolar methanol(1) + benzene(2) at 350 K and 1500 mmHg is to be adiabatically flashed to atmospheric pressure. The two-parameter Margules model is to be applied with A12 = 1.85, A21 = 1.64. Express all flash equations in terms of Ki values and Ki values in terms of Modified Raoult’s law.

a. List all the unknown variables that need to be determined to solve for the outlet.

b. List all the equations that apply to determine the unknown variables.

11.5. In the system A + B, activity coefficients can be expressed by the one-parameter Margules equation with A = 0.5. The vapor pressures of A and B at 80°C are PAsat = 900 mmHg, PBsat = 600 mmHg. Is there an azeotrope in this system at 80°C, and if so, what is the azeotrope pressure and composition?

11.6. The system acetone(1) + methanol(2) is well represented by the one-parameter Margules equation using A = 0.605 at 50°C.

a. What is the bubble pressure for an equimolar mixture at 30°C?

b. What is the dew pressure for an equimolar mixture at 30°C?

c. What is the bubble temperature for an equimolar mixture at 760 mmhg?

d. What is the dew temperature for an equimolar mixture at 760 mmhg?

11.7. The excess Gibbs energy for a liquid mixture of n-hexane(1) + benzene(2) at 30°C is represented by GE = 1089 x1x2 J/mol.

a. What is the bubble pressure for an equimolar mixture at 30°C?

b. What is the dew pressure for an equimolar mixture at 30°C?

c. What is the bubble temperature for an equimolar mixture at 760 mmHg?

d. What is the dew temperature for an equimolar mixture at 760 mmHg?

11.8. The liquid phase activity coefficients of the ethanol(1) + toluene(2) system at 55°C are given by the two-parameter Margules equation, where A12 = 1.869 and A21 = 1.654.

a. Show that the pure saturation fugacity coefficient is approximately 1 for both components.

b. Calculate the fugacity for each component in the liquid mixture at x1 = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0. Summarize your results in a table. Plot the fugacities for both components versus x1. Label your curves. For each curve, indicate the regions that may be approximated by Henry’s law and the ideal solution model.

c. Using the results of part (b), estimate the total pressure above the liquid mixture at 55°C when a vapor phase coexists. Assume the gas phase is ideal for this calculation. Also estimate the vapor composition.

d. Comment on the validity of the ideal gas assumption used in part (c).


a. The acetone(1) + chloroform(2) system can be represented by the Margules two-parameter equation using A12 = –1.149, A21 = –0.862 at 35.17°C. Use bubble-pressure calculations to generate a P-x-y and y-x diagram and compare it with the selected values from the measurements of Zawidzki, Z. Phys. Chem., 35, 129(1900).

b. Compare the data to the predictions of the MAB model.



a. Fit the Margules two-parameter equation to the methanol(1) + benzene(2) system T-x-y data below at 90°C (Jost, W., Roek, H, Schroeder, W., Sieg, L., Wagner, H.G. 1957. Z. Phys. Chem. 10:133) by fitting to x1=0.549. Plot the resultant fit together with the original data for both phases.

b. Compare the data with the predictions of the MAB model.



a. Fit the Margules two-parameter equation to the n-pentane(1) + acetone(2) system P-x-y data below at 1 bar (Lo et al. 1962. J. Chem. Eng. Data 7:32) by fitting to x1=0.503. Plot the resultant fit together with the original data for both phases.

b. Compare the data with the predictions of the MAB model.


11.12. For a particular binary system, data are available:

T = 45°C P = 37 kPa x1 = 0.398 y1 = 0.428

In addition, Image and Image. From these data,

a. Fit the one-parameter Margules equation

b. Fit the two-parameter Margules equation

11.13. 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, J. Amer. Chem. Soc., 71, 1793(1949). Estimate the bubble pressure at 55°C and x1 = 0.1, using

a. The one-parameter Margules equation

b. The two-parameter Margules equation

11.14. 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, P = 34.98 kPa. Estimate the dew pressure at 298 K and y1 = 0.6, using

a. The one-parameter Margules equation

b. The two-parameter Margules equation

11.15. 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 one-parameter Margules equation.

a. Estimate the activity coefficient of component 1 at x1 = 0.75 and 80°C. [Hint: The relative volatility (given in problem 11.2) is unity at the azeotropic condition.]

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

11.16. Ethanol(1) + benzene(2) form an azeotropic mixture. 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 MAB model.

b. Prepare a y-x and P-x-y diagram for the system at 45°C assuming the one-parameter Margules model and using the experimental pressure at xE = 0.415 to estimate A12.

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

11.17. The acetone + chloroform system exhibits an azeotrope at 64.7°C, 760 mmHg, and 20 wt% acetone.

a. Use the MAB model to predict the T-x-y diagram at 1 bar.

b. Use the Margules one-parameter model to estimate the T-x-y diagram at 1 bar.

11.18. For the Margules two-parameter model 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 using data at 78.15°C:


11.19. Using the data from problem 11.18, fit the two-parameter Margules equation, and then generate a P-x-y diagram at 78.15°C.

11.20. 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?

11.21. Suppose a vessel contains an equimolar mixture of chloroform(1) and triethylamine(2) at 25°C. The following data are available at 25°C:


a. If the pressure in the vessel is 90 mmHg, is the mixture a liquid, a vapor, or both liquid and vapor? Justify your answer.

b. Provide your best estimate of the volume of the vessel under these conditions. State your assumptions.

11.22. Ethanol(1) + benzene(2) form azeotropic mixtures.

a. From the limited data below at 45°C, it is desired to estimate the constant A for the one-term Margules equation, GE/RT = Ax1x2. Use all of the experimental data to give your best estimate.


b. From your value, what are the bubble pressure and vapor compositions for a mixture with x1 = 0.8?

11.23. An equimolar ternary mixture of acetone, n-butane, and ammonia at 1 MPa is to be flashed. List the known variables, unknown variables, and constraining equations to solve each of the cases below. Assume MAB solution thermodynamics and write the flash equations in terms of K-ratios, with the equations for calculating K-ratios written separately. (Hint: Remember to include the activity coefficients and how to calculate them.

a. Bubble temperature

b. Dew temperature

b. Flash temperature at 25mol% vapor

b. Raised to midway between the bubble and dew temperatures, then adiabatically flashed.

11.24. 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. One-parameter Margules equation

b. Two-parameter Margules equation

11.25. Fit the specified model to the methanol(1) + benzene(2) system P-x-y data at 90°C by minimizing the sum of squares of the pressure residual. Plot the resultant fit together with the original data for both phases (data are in problem 11.10), using

a. One-parameter Margules equation

b. Two-parameter Margules equation

11.26. Fit the specified model to the methanol(1) + benzene(2) system T-x-y data at 760 mmHg by minimizing the sum of squares of the pressure residual. Plot the resultant fit together with the original data for both phases (Hudson, J.W., Van Winkle, M. 1969. J. Chem. Eng. Data 14:310), using

a. One-parameter Margules equation

b. Two-parameter Margules equation


11.27. VLE data for the system carbon tetrachloride(1) and 1,2-dichloroethane(2) are given below at 760 mmHg, as taken from the literature.23


a. Fit the data to the one-parameter Margules equation.

b. Fit the data to the two-parameter Margules equation.

c. Plot the P-x-y diagram at 80°C, based on one of the fits from (a) or (b).

11.28. When only one component of a binary mixture is volatile, the pressure over the mixture is determined entirely by the volatile component. The activity coefficient for the volatile component can be determined using modified Raoult’s law and an activity coefficient model can be fitted. The model will satisfy the Gibbs-Duhem equation and thus an activity coefficient prediction can be made for the nonvolatile component. Consider a solution of sucrose and water. The sucrose is nonvolatile. The bubble pressures of water (1) + sucrose (2) solutions are tabulated below at three temperatures.


a. Fit the one-parameter Margules equation to the water data at the temperature(s) specified by your instructor. Report the values of A12.

b. Prepare a table of γ1 values and plot of the experimental and fitted/predicted ln γ1 versus x1 for water and sucrose over the range of experimental compositions for the temperature(s) specified by your instructor.

c. Prepare a table of values and on the same plot as (b) add a curve for ln γ2* for the temperature(s) specified by your instructor.

d. Prepare a table of values and a plot of osmotic pressure (in MPa) for the solution versus C2 (g/L) at 25°C. The density at 25°C can be estimated using ρ(g/mL) = 0.99721 + 0.3725w2 + 0.16638w22 where w2 is wt. fraction sucrose. Include a curve of the osmotic pressure expected for an ideal solution.

e. Calculate the osmotic pressure (MPa) using the activity of water modeled with the one-parameter Margules equation at 25°C fitted in part (a). Add it to the plot in part (d).

f. Calculate the second and third osmotic virial coefficients (for concentration units of g/L) at 25°C by fitting the calculations from part (d). Add the modeled osmotic pressure to the plot from (d).

g. From the temperature dependence of the one-parameter Margules parameter fitted in (a), show that the parameter may be represented with f(1/T(K)). Then provide a model for the excess enthalpy and the parameter value(s) that represent the experimental data.

11.29. Red blood cells have a concentration of hemoglobin (Mw ~ 68000) at 0.3 M. The osmotic pressure a body temperature (37°C) is 0.83 MPa. Water can permeate the cells walls, but not hemoglobin.24

a. Using only the second osmotic coefficient, determine the coefficient value (L/g), and determine the activity of water at the conditions given above.

b. Calculate the ideal solution osmotic pressure at the conditions given above.

c. Suppose we were to transfer red blood cells in a laboratory solution at 37°C (blood banks need to do this). We want the external glucose solution to match the red blood cell’s internal osmotic pressure to avoid swelling or shrinking of the cells. If glucose has an osmotic pressure of 2 MPa at 0.7 M and 37°C, what glucose concentration (g/L) would match the internal osmotic pressure to keep the blood cells stable? What is molality of the resulting glucose solution? Comparing molalities, what can you infer about the solution non-idealities of the glucose solution compared to the hemoglobin solution?

11.30. Osmotic pressure of bovine serum albumin (BSA) has been measured at 298.15 K and various pH values by Vilker, V.L., Colton, C.K., Smith, K.A. 1981. J. Colloid Int. Sci. 79:548, as summarized in the table below. The investigators report the BSA molecular weight in their sample as 69,000.


a. Regressing all data, determine the second and third osmotic virial coefficients for pH 7.4.

b. Regressing all data, determine the second and third osmotic virial coefficients at pH 4.5.

11.31. Boric acid is a common supplement to make ophthalmic solutions isotonic. It is entirely undissociated at normal ophthalmic conditions.

a. Estimate the concentration (wt%) of boric acid to prepare a solution that is isotonic with human blood.

b. Estimate the concentration (wt%) of boric acid that should be added to a 0.025wt% solution of Claritin to make it isotonic. The molecular formula of Claritin is listed at as C22H23ClN2O2.

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