Chapter 19. Molecular Association and Solvation

The satisfaction and good fortune of the scientist lie not in the peace of possessing knowledge, but in the toil of continually increasing it.

Max Planck

When specific chemical forces act between molecules, there is a possibility of complex formation. Chapter 17 dealt with systems where the interactions were strong enough to create new molecules. The interactions of complexes are much weaker but affect solution thermodynamics in a way that is fundamentally different from van der Waals interactions. The solution thermodynamics of complex formation is best represented as simultaneous phase and reaction equilibrium. The complexes usually cannot be isolated, but their existence is certain from measurements such as spectroscopic studies. Hydrogen bonding is an example of this type of behavior, as well as Lewis acid/base interactions. When complexation occurs between molecules that are all from the same component, the phenomenon is called association. For example, acetic acid dimerizes in pure solutions. When complexation occurs between molecules that are from different components, the phenomenon is called solvation. Complexation can occur in familiar ways, like hydrogen bonding, or in less familiar settings like charge transfer complexes. Mathematically, these phenomena can be treated in the same manner, which we will refer to as the chemical contribution to the Helmholtz energy, Achem. By taking the limit of the resultant formulas as the complexation energy approaches infinity, we can also derive a contribution of the Helmholtz energy that characterizes the formation of dimers and chains, Achain. Until now, we have described nonspherical contributions with empirical corrections, like κ(ω) of the Peng-Robinson equation. Chemical theory provides a fundamental description of how nonspherical molecules have nonzero acentric factors. With these contributions, the Helmholtz energy and compressibility factor become,

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These equations comprise a new perspective on how equations of state should be formulated. Rather than limiting molecular interactions to a spherically symmetric perspective like the square-well model, chemical theory accounts for orientationally specific interactions. For example, the proton acceptors in water do not interact with the proton donors in the same way that they interact with other proton acceptors. Clearly, the orientations of water molecules must matter. Chemical theory provides a rigorous and practical means to recognize this.

Imagine for a moment how complex the analysis of complexation could be. For example, alcohols have proton acceptors and donors that can bond in chains of dimers, trimers, ... Each oligomer might need a specific activity coefficient and an independent reaction equilibrium constant. Then similar specifications could be necessary for all solvation interactions. Fortunately, the reaction equilibrium constants tend to follow predictable patterns, which we need to understand. Furthermore, the oligomer activities can be related through the equilibrium constraints. In the end, the model equations are only slightly more complex than the Peng-Robinson equation.

Furthermore, the mathematical formalism of chain formation can be extended by simply extrapolating the complexation energy to infinity. This provides a self-consistent theory of polymeric molecules. In previous discussions, we assumed that the shape factor in a model like ESD was somewhat arbitrarily imposed. In the present discussion, the shape factor is derived as a natural outcome of chain formation. This opens the door to a discussion of intramolecular interactions, whereas previous discussions have focused on intermolecular interactions. To get an idea of what lies beyond that door, note that proteins, DNA, and RNA are polymers with very specific intramolecular arrangements. Factor in the complexity of pH and other polyelectrolyte interactions, and you can understand why this text really is merely an introduction.

Chapter Objectives: You Should Be Able to...

1. Explain the relations between reaction and phase equilibria that yield the model equations for the PC-SAFT and ESD models.

2. Solve the model equations for the mole fraction of monomer, dimer, and so on, and the resultant fugacity expressions to obtain K-values.

3. Critically assess the intramolecular characterization implicit in PC-SAFT and ESD models and suggest deficiencies that should be addressed in future research.

19.1. Introducing the Chemical Contribution

Evidence of complexation is often subtle and could be overlooked or ignored in many situations, but its effects are irrefutable in other situations, so a question arises about ignoring facts because they are inconvenient. For example, dimerization of carboxylic acids is observable in the saturated vapor compressibility factor of small acids, but less so for acids of 10 carbons or more. On the other hand, the trend in LCEP discussed in Chapter 16 shows that dimerization is present even in long-chain fatty acids. Another commonly cited example of association is HF vapor, important in petroleum refining and the manufacture of refrigerants, modeled as (HF)n with n predominately 2 or 6. Evidence of this stoichiometry is found in vapor density data. Alcohols are also common substances in the chemical process industry that exhibit association. Spectroscopic data are the best source for characterizing complex formation and further information is available.1 On the other hand, infrared spectra of the hydroxyl stretch, for example, can be difficult to interpret because they include a broad band of stretches indicative of the various oligomers that form in chains. Solid-liquid phase boundaries are indicative of strong complex formation, and may be used to infer complex stoichiometries. For example, the phase diagram for NH3 + water in Fig. 14.14 shows two complexes that form in the solid phase, and might be expected to appear in fluid phases in addition to ionic species. Acetone and chloroform show a 1:1 compound in the solid phase.2 Altogether, the chemical evidence and the results of molecular simulations are converging on the outlook that complexation is an effect that can and should be included in any systematic theory of molecular interactions and phase behavior.

Hydrogen Bonding

Hydrogen bonding is the most common phenomenon leading to association or solvation. As an example, consider the saturated vapor phase compressibility factor of acetic acid. Table 19.1 shows that, even though the pressure is very low for the saturated vapor at low temperature, the vapor compressibility factor of acetic acid deviates considerably from unity, when we would expect the ideal gas law to hold and Z = 1. For comparison, benzene under the same conditions is very nearly an ideal gas.

Table 19.1. P-V-T Evidence of Association from the Compressibility Factors of Saturated Vapors

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In the case of acetic acid, the small compressibility factor is due to the dimerization of the carboxylic acid. Even though the pressure is very low, the compressibility factor approaches 0.5 because the number of molecules in the vapor is actually half the apparent amount, owing to the dimerization.3 Note that the compressibility factor increases as the temperature is increased. Why should the conversion to dimer be more complete at low temperature? Because association is an exothermic reaction and the van’t Hoff relation clearly dictates that conversion of exothermic reactions decreases with increasing temperature. Another implication of associating fluids relative to nonassociating fluids is that higher temperatures are needed to make the associating network break into a gas, and this means that Tc for an associating component will be significantly higher than a nonassociating component of similar structure (e.g., H2O versus CH4). The chemical association of acetic acid during dimerization is illustrated in Fig. 19.1. Note that the structure forms two hydrogen bonds simultaneously which makes the dimerization quite strong. Note also that a property of hydrogen bonding is that the O-H-O bond angle is nearly linear. Although the ring has eight atoms, the carboxylic acid structure is close to a six-sided ring, not an eight-sided ring.

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Figure 19.1. Schematic of association in acetic acid.

Hydrogen bonding is also common between different species. For example, the hydroxyl hydrogen in phenol is fairly acidic while trimethyl amine is basic. The trimethyl amine would form a solvation complex in this case, even though it is incapable of association. Hydrogen bonding in solvating systems can enhance miscibility. For example, water and triethylamine are miscible in all proportions below 18°C (and above 0°C where water freezes). However, above 18–19°C, the increased thermal energy breaks the hydrogen bonds resulting in an immiscibility region which increases with increasing temperature4 until the hydrogen bonding energy becomes too weak to influence the phase behavior. To put this into perspective, water miscibility with triethylmethane (aka 3-ethylpentane) is practically negligible at all conditions.

Usually, hydrogen bonding refers to protons bonding when they are attached to O, N, or F. In rare instances, protons bound to carbons can form a complex and it is common to refer to this as hydrogen bonding as well.1 The hydrogen on a highly chlorinated molecule can form a complex with a Lewis base, such as a carbonyl, ether, or amine group. Well-known examples are chloroform + acetone, chloroform + diethyl ether, and acetylene + acetone, as depicted in Fig. 19.2. The chloroform + acetone in homework problem 11.15 should be reconsidered in light of complex formation.

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Figure 19.2. Schematic of solvation in several pairs where association of the pure components is negligible.

Charge-Transfer Complexes

Solvation can also occur in the absence of “hydrogen bonding,” per se. Similar kinds of interactions can be called “charge-transfer complexes,” in which case one component is electron-rich (loosely bound electron) and another is strongly electron-attracting (low energy vacant orbital). For example, common electron donors are benzene rings with donating groups like –OH, –OCH3, –N(CH3)2. Common electron acceptors are compounds with several nitro groups (1,3,5-trinitrobenzene), quinones, and compounds with several CN groups (tetracyanoethane, 2,3-dicyano-1,4-benzoquinone). For example, nitrobenzene and mesitylene form a complex when they are mixed as shown in Fig. 19.3. Tetrahydrofuran (THF) and toluene form a charge transfer complex when mixed together. When a charge-transfer complex forms, the energy typically corresponds to about 5 kJ/mole, and the strength is the same order of magnitude as a hydrogen bond. Chemically, these situations are all a bit different, but the mathematics is the same.

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Figure 19.3. Schematic of charge-transfer complexation.

We discussed acidity and basicity in connection with the MOSCED and SSCED models in Chapter 11. Recall that a measure of the relative acidity/basicity is available from spectroscopic measurements tabulated in the Kamlet-Taft acidity/basicity parameters.5 The Kamlet-Taft parameters are determined by comparing spectroscopic behavior of probe molecules in the solvents of interest. A small table of parameters has been included on the back flap.6 These acidity and basicity parameters were designed for the SCED perspective, but they can provide guidelines for the energies of hydrogen bonding and complex formation too. The primary difference is that the SCED perspective does not alter the underlying solution model in a fundamental sense. To clarify, the similarity of the SSCED model to the Scatchard-Hildebrand means that the same solution model would be obtained with proper choices of kij. The kij values of the SSCED model are more predictable, but they do not alter the skewness of the Gibbs excess energy any more than the kij values of the Scatchard-Hildebrand model. Recognizing hydrogen bonding and complexation as mild reactions does alter the skewness of the Gibbs excess energy. In this sense, complexation theory provides an alternative to local composition theory. Whereas local composition theory correlates qualitatively with molecular simulation results, complexation theory correlates quantitatively. This means that a systematic step can be taken in connecting the molecular interactions with their macroscopic behavior.

Preliminary Considerations of Stoichiometry and Notation

In a generalized binary solution,7 the ith complex can be represented by the general form Aai Bbi, where the values of ai and bi are integers which will depend on the particular system (note that for an associated specie, either ai or bi is zero, but that won’t affect the proof). The integers ai and bi are the stoichiometric coefficients:

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For example, a hypothetical system is shown in Fig. 19.4 that exhibits both association and solvation where the components A and B are added to the solution in quantities nA and nB.

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Figure 19.4. Illustration of mixing of species A with species B to form a solution with both association and solvation. The labels in the containers indicate the true species present, not the relative concentrations. The species have been chosen for illustrative purposes.

The concentrations Image and Image are the mole fractions that are experimentally important for macroscopic characterization and are the conventional mole fractions. Since these mole fractions do not represent the true species in solution, they are also called the apparent mole fractions. The mole fractions in the actual solution are called the true mole fractions, and also are denoted by x’s; for example, Image, Image. Note that xA (the apparent mole fraction) is not the same as (the A monomer mole fraction). Because the monomer mole fraction will end up being so important in later proofs, we give the special subscript M to help distinguish this as the monomer in the true mixture. Therefore, xA1 ≡ xAM, and xB1 ≡ xBM. The true species that are present in solution will usually be inferred from fitting experimental data, and the true mole fractions are usually modeled quantities rather than experimental quantities since they are subject to the assumptions of the model. The nomenclature is nevertheless the established convention in the literature. Note that the implication that the mixture of this discussion is a liquid mixture is not restrictive. The same balances and notation will be used to refer to vapor mixtures, but y’s will be substituted for the x’s.


Image The subscript M is used for monomer.


19.2. Equilibrium Criteria

Mole Balance

One may wonder how quantification of the phenomena can be approached in a generalized fashion, but the criteria are presented clearly by Prigogine and Defay (1954) whose proof we reproduce here with modified notation. The first balances that must be satisfied are the material balances. For a binary solution created from nA moles of A, and nB moles of B, where ni is the moles of each specie formed by Eqn. 19.3,

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where the summations are over all i true species found in the solution. For example, in the binary solution of Fig. 19.4, Eqn. 19.4 becomes

nA = 1nAM + 0nBM + 2nA2 + 1nAB2

nB = 0nAM + 1nBM + 0nA2 + 2nAB2

Chemical Potential Criteria

The chemical potentials of the true species are designated by μAM, μBM, μAaiBbi and so forth, and the apparent chemical potentials by μA and μB. Applying the principles of chemical equilibria to binary Eqn. 19.3, we find

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If the total differential of G (Eqn. 17.7) is evaluated at constant T and P, allowing the species to come to equilibria,

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we find by incorporating Eqn. 19.5 and the differential of Eqn. 19.4

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On the other hand, for any binary solution at constant P, T, according to the apparent components

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By comparing 19.6 and 19.7 we conclude

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Therefore, the apparent (conventional) chemical potential is quantified by a model that calculates the chemical potential of the true monomer species. It should be noted that this proof is independent of the number or stoichiometry of species that are formed in solution.

Fugacity Criteria

The chemical potential criteria may be extended to fugacity. For the apparent chemical potential,

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and for the monomer,

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Note that for a species that associates, the standard state for the monomer is a mixture state since, even when A is pure, there is a mixture of true associated species. Applying Eqn. 19.8,

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where the standard state is pure component A.

In the event that component A does not associate, the true solution is completely a monomer when A is pure, and

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which leads to

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The situation when A associates is slightly more complex. Recognizing that the apparent state neglects complexation, the chemical potential of monomer μAM in Eqn. 19.10 can be calculated,

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where the lower limit is the hypothetical state of pure monomer, and the upper limit is the monomer state that actually exists in a pure associating solution of A. Integrating both sides, and recognizing the lower limit of each integral as the apparent standard state

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Combining with Eqn. 19.11 again results in Eqn. 19.13. Note that a parallel proof would show

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Activity Criteria

Returning to Eqn. 19.9, it can be rewritten in terms of the apparent activity and activity coefficient.

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Defining an activity coefficient, αAM, of the true monomer species, the chemical potential is

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Using Eqn. 19.8 to equate Eqns. 19.15 and 19.16,

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where the exponential term is a constant at a given temperature. The symmetrical convention of apparent activity requires Image. For a nonassociating species, the exponential term of Eqn. 19.17 is unity by Eqn. 19.12, and thus Image, Image. For an associating species Image which is the true mole fraction of monomer in pure A, which is not unity. Therefore, the exponential term is simply the reciprocal of the limiting value of the monomer activity Image. As such, we write

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where Image for a nonassociating component. A parallel proof would show that

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These equations show how the activity coefficient could be less than one even for an “ideal” solution. For example, acetone and chloroform might form an ideal solution in the sense that αAM=1=αBM at all concentrations. Complexation would result in xAM< xºAM when B was present, such that γA < 1. We explore this prospect extensively in Section 19.4.

19.3. Balance Equations for Binary Systems

The balance equations to be solved take the same form for both vapors and liquids. The liquid equations will be shown, and the reader should recognize the vapor equations by analogy. First, the true mole fractions must sum to unity:

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In a binary system, a balance equation can be written for either component to match the apparent mole fraction:

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Dividing numerator and denominator by the true number of moles, nT,

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Rearranging Eqn. 19.21 to facilitate implementation of the balance, multiply by the denominator,

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and collecting the true mole fraction results in a form of Eqn. 19.21 that is easier to implement:

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Eqns. 19.20 and 19.22 are not yet ready to implement because all of the true mole fractions are unknown and only two equations have been developed. In the next section, we show that the true mole fractions can be written in terms of an equilibrium constant and the monomer mole fractions, which will provide sufficient information once the equilibrium constants are known.

19.4. Ideal Chemical Theory for Binary Systems

The simplest method of modeling complex behavior is to neglect the nonidealities by modeling a vapor phase as an ideal gas mixture including the complexes (true fugacity coefficients equal to 1), and to model a liquid phase as an ideal solution containing complexes (true activity coefficients equal to 1). This approach is called Ideal Chemical Theory and can be used to calculate the non-ideal apparent fugacity or activity coefficients. Two brief observations at the outset help to put the chemical perspective into context. First, in reference to systems that can only solvate (not associate), the observed activity coefficients must be less than one because xAM<x°AM when B is present. Second, for systems in which one component associates and the other can neither associate nor solvate, the observed activity coefficients must be greater than one because xAM> x°AM when B is present owing to the interference of B with A from dilution. If you combine these two observations for mixtures that can solvate and associate, you can see how an entire range of activity coefficients may be obtained.

Modeling complex formation with ideal chemical theory, Eqn. 19.3 can be expressed in terms of an equilibrium constant:

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Plugging into Eqns. 19.20 and 19.22, the equations to be solved are obtained:

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Once the Ki are known, then xAM and xBM can be determined by solving these two equations at a specified apparent concentration. Subsequently, all true mole fractions, xi (Eqn. 19.23) and the apparent mole fractions γA, γB (Eqns. 19.18 and 19.19), can be calculated. If γA, γB are known from experiment, and the complex stoichiometry is known, Ki values can be adjusted to fit the data using optimization methods. A spreadsheet is provided for solving for the true species for given values of Ki in the programs Ichemt.xlsx and Ichemt.m.

For ideal chemical theory applied to the vapor phase, the xi are replaced with yi and Eqn. 19.3 is expressed as

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Eqns. 19.20 and 19.22 then become

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These equations are marked as ideal gas equations since they are ideal gas equations from the perspective of the true solution. As with the liquid-phase calculation, if the Ki values are known, yAM and yBM can be determined.


Example 19.1. Compressibility factors in associating/solvating systems

Derive a formula to relate the true mole fractions to the compressibility factor of a vapor phase where the true species follow the ideal gas law.

Solution

A vessel of volume V holds no apparent moles. However, experimentally, in the same total volume, there would be a smaller number of true moles nT. Applying the ideal gas law,

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Experimentally, we wish to work in terms of the apparent number of moles,

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Note that this equation is labelled as an ideal gas equation because the true species follow the ideal gas law, even though from the perspective of the apparent species, the ideal gas law will not be followed. From the total mole balances, Image, and Image; therefore,

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Dividing numerator and denominator by nT,

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Therefore, once the true mole fractions have been determined, the compressibility factor can be calculated. Determining the true mole fractions requires solving the reaction equilibria, as discussed in the next example.



Example 19.2. Dimerization of carboxylic acids

P-V-T measurements of acetic and propionic acid vapors are available.a The equilibrium constants for acetic and propionic acids at 40°C are 375 bar–1 and 600 bar–1 respectively. At a pressure of 0.01 bar, determine the true mole fractions, the compressibility factor, and the fugacity coefficients.

Solution

Beginning with Eqn. 19.27, letting A be the acid of interest,

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Eqn. 19.28 is not required since the system is a single component. This simple equation can be solved with the quadratic formula,

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At P = 0.01 bar, for acetic acid, yAM = 0.4, YA2 = 0.6, and even at this low pressure, Eqn. 19.30 gives Z = 0.625. For the fugacity coefficient, starting with Eqn. 19.13,

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Since the system is pure, yA = 1 and the fugacity coefficient on the left-hand side will be for a pure species. Since the model uses ideal chemical theory, Image. Therefore

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The same procedure can be repeated for propionic acid, however it will be even more nonideal. The answers are: yAM = 0.333, Z = 0.6, Image.


a. McDougall, F. H., 1936. J. Amer. Chem. Soc. 58:2585; 1941. 63:3420.


Example 19.3. Activity coefficients in a solvated system

1,4-dioxane (component B) is a cyclic 6-member ring, C4H8O2, with oxygens in the 1 and 4 positions. When mixed with chloroform (component A) the oxygens provide solvation sites for the hydrogen on chloroform. Since there are two sites on 1,4-dioxane, two complexes are possible, AB and A2B. McGlashan and Rastogia have studied this system and report the liquid phase can be modeled with ideal chemical theory using KAB = 1.11, KA2B = 1.24 at 50°C. Calculate the true mole fractions and activity coefficients across the composition range.

Solution

We will use the program Ichemt.xlsx to solve Eqns. 15.21 and 15.22. For the A2B compound, ai = 2, bi = 1, Ki = 1.24. For the compositions, we enter increments of 0.05 for the apparent compositions. Near the endpoints, we enter xA = 0.001 and xA = 0.999 to avoid numerical underflows and overflows. The activity coefficients are easily calculated using γA = zA/xA, γB = zB/xB since neither component exhibits association. The results are shown in Fig. 19.5. Note that the solvation causes negative deviation from Raoult’s law. Also note the relation between the complex stoichiometry and the maxima in the complex concentration. Can you rationalize why the infinite dilution activity coefficient of 1,4-dioxane is smaller than the infinite dilution activity coefficient of chloroform?

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Figure 19.5. Ideal chemical theory applied to the chloroform + 1,4-dioxane system as calculated in Example 19.3.


a. McGlashan, M. L., Rastogi, R. P. 1958. Trans. Faraday Soc. 54:496.


Image Ichemt.xlsx Ichemt.m


19.5. Chemical-Physical Theory

The assumptions of ideal chemical theory are known to be oversimplifications for many systems and physical interactions must be included. For a liquid phase, the activity coefficients of the true species can be reintroduced. Then

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Utilizing this result with Eqns. 19.20 and 19.22, the following equations are obtained:

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Since most activity coefficient models require two parameters per pair of molecules, the number of parameters becomes large. In addition, any parameters for the complex must be estimated or fit to experiment since the complex cannot be isolated. Solution of the equations is more challenging because the true activity coefficients must be updated with each iteration on xAM and xBM.

For chemical-physical theory applied to the vapor phase,

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Eqns. 19.20 and 19.22 then become

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The physical properties of the complex must also be modeled with this approach, and the same challenges for solving the equations are present as discussed above for chemical-physical theory of liquid phases.

An interesting study has been performed by Harris8 for acetylene in n-hexane, butyrolactone, and n-methyl pyrrolidone at 25°C. In this study, a simplified van Laar model was used to model the physical deviations, which resulted in one physical parameter. Naturally, the acetylene + n-hexane does not exhibit solvation, but the other binaries do, with the pyrrolidone showing the strongest complexation. Further, the n-hexane system has positive deviations from Raoult’s law across the composition range, the pyrrolidone shows negative deviations, and the lactone shows both positive and negative deviations. All three systems are accurately modeled using two parameters each—one chemical parameter and one physical parameter.

Another approach to the chemical-physical theory is to use the Flory-Huggins theory for the physical contributions. This is the approach of Coleman and Painter in modeling polymer solutions. The Coleman-Painter model leads to complications in the extension to ternary mixtures, however, owing to several details in their perspective on chemical networks.9

Multicomponent chemical-physical theory can be achieved most elegantly with Wertheim’s theory which we will discuss in the next section.10 Wertheim’s theory characterizes chemical interactions from the perspective of the acceptor or donor sites instead of the species. This simplifies to the counting of nonbonded sites, especially for multicomponent systems, and the nonbonded sites can be related to the monomer fraction, which suffices to define the solution thermodynamics. Wertheim’s theory requires a complementary physical theory for the nonchemical attractive and repulsive interactions. Briefly, chemical interaction is short-ranged, so variations in bonding are affected by the frequency of species coming into contact. Repulsive interactions dominate the frequency of contact (specifically, g(σ)). We can estimate g(σ) for spherical molecules with the Carnahan-Starling equation. For nonspherical molecules, we can imagine that they are composed of spherical segments. Then the role of the attractive contribution is like that of a spherical molecule, to provide a disperse field of attractive energy that acts between spherical entities and reduces the pressure. This leads to a remarkably compact and self-consistent model of chemical-physical equilibria.

Before we begin our discussions of Wertheim’s theory, let us mention an additional approach to chemical-physical theory is provided by Heidemann and Prausnitz.11 They showed that reasonable assumptions about the van der Waals parameters of monomers, dimers, trimers, and so on leads to a closed form solution for the compressibility factor and fugacity coefficient. Similar to Wertheim’s theory for pure fluids, the Heidemann-Prausnitz method provides a complete chemical-physical theory, describing all variations with density, temperature, composition, and chemistry. However, similar to the Coleman-Painter theory, this method has complications in the extension to multicomponent mixtures. Suresh and Elliott12 showed that the Heidemann-Prausnitz method is equivalent to Wertheim’s theory subject to certain assumptions about the change in heat capacity due to reaction. In the interest of covering the most general method, we focus now on Wertheim’s theory, but we introduce concepts using the Heidemann-Prausnitz perspective as a simple way of illustrating several of the more striking results derived from Wertheim’s theory. This is necessary because the rigorous proofs of Wertheim’s theory of Wertheim’s original publications go beyond the introductory scope envisioned here.

19.6. Wertheim’s Theory for Complex Mixtures

The general approach is exactly what you would expect: Write all the reaction and phase equilibrium constraints and then solve the nonlinear system of equations. Making this approach into a practical alternative to, say, the Peng-Robinson model requires several clever observations, approximations, and rearrangements, however. Wertheim’s theory is based on the contribution to the Helmholtz energy. In the end, Achem is recognizable as a distinct contribution with a firm foundation in experimental observation and molecular simulation that adds just one intermediate (but robust) step in solving for the density given temperature and pressure.

Wertheim’s theory has the same objective as this chapter: to develop a theory for the chemical contribution to the Helmholtz energy, Achem, and consequently Zchem, through the derivative relations in Chapter 6. Because the volume derivative of A results in P, the volume (or density) derivative of A can be used to calculate the contribution of chemical interactions to Z. Wertheim’s theory refers to the concepts of monomers and dimers discussed previously, but develops a self-contained and self-consistent model based on a given equation of state for the nonchemical contributions. In Chapter 7, we demonstrated that an equation of state expression for Z could be written in terms of the repulsive Zrep and attractive Zatt contributions. (i.e., Arep and Aatt, aka the “physical” contributions). Wertheim’s original development uses sophisticated statistical mechanics beyond the scope of this introduction. Nevertheless, we can understand his results in terms of contributions to the reaction and equilibrium equations. Whenever we arrive at a set of terms that seems difficult to simplify, we can apply Wertheim’s result as a “clever guess,” and show how this result leads to a self-consistent interpretation for specific physical contributions (e.g., the van der Waals model). Once we have an expression for Achem expressed in terms of ρ0 and T, it can be added to the physical contributions and the equation of state can be applied like any other equation of state.

Because the notation is complicated, we develop this section using pure components, and later generalize the results. Wertheim’s theory is applied to equations of state, so we use the notation x to represent mole fractions in both the vapor and liquid phases and the state of aggregation will be determined by the size of Z. Also, we omit the “A” from xAM when there is only one component. The starting point for Wertheim’s theory is to rearrange the analysis in terms of the true numbers of bonding sites in the fluid. The extent of association is then characterized in terms of the fraction of bonding sites not bonded:

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This “fraction of bonding sites not bonded” is closely related to the fraction monomer, xM. The relevant mass balances are discussed below. To understand Wertheim’s theory, you must understand what is meant by a hydrogen bonding site. A key element of Wertheim’s perspective is to characterize the bonding sites as small, off-center “blisters” of attractive energy. This gives orientational specificity because the sites can only bond if the angle from the left repulsive site to the bonding sites to the right repulsive site is close to 180°; any other orientation would be inconsequential. Furthermore, the smallness of the blister relative to the repulsive site means that three sites cannot bond simultaneously because it would require the third repulsive to overlap with the two that were already bound as shown in Fig. 19.6. This captures the short-range nature, orientational specificity, and steric hindrance that we recognize in hydrogen bonding, and complexation in general.

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Figure 19.6. Wertheim’s perspective on bonding sites. The shaded portions represent the blisters. The lower two molecules are happily bonded, but the upper molecule can’t join the same bond.

Dimer Formation

We begin with association to form a dimer. For the formation of a dimer (denoted with subscript D), Eqn. 19.4 and the sum of true mole fractions can be combined and rearranged to relate the apparent moles n0 and total true moles nT to the fraction of unbonded sites X:

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Dividing Eqn. 19.38 by n0 gives 1 – nT/n0 = (1 – X)/2, where X is the fraction of bonding sites not bonded:

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Noting that the solution must satisfy the mass balance, Eqn. 19.20, xM + xD = 1. Together with the reaction equilibrium (law of mass action, Eqn. 19.34) we may write,

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Substituting Eqn. 19.40 gives 1 – 2X/(1 + X) = 4X2PKaM2D)/(1 + X)2, which can be rearranged to

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Defining

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may seem odd at first glance, but it is one of the major simplifications derived from Wertheim’s theory. We outline his analysis below, but a key step was when Wertheim showed that this conglomeration of symbols can be simplified to

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where g(σ) is the radial distribution function at contact distance σ, εC is the bond energy of the complex, and KC is the bonding volume related to the size of the “blisters.”13 Eqn. 19.42 shows that X can be solved directly from the density and temperature since Δ is a function of ρ and T. Since the density and temperature must be specified in applying the physical contributions of the equation of state anyway, the interjection of this contribution does not complicate matters in the way that solving simultaneous phase and reaction equilibria does. Imagine how cumbersome this model might become if all complexation required iterative solution. The principles are the same, but the feasibility is radically altered.

Aside from the advanced statistical mechanical analysis of Wertheim’s paper, we can appreciate the phenomenology of his analysis in two ways. First, we can recognize Δ as an equilibrium constant of a reversible reaction, the ratio of forward and reverse rates. The forward reaction is proportional to the probability of the sites finding each other. This probability is zero if the density is zero, and it is enhanced by g(σ). The reverse reaction is inhibited by the strength of association. The stronger the bonding energy, the slower the dissociation.14 Second, we can apply the van der Waals model with some simple assumptions. If we assume that bD = 2bM, aDD = 4aMM, and aDM = 2aMM, we obtain a similar result, in the manner of Heidemann and Prausnitz as shown in Fig. 19.7. Applying these assumptions to the fugacity coefficients of the van der Waals equation and substituting into Eqn. 19.42, we obtain,

Image
Image

Figure 19.7. Illustration that aDD = 4 aMM and aDM = 2 aMM are reasonable by adding the number of pair interactions.

Simplifying gives,

Image

Note that the vdW EOS corresponds roughly to g(σ) = 1/(1 – ηP). Also, the equilibrium constant, Ka, can be referenced to the critical temperature and written as,

Image

The “best” expression for ΔCp/R is debatable. Experimental measurements are unlikely to provide sufficient precision to resolve the debate. From a practical perspective, we would prefer a compact expression for Ka. From a theoretical perspective, Wertheim’s analysis is the most sophisticated. Suresh and Elliott15 used Tc as a reference temperature, and showed that Wertheim’s analysis is consistent with the assumption that,

Image

Substitution shows the resulting relation between Ka and a portion of Eqn. 19.44,

Image

where (KacRTc)/(P°) = KC and –ΔHTc / R = εC / k. The superscript “C” denotes a C-type association but the relations can be extended to an AD-type (though an arbitrary reference temperature must be selected for AD interactions).16 Substituting this expression for Ka in 19.46 and g(σ) for 1/(1 – ηP) results in Eqn.Eqn. 19.44, eliminating the need to solve iteratively for the density, monomer and dimer concentrations, and fugacity coefficient ratio. An additional step is required to transform the extent of reaction (implicit in X) into a remarkably simple thermodynamic contribution, Achem.

Low Density

The next objective is to evaluate the impact on Helmholtz energy, Achem, the change in Helmholtz energy due to bonding. As discussed in the introduction, Achem is the chemical contribution as it pertains to an equation of state. Let’s begin by rewriting the Gibbs energy for a single bond at low density, noting that PV = nTRT then.17

Image

Or, on a apparent molar basis, (dividing by n0),

Image

In rearranging, note that Gchem = Δμchem = RT ln Image, where Image is the total fugacity with association fully recognized and f0 is the apparent fugacity based on zero association.

Image

Further noting that

Image

Substituting Eqns. 19.39 and 19.53 into Eqn. 19.51, and recalling X = nM/n0, we have

Image

This turns out to be a very powerful equation.

All Densities

The remarkable aspect of Eqn. 19.54 is that it is accurate for all densities and extents of association, although it has been derived here only for binary association at low density. In fact, the significance of Wertheim’s work is that he provides a rigorous statistical mechanical derivation of this identity at all conditions. Once again, we can support this result phenomenologically through the van der Waals model. Adapting Eqn. 19.38,

Image

This equation shows that there is no overlap of repulsive sites when a hydrogen bond occurs, so the volume occupied by molecules is the same regardless of association. Similarly,

Image

We can express the fugacity of the fluid in two ways, noting that fM = f where fM is defined by the monomer fugacity in the mixture, that is, ln(φM) = ln(fM/xMP), and f is defined by ln(f/P) = (G – Gig)/RT for the “pure” fluid based on the apparent perspective. The expression in terms of φM relates to the true species and uses the fugacity expression for a component in a mixture. The expression in terms of f/P emphasizes that we are still discussing a single pure component,

Image

By Eqns. 19.55 to 19.56,

Image
Image

Equating 19.58 to 19.59, we can immediately cancel terms of ln(1 – ηP) and ρ0aMM. Also noting that ln(PV/n0RT) – ln(PV/nTRT) = ln(nT/n0),

Image

noting that xM·nT/n0 = X, and from Eqn. 19.39 (nT/n0 = (1+X)/2), we obtain,

Image

Recall that ZP = d(A/RT)/dηP. Taking Eqn. 19.54 as a trial solution and checking that it is consistent with Eqn. 19.61,

Image

We can evaluate ∂X/∂ηP through Eqn. 19.42 by differentiating implicitly.

Image

Multiplying and dividing the left side by X and replacing X2Δ with 1 – X, then multiplying and dividing the right side by Δ and replacing X2Δ with 1 – X, we obtain,

Image

Recalling the definition of Δ from Eqn. 19.44,

Image

Substituting Zchem = (–1 + 1/X) / [2(1 – ηP)] gives

Image

Hence we have recovered Eqn. 19.54 and verified our trial solution, using the van der Waals model, without the assumption of low density.The online chapter notes include a demonstrationthat Eqn. 19.54 is also recovered with the ESD model. Altogether, we can thoroughly appreciate the results of Wertheim’s analysis, even if the rigors of Wertheim’s statistical mechanics exceed our current scope. We can derive the framework of the simultaneous reaction and phase equilibria and see the crucial terms requiring simplification. At that point, Wertheim’s “clever guesses” provide a tremendous simplification of an immensely complex problem, all the more remarkable when recognizing that they are thoroughly grounded in a rigorous fundamental analysis.

Given Eqn. 19.54 for the free energy and Eqn. 19.42 to solve for X, the problem is essentially solved. Z = 1 + Zrep + Zatt + Zchem can be solved at a given T and P by iterating on ρ. Then the free energy equation yields the fugacity. The algorithm to solve for apparent density is illustrated in Fig. 19.8. Relative to the numerical solution of binary association in a pure fluid, this result might not seem so impressive. Nevertheless, further analysis shows that the extension to chain association of multiple components requires the same effort from Wertheim’s perspective, whereas the numerical solution quickly becomes overwhelming.

Image

Figure 19.8. Flow sheet for calculating density by the van der Waals associating fluid model.


Example 19.4. The chemical contribution to the equation of state

Assuming Δ is about 1000 at 300 K and ρ = 1.04 g/cm3, estimate Achem/RT, Zchem, and xM of liquid acetic acid. You may assume that bM = 23 cm3/mol and the van der Waals model for Z.

Solution

Referring to Eqn. 19.66, Achem/RT = ln(X) + (1 – X)/2 and Zchem = –(1 – X) / [2(1 – ηP)]

Image

Referring to Eqn. 19.39, xM = 2X/(1 + X) = 0.06037

Comparing xM to X, the true solution is 94% dimer and 97% of the acid molecules exist in the dimer form. It is also interesting that Zchem < –0.5 for this liquid phase. The amount of dimerization would decrease at lower density, and for the gas phase it would be significantly lower, with Zchem smaller in magnitude.


Chain Association

To extend the analysis from dimer formation to model chain formation, the primary adjustment is to assign two sites per molecule, consistent with one proton acceptor (A) and one proton donor (D), as we might expect for an alcohol. We can easily count the number of acceptors and donors in such linear chains by noting that one unbonded acceptor is left in each bonded chain, referring to Fig. 19.9. The equations for donors are entirely symmetrical and are omitted for simplicity. Note that nA (the mole number of acceptors not bonded) is something quite different from nA (the mole number of an “A-mer”). The extent of association is then characterized in terms of the fraction of acceptor sites not bonded, XjA. To see the relationship, consider the mass balances We obtain,

Image
Image

Figure 19.9. Wertheim’s theory of chain association in a two-site model.

But the total number of acceptors is given by noting that there are “j” total acceptors per j-mer,

Image

Note that no refers to the same apparent number of moles discussed previously. Therefore,

Image

There is a further simplification that results from treating the bonding sites instead of the bonding molecules. The fraction of sites bonded can be perceived as a simple product of the bonding probabilities. First, note that the fraction of monomers bonded, xAD, and the fraction of monomers not bonded, XA, must sum to unity.

Image

xAD is the fraction of acceptors that are bonded, regardless of whether they are bonded in monomers, dimers, trimers, ... In principle, the second term is an infinite sum. From an acceptor site perspective, however, we assume that the thermodynamic change from the unbonded state to the bonded state is the same, regardless of the degree of polymerization for that i-mer. That is, adding one more monomer to the end of a chain has the same equilibrium constant regardless of the chain length. Chemically, we have

Image

That transition can be represented by

Image

where XD is the fraction of unbonded donors and ΔAD = ρg(σ) KAD[exp(εAD/kT) – 1] adapts Eqn. 19.44 to AD interactions. The term on the left is the fraction of acceptors that are bonded, and the term on the right expresses the observation that acceptors and donors must be unbonded in order to be available for bonding. By noting that one donor bonds for every acceptor, we see that XA = XD. Then we write Eqn. 19.72 in terms of XA, (XA)2, and ΔAD and we can solve the quadratic equation to obtain

Image

The extension of the Helmholtz energy to chain formation simply applies the same formula developed for dimer formation. This formula accounts for the change in entropy and energy each time a bond is formed. Whether the bond is formed as part of a dimer or part of a chain, the reduction in entropy by forming a bond is the same. So is the energy released by the bond formation. In terms of acceptors and donors for a pure fluid, Eqn. 19.54 becomes,

Image

Eqns. 19.73, 19.74, and 19.44 characterize the chemical contribution for molecules like alcohols. Given a temperature and density, Eqn. 19.44 gives ΔAD, then Eqn. 19.73 gives XA, then Eqn. 19.74 gives A chem, then Eqn. 19.1 gives (A – Aig). Altogether, just one extra step (Eqn. 19.73) is required to compute the Helmholtz energy relative to the van der Waals model, but the rigor of the chemical perspective is greatly enhanced. In the MOSCED model, for example, the contribution involving (αi – αj)(βi – βj) is entirely empirical. It is contrived to give the right sign when mixing acids and bases, but there is no basis for it in theory, and it does not alter the skewness of the excess Gibbs energy. On the other hand, Wertheim’s theory is based on a rigorous derivation relating the molecular scale bonding volume and energy to the macroscopic properties, and fundamentally altering the behavior of the Gibbs energy.

Extension to Mixtures

Analyzing the impact of the chemical contribution on excess Gibbs energy requires extension of Wertheim’s theory to mixtures. The beauty of Wertheim’s perspective is that the extension of the reaction equilibrium relation (Eqn. 19.72) is entirely straightforward. One donor must bond for each bonded acceptor, whether the molecules are mixed or pure. The only issue is which molecule possesses the acceptor and which possesses the donor, but that is a notational detail. Furthermore, the fraction of bonded acceptors on the ith molecule must be in equilibrium with the unbonded donors and acceptors. The only difference is that the transition of an acceptor to being bonded can be effected by any donors, including those on other molecules. We simply need to sum all the transition probabilities and the extension becomes:

Image

where xi is the apparent mole fraction of component i and Nd,i is the number of hydrogen bonding segments in component i. Nd,i accounts for the prospect that one molecule could have several donors, like the hydroxyl sites in polyvinyl alcohol. The extension of Eqn. 19.44 to mixtures becomes

Image

The ordering of the subscripts and superscripts in Eqn. 19.76 provides the notational detail that permits accounting for which bonding site resides on which molecule. For example, by writing εijAD, it is implied that the acceptor is on the ith component and the donor is on the jth. To designate the energy of a donor on the ith component with an acceptor on the jth, we should write εijDA. To clarify, this distinction might be important in a mixture of alcohols (i) and amines (j), for example. An amine is usually a weak proton donor (indicated by its low acidity, αj) but a strong proton acceptor (indicated by its basicity, βj), whereas a typical alcohol has roughly equal acidity and basicity (αi = βi). If we suppose that βi ~ βj, then εiiAD ~ εijAD > εijDA ~ εjjDA. This would mean strong solvation for the amines and negative deviations from Raoult’s law, as observed experimentally. Eqns. 19.75 simply states that the bonding probability for an acceptor on the ith species increases when there are donors on other molecules, and it decreases proportional to the mole fraction when the donor species are diluted by nonassociating species. The precise extent of chemical interaction is controlled by ΔijAD, which could range from zero (for alkanes in water, for example) to a substantial quantity (when mixing carboxylic acids, for example).

The extension of Achem to any number of bonding sites or components becomes,

Image

Briefly, this equation indicates that the change in Helmholtz energy due to bonding is the same regardless of how those bonds are formed. In other words, the reduction in entropy due to bond formation is universal when the packing fraction is unchanged. We know that entropy is the primary contribution because energy does not appear explicitly in Eqn. 19.77. Bonding energy affects Achem implicitly through Δ, because a larger energy gives a larger value of Δ and a smaller value of XiB. We have included XC here to represent dimerization (e.g., carboxylic acid bonding) as something distinct from AD interaction. A’s can only bond with D’s, and this leads to chain formation. On the other hand, C’s can only bond with C’s, confining these sites to dimerization. Although we wrote the equation for these three types of bonds, there is really no limit and Eqn. 19.77 can be extended straightforwardly to many situations. In the description here, XC would not affect XA because C’s can only interact with C’s. If a carboxylic acid (e.g., acetic acid) is to interact with an alcohol (e.g., water), A’s or D’s would need to be included as part of the carboxylic acid segment, in addition to the C’s, as illustrated in Fig. 19.10.18

Image

Figure 19.10. A bonding site model for acetic acid.

Eqns. 19.7519.77 provide a powerful and versatile complement to our treatment of phase equilibria. In Chapters 1012, we might have alluded to hydrogen bonding, for example, as a reason why oil and water do not mix, but our models did not truly recognize it as bonding. The van der Waals models and local composition theories treat attractive energy as spherically symmetric, like the square-well potential. But complexation is stereospecific and this alters the description of the Helmholtz energy. The Helmholtz energy of hydrogen bonding is as different from that of the van der Waals model as Eqn. 19.77 is from /RT.

19.7. Mass Balances for Chain Association

The thermodynamics and phase behavior are sufficiently described by Eqns. 19.75 and 19.77, but you may be curious about the true mole fractions of the species. Furthermore, it is interesting to see how this “fraction of acceptor sites not bonded” is closely related to the fraction monomer, xM. This turns out to be a bit subtle, and it should not distract you from the primary issue of phase behavior. If you are interested, we can use material balances to obtain two simple relations between the true number of moles in the solution, nT, and the apparent number of moles that we would expect if there was no association,19 no. Note that no is the number of moles one would compute based on dividing the mass of solution by the molecular weight of a monomer as taught in introductory chemistry. For example, in 100 cm3 of water one would estimate

no = 100 cm3 · 1.0 g/cm3/(18 g/mole) = 5.556 moles

But how many moles of H2O monomer do you think truly exist in that beaker of water? We will return to this question shortly. Note that each i-mer contains “i” monomers, such that the contribution to the apparent number of moles is i·ni. Note also that the true mole fractions, xi, are given by ni/nT, but it may not look so simple at first.

We begin by noting that Ki = Ki–1 implies that xi = xM(xi–1Δ). This leads to a recursive relation originating with the monomer, so xi = xM(xMΔ)i.

Image

Substituting Eqn. 19.4119.44,

n0 = nT Σ i·xM(xMΔ)i = xM nT [1 + 2(xMΔ) + 3(xMΔ)2 + 4(xMΔ)3 + ...]

This series may not appear to be familiar but it is a common converging series. Referring to series formulas in a math handbook, we find that

Image

Since the mole fractions must sum to unity, we can write a second balance, for xi,

Image

and again recognizing the series,

Image

Substituting xM for (1 – xMΔ) in Eqn. 19.79 results in,

Image

This equation makes clear that the properties of the mixture are closely related to the properties of the monomer.


Example 19.5. Molecules of H2O in a 100 ml beaker

Assuming Δ is about 100 at room temperature and ρ = 1 g/cm3, estimate the moles of H2O monomer in a 100 ml beaker of liquid water.

Solution

Note that the problem statement requests moles of H2O, not (H2O)2 or (H2O)3, and so on, so we are interested in the true number of H2O monomer moles. We know n0 = 5.556 by applying the monomer molecular weight, but the number of monomer moles nM = xM·nT will be significantly less. Proceeding, using Eqn. 19.73,

Image

Therefore, the true number of moles is 100 times less than the apparent number of moles.


19.8. The Chemical Contribution to the Fugacity Coefficient and Compressibility Factor

The solution to phase equilibrium problems can be achieved in the manner of Chapter 15 (Eqn 15.20), where Eqns. 19.1 and 19.2 describe the enhanced equation of state. Eqns. 19.7519.77 completely characterize the temperature, density, and composition dependence of the chemical contribution to Helmholtz energy. The Zchem contribution is implied, but requires differentiation as in RT·Zchem = –V∂(AAig)/∂V. Similarly, the fugacity coefficient is implicitly determined through differentiation. Nevertheless, the differentiation can be complicated relative to the fugacity coefficient of the van der Waals model. The summation of Eqn. 19.77 means that terms like ∂Xi/∂nj contribute and Eqn. 19.75 implies a nonlinear system of equations that must be solved to determine these contributions. For example, consider a mixture of three alcohols, with Image and Nd,i=1 for all i. Eqn. 19.75 implies that

Image

The only way to fully determine all ∂Xi/∂nj is to apply Eqn. 19.75 eight more times to obtain nine equations for the nine unknown values implied by ∂Xi/∂nj. Once again, chemical theory seems to become impractical.

Fortunately, this particular nonlinear system of equations possesses subtle but advantageous properties. Briefly, there are many symmetries in the calculus that lead to surprising simplifications when cleverly manipulated. Michelsen and Hendriks showed that Achem can be rewritten as the stationary point of a generalized function Q.20 The term “stationary point” refers to a condition where Image, for all i and B. At this time, there does not appear to be any underlying thermodynamic significance to the function Q or its stationary point. We apply it here merely as a mathematical device. The beauty of the generalized function is that derivatives with respect to Image can be separated from derivatives with respect to V or nj. By chain rule, and using Image at the stationary point,

Image
Image

The generalized function, Q, can be inferred by adding and subtracting the defined term

Image

from Achem. The term h can then be rearranged through Eqn. 19.75 to obtain a function that is quadratic in acceptor/donor contributions by explicitly recognizing A/D contributions in place of the generic B. The algebra proceeds as,

Image

Expanding h in terms of Image and Image explicitly, and substituting for Image and Image,

Image

Now we recognize the generalized function with removal of the stationary point constraint.

Image

Note that the summation over Image goes to zero when differentiated at the stationary point, so the term involving –h/2 is the only one that matters. Then we can take advantage of Eqns. 19.84 and 19.85 to obtain,

Image

Differentiating Eqn. 19.88,

Image
Image
Image

The derivative of ΔijAD is straightforward, similar to Eqn. 19.54. Once again, the chemical contribution appears at first to be hopelessly complicated, but clever insights reduce the computational complexity to a level comparable with the Peng-Robinson model.

The computational complexity of Eqns. 19.7519.77 can be further reduced in the special case where Image, which we refer to as the square root combining rule (SRCR). In general, Eqns. 19.7519.77 require an iterative solution, as illustrated in Example 19.7. An initial guess for the iterations is

Image

where Image for all i,B. Note: Image. This simplification is accurate for alcohols and hydrocarbons, but not for alcohols and amines.21

This concludes our analysis of chemical contributions to phase equilibrium. Eqns. 19.7519.77 and 19.9119.93 permit solution of Eqns. 19.1 and 19.2 for mixtures as well as pure fluids and computation of the fugacity coefficients to perform any phase equilibrium determination. Wertheim’s theory of solution thermodynamics is more challenging than that of van der Waals or local compositions, but it replaces the empirical conjectures of those models with rigorous analysis that has been verified with molecular simulations. The perspective offered by Wertheim’s theory suggests further application of the basic equations to describe nonspherical molecules. This extension provides a self-consistent and rigorous description of the thermodynamics of all sizes of molecules, as discussed in the following section.

19.9. Wertheim’s Theory of Polymerization

Now that we have an accounting for the thermodynamics of bond formation, it is natural to wonder what happens to the thermodynamics as the bond energy approaches infinity. This would be a natural limit for covalent bond formation. Having a theoretical basis for nonspherical molecules would be a big step forward, considering that all theories discussed until now have been based on spherical molecules. Of course, we added correction terms like α(T, ω) to the Peng-Robinson model, but this was done with no theoretical basis. Wertheim’s theory provides an opportunity to develop meaningful guidelines for shape effects.

The key step is to find the contribution to the equation of state from forming a bond in the limit of infinite bond energy. The result for binary association, Eqn. 19.54, is convenient to illustrate the key points. At first glance, the limit may not seem obvious, because the X term in Achem must approach zero and the log term would then be undefined. This issue can be resolved by substituting, say, 1 – X = X2Δ. We use Abond to denote the covalent nature of the bonds.

Image

Example 19.6. Complex fugacity for the van der Waals model

A sample calculation with a specific reference equation of state should clarify these results. Let K12AD = K22AD = 0.72 cm3/mol and ε12AD = ε22AD = 20 kJ/mol, b1=27.5 and b2=20.4 cm3/mol.

a. Derive Zchem and ln(φkchem) adapting the definition of Δ from Eqn. 19.44.

b. Evaluate the expressions for trimethylamine(1) + methanol(2) at x1 = 0.5, ρ = 0.0141mol/cm3, and T = 300K.

Solution

a. ΔijAD = ρKAD(exp(βεijAD) – 1)/(1 – ηP) = Δ.

Eqn. 19.65 shows that ηP∂Δ/∂ηP = Δ/(1 – ηP)

Substituting gives Zchem = –0.5h/(1 – ηP). For ln(φkchem) Eqn. 19.93 requires n∂(ΔijAD/n)/∂nk, Image

Similarly, n∂(ΔijDA/n)/∂nk = (ΔijDAbkρ)/(1 – η). Substituting this result into Eqn. 19.93 gives,

Image

b. Evaluating these expressions, Δ11DA = Δ12DA= 0 because trimethylamine (TMA) has no donors, so X1D = 1. It may seem odd to represent X1D = 1 when there are no donors, but site occupation is impossible when Δ = 0 for that site.

Image
Image
Image

To solve Eqns. 19.93 for X1A, X2A, and X2D,

Image
Image
Image

This gives three equations. Rearranging (below) shows that X1A = X2A. We can replace these to obtain a quadratic equation in terms of X2D. Usually, we would need to iterate to solve for X2D.

Image
Image
Image
Image
Image
Image
Image

This shows that X2D is almost completely bonded.

By Eqn. 19.88,

h = x12X1AX1DΔ11AD + 2x1x2X1AX2DΔ12AD + x22X2AX2DΔ22AD + x12X1DX1AΔ11DA + 2x1x2X1DX2AΔ12DA + x22X2DX2AΔ22DA

Substituting x1, x2, and X gives

h = 0 + 2·x1x2·X1AX246.4 + x22X2AX2D·46.4 + 0 + 0 + x22X2DX2A·46.4 = 0.960
Zchem = –0.5h/(1 – ηP) = –0.480/(1 – 0.338) = –0.725

By Eqn. 19.92,

Image

There are several points of interest in this result. The acceptors in this mixture outnumber donors by two to one. Therefore, it is impossible that XiA< 0.5, and, in fact, X2D ~ 2·(X1A – 0.5) because the lack of donor saturation is reflected twice, in X1A and X2A. The compressibility factor is depressed in a simple way that sums over all donors and acceptors, but the fugacity is depressed more for the alcohol than for the amine. There are three ways for the alcohol to interact, but only one for the amine, so the depression of the fugacity is much greater. On the other hand, the fugacity of the alcohol is depressed less in the mixture than in the pure fluid because relatively fewer acceptors are bonded (ln(φ2chem)= –6.105 at x2 = 1). So the activity of the alcohol in the mixture is relatively enhanced by hydrogen bonding while the activity of the amine is depressed at all compositions.



Example 19.7. More complex fugacity for the van der Waals model

Evaluate the expressions for Zchem and ln(φkchem) of trimethylamine(1) + methanol(2) at x1 = 0.4, ρ = 0.0141 mol/cm3, and T = 300 K. Let K12AD = K22AD = 0.72 cm3/mol and 1.25ε12AD = ε22AD = 20 kJ/mol, b1 = 27.5, and b2 = 20.4 cm3/mol.

Solution

The difference between this example and the previous is that ε12AD ≠ ε22AD, indicating that the solvation is slightly weaker than the alcohol association. Because of this lack of symmetry, an iterative solution for X is required. Recalling part (a) of the previous example,

Zchem = –0.5h/(1 – ηP)

For ln(φkchem): –∑ xj(XkA XjDΔkjAD + XkDXjA ΔkjDA) – 0.5hbkρ/(1 – ηP)

Substituting the mole fractions and solving for Δ’s,

b = 0.4·27.5 + 0.6·20.4 = 23.4; ηP = 0.0141·23.4 = 0.328. This is slightly less than Eqn 19.99.

Δ22AD = ρKAD(exp(βε22AD) – 1)/(1 – η) = 45.8. Δ12AD = 9.21; Δ11DA = Δ12DA = 0.

1 – X1A = 0.5X1AX2DΔ12AD; 1 – X2D = 0.5X1AX2DΔ12AD + 0.5X2AX2DΔ22AD;

1 – X2A = 0.5X2AX2DΔ22AD;

X1A = 1/(1 + 0.4X2DΔ12AD); X2A = 1/(1 + 0.6X2DΔ22AD);

X2D = 1/(1 + 0.4X1AΔ12AD + 0.6X2AΔ22AD);

Unlike the previous example, an explicit solution is not found. The previous example was contrived to achieve an exact solution, but this is rarely possible. Normally, we must iterate to achieve a numerical solution. It is convenient to guess X2D, then compute X1A and X2A, then check the new value of X2D based on X1A and X2A. To initialize X2D, a reasonable estimate can be based on a variation of the solution for ΔkjAD =(ΔkkADΔjjAD)1/2.

Applying Eqn. 19.94,

(–1 + 1/X2D) ≈ Σ xjΔ2jAD/[1 + (ΔjjAD)1/2] = 0.122 X1A = 0.597; X2A = 0.230; X2D = 0.105;

Five more iterations give, X1A = 0.678; X2A = 0.297; X2D = 0.0855. Five iterations is actually a large number because this particular mixture deviates substantially from the SRCR.

X1A = 1/(1 + 0.4·0.0857·9.21)= 0.678

X2A = 1/(1 + 0.6·0.0857·45.8)= 0.297

X2D = 1/(1 + 0.4·0.678·9.21 + 0.6·0.297·45.8) = 0.0855

Substituting into Eqn. 19.88

h = x12X1AX1DΔ11AD + 2x1x2X1AX2DΔ12AD + x22X2AX2DΔ22AD

+ x12X1DX1AΔ11DA + 2x1x2X1DX2AΔ12DA + x22X2DX2AΔ22DA

h = 0 + 2·x1x2·X1AX2D·9.21 + x22X2AX2D·45.8 + 0 + 0 + x22X2DX2A·45.8 = 1.097

Zchem = –0.5h/(1 – ηP) = –0.480/(1 – 0.328) = –0.816

ln(φ1chem) = ln(X1A) – [x1(0 + 0) + x2(X1AX2DΔ12AD + 0)] – 0.5hbkρ/(1 – ηP) = –2.228

ln(φ2chem) = ln(X2A) + ln(X2D) – [x1(0 + X2DX1AΔ21DA)

+ x2(X2AX2DΔ22AD + X2DX2AΔ22DA)] – 0.5hbkρ/(1 – ηP) = –5.855

These results show that a 20% change in ε12AD compared to Example 19.6 gives a 500% change in Δ12AD. That is fairly sensitive. This change in Δ12AD is primarily responsible for the increase in X1A from 0.520 to 0.679 and the decrease of X2A from 0.520 to 0.298. Overall, the chemical contributions are slightly stronger because the composition of amine has been reduced.


Eqn. 19.95 is helpful when Δ→∞ because Z can be obtained by differentiation of A. Taking the derivative,

Image

From a model for Δ, the bonding contribution to the EOS results. For example, if Δ is given by the van der Waals model,

Image

Generalizing this result to a chain with m segments, there are (m–1) bonds per chain. For example, continuing with the vdW model,

Image

This is essentially Wertheim’s theory of polymerization, although Wertheim specifically treated the case resulting in a mixture with a range of molecular weights and average degree of polymerization of <m>.22

19.10. Statistical Associating Fluid Theory (The SAFT Model)

Shortly after Wertheim’s work appeared, Chapman et al. formulated an equation of state that incorporated the bonding contribution and complexation as well as the disperse repulsive and attractive terms. Their perspective was to treat any solution in the conventional way as a fluid of independent spheres, then to add the bonding contribution required to assemble the spheres into chains. Then the equation of state becomes

Image

Adding and subtracting (1 – m) to isolate the ideal gas limit,

Image

Recognizing the significance of Wertheim’s statistical mechanical theory for associating (and solvating) systems, Chapman et al. named their model SAFT. In principle, any equation of state can be applied for the dispersion interactions, but Chapman et al. adopted the Carnahan-Starling model for the hard-sphere systems, including the Mansoori-Carnahan-Starling-Leland (MCSL) model for hard-sphere mixtures.23 That choice has remained consistent in most variations of the SAFT model, but several alternatives have been adopted to describe the attractive dispersion interactions, Zatt. The original version suggested using second order perturbation contributions of the Lennard-Jones fluid for Zatt.24 Huang and Radosz adopted a 20-parameter equation of state for argon (HR-SAFT).25 More recently, Gross and Sadowski took a slightly different approach.26 They treated the hard-sphere and chain contributions in the usual manner of SAFT, but treated Zatt by a second order perturbation theory that takes the tangent-sphere-chain as the reference fluid, instead of the tangent spheres themselves. They refer to their method as Perturbed Chain SAFT (PC-SAFT). In the conventional SAFT approach, Zatt/m would be a universal curve, but PC-SAFT shows a mild variation in this quantity with chain length. We focus our discussion on PC-SAFT for the most part.


Example 19.8. The SAFT model

Chapman et al. (1990)a suggested that second order perturbation theory could be applied for the segment term of the SAFT model, with the hard-sphere contribution described by the Carnahan-Starling (CS) equation and the Aatt given by:

Aatt/RT = A1βε + A2(βε)2

A1 = –11.61ηP – 8.28ηP2 – 5.24ηP3 + 34.21ηP4

A2 = –25.76ηP + 181.87ηP2 – 547.17ηP3 + 529.00ηP4

Express this model as an equation of state for alcohols, including Zchain; that is Z = Z(mP,βε).

Solution

The CS equation is given by Zhs–1 = 4ηP(1 – ηP/2)/(1 – ηP)3. This corresponds to g(σ) = (1 – ηP/2)/(1 – ηP)3 and, by Eqn. 19.44, Δ = ρ(1 – ηP/2)KAD(exp(βεijAD) – 1)/(1 – ηP)3. Then, ∂lnΔ/∂lnηP = ηP{Δ/η – Δ/(2 – ηP) + 3Δ/(1 – ηP)} = Δ{1 + (5ηP – 2ηP2)/[(2 – ηP)(1 – ηP)]}.

Substituting,

Zchain = Zbond + (m – 1) = –(m – 1)(Zbond – 1) = –(m – 1)(5ηP – 2ηP2)/[(2 – ηP)(1 –ηP)]

Zchem = –0.5h∂lnΔ/∂lnηP = –(1 –XA) ∂lnΔ/∂lnηP = –(1 – XA){1 + (5ηP – 2ηP2)/[(a – ηP)(1 – ηP)]}; and Zseg = Zhs – 1 + Z1βε + Z2(βε)2, where Z1 = –11.61ηP – 16.56ηP2 – 15.72ηP3 + 136.84ηP4 and Z2 = –25.76 ηP + 363.74ηP2 – 1641.51ηP3 + 2116.00ηP4.

Putting it all together, Z(mP,Tr)= 1 + m[4ηP(1 – ηP/2)/(1– ηP)3 + Z1βε + Z2(βε)2] – (m – 1)(5ηP – 2 ηP2)/[(2 – ηP)(1 – ηP)] – (1 – XA){1+(5ηP – 2ηP2)/[(2 – ηP)(1 –ηP)]}.

where XA = [–1 + (1 + 4Δ)1/2]/(2Δ), Δ = ηP(1 – ηP/2)(KAD/b)(exp(Hβε) – 1)/(1 – ηP)3, H = εijAD/ε. Since all the terms can be computed based on m,ηP, βε, the equation of state is complete.


a. Chapman, W.G., Gubbins, K.E., Jackson, G., Radosz, M. 1990. Ind. Eng. Chem. Res. 29:1709.

The tangent-sphere-chain that lays the foundation of all SAFT models is well defined and relatively simple to treat by molecular simulation. This makes it possible to evaluate the accuracy of Wertheim’s theory for the hard chain reference system. With only slightly more effort we can also evaluate the accuracy for a reference fluid of fused sphere chains with 110° bond angles, as in n-alkane chains. As shown in Fig. 19.11, the comparison is quite favorable in both cases, showing that Wertheim’s theory and the related SAFT models have a solid theoretical foundation that is validated by molecular simulation.

Image

Figure 19.11. Comparison of molecular simulations, the van der Waals equation, and the ESD equation of state for Zrep. Nd is the number of spheres in a chain.

The PC-SAFT model has the same form as Example 15.8 except for Zattseg.

Image
Image
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Eqns. 19.117 and 19.118 include 42 coefficients listed in the original reference.

You might wonder whether there is a simpler form of the SAFT model that is more sophisticated than the van der Waals model, but not as complicated as the PC-SAFT model. Such a model would be convenient for illustrating the key advantages of an association model without losing the simplicity of a cubic model like the PR model. One alternative is simply to add the association contribution of Example 15.8 to the PR model. This is the basis of the CPA model of Kontogeorgis et al.27 This is a feasible model and it has been applied in many practical settings, but it is not entirely faithful to the Wertheim perspective in that it uses g(σ) from one model and Zhs from another, while ignoring Zchain completely. Another alternative is to reconsider the ESD model in light of the SAFT analysis. Then we can rewrite the ESD model as a “simplified SAFT” model:28

Image

In this form, we recognize that g(σ)=1/(1–1.9ηP) provides consistency as a SAFT model. Then,

Image
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19.11. Fitting the Constants for an Associating Equation of State

To this point in the discussion, we have assumed that the constants needed for a fluid are available. However, association models add complexity in the sense that two association parameters must be characterized in addition to the usual size (b), energy (a or ε), and shape (k, m, q, or c). One simple approach is to assign standardized values to the bonding volume and energy. For example, alcohols can be assigned an energy of 17 kJ/mol. Aldehydes, amides, amines, and nitriles can be assigned an energy of 5.2 kJ/mol. Given the bonding energy and volume, three parameters remain to be determined in a manner equivalent to three parameter corresponding states.

The simplest case is when the association energy is zero. Then the critical method can be applied in the usual way. For the ESD model, this is especially simple, because it is cubic. The approach of setting (Z–Zc)3 = 0 can be applied over a range of values of c from 1 to infinity. For each value of c, the acentric factor can be computed once the critical point has been determined. Then the value of c can be regressed as a function of acentric factor, and values of b and ε can be correlated as functions of c. This results in the following correlations,

Image
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where a = 1.9(9.5qk1) + 4ck1 and k1 = 1.7745

An interesting implication of this result is that Zc in the infinite chain limit. Lue et al. showed that this is a general result for all SAFT models, despite the experimental observation that Zc appears to approach zero for long chains.29 They attributed this deficiency to inaccurate characterization of the intramolecular interactions by SAFT models at low density and high temperature.

At a slightly higher level of complexity, the bonding energy and volume can be treated as adjustable parameters and regressed to minimize deviations in vapor pressure and density. This is the predominate method for most SAFT models. In fact, the critical point method has been systematically avoided for SAFT models other than the ESD model. The regression method requires extensive pure component data. Unfortunately, sufficient data exist for relatively few compounds to regress optimal values or even critical values, and those regressions have already been performed and the results are available. Therefore, the important problem is to characterize the constants when data are few or nonexistent.

Emami et al. have formulated a convenient method that requires little or no experimental data.30 Their method has been developed for the ESD, HR-SAFT, and PC-SAFT models. The method refers to standard literature correlations for ΔHvap and ρliq298.15 and provides UNIFAC group contribution correlations for the shape factors. This method is facilitated by spreadsheets that are available in Chapter 15 supplements on the textbook’s web site.

Implementations of ESD, HR-SAFT, and PC-SAFT are available from the various authors. A convenient set of implementations that also provides the capability to generate global phase diagrams is available from Cismondi et al.31

19.12. Summary

A simple way of remembering the qualitative conclusions of this analysis can be derived by considering the behavior of the fugacity coefficient. One can easily demonstrate that the fugacity coefficient of the monomeric species is insensitive to the extent of association if it is expressed on the basis of the true number of moles in the associated mixture. But all of our phase equilibrium algorithms are based on the fugacity divided by the apparent mole fraction; for example, the flash algorithm is the same for any equation of state. The relation between the two fugacity coefficients is given by

Image

This means that we must simply multiply the fugacity coefficient from the usual equation of state expression by the ratio of true to apparent mole fraction. Since this ratio is always less than one, we see that the effect of association is to suppress the effective fugacity of the associating species.

For mixtures, elevation of the monomer mole fraction by breaking the association network accounts for VLE quite accurately. Fig. 19.12 illustrates the benefit of a chemical physical model relative to a purely physical equation like the Peng-Robinson equation. The figure depicting the methanol + cyclohexane system shows the improved accuracy in representing simultaneous LLE and VLE when hydrogen bonding is recognized. Notice the change in the skewness of the curves when hydrogen bonding is applied. The hydrogen bonding model is accomplishing this change in skewness as a clear and understandable explanation of the physics. By contrast, the van Laar model in Chapter 11 altered the skewness by adjusting constants that ignore the physics. We would expect that the stronger physical basis would provide greater capability for extrapolations to multicomponent mixtures. Unfortunately, remarkably few multicomponent studies have been performed to date. Hence, there is no single recommended method for treating nonideal multicomponent solutions at this time.

Image

Figure 19.12. T-x-y diagram for the system methanol + cyclohexane. Data from Soerensen, J. M.; Arlt, W. Liquid-Liquid Equilibrium Data Collection; DECHEMA: Frankfurt/Main, 1979 Vol. V, Part 1.

From a theoretical perspective, however, we may still feel uncomfortable with having made several sweeping assumptions with little justification besides their making the equations easier to solve. This may not seem like much of an improvement over local composition theory. On the other hand, the assumptions could be reasonably accurate; they simply need to be tested. As in the case of local composition theory, molecular simulations provide an effective method of testing the assumptions implicit in the development of a theory. Fig. 19.13 shows a comparison to molecular simulation results and to Wertheim’s theory.32 It can be seen that the above assumptions lead to reasonably accurate agreement with the molecular simulations and therefore they represent at least a self-consistent theory of molecular interactions.

Image

Figure 19.13. DMD-B simulation of hard dumbbell methanol with reduced bond length l/σ = 0.4, at T = 300 K and NAεHB/R = 2013 K. TPT1 theory is an adaptation of Wertheim’s theory.

This is not to say that chemical theory completely solves all problems. Local composition effects are real and should be incorporated into the mixing rules. Evidence supporting this step can be found in the anomalous behavior of the methane + hexane system. If such local composition effects are so prominent for nonassociating solutions, they should be accounted for at all times. As an example of other problems, the association network of water seems to be different enough from that of alcohols that a more sophisticated model will be necessary to represent difficult solutions like hydrocarbons + water to the high degree of accuracy (ppm) required by organizations like the Environmental Protection Agency. Furthermore, the solvation between different species can be extremely complicated and require substantially more investigation to develop reliable engineering models. Finally, it is well known that “nonadditive” effects play a significant role in aqueous and alcoholic solutions.33 That is, the energy of network formation changes in a way that cannot be understood based only on a simple potential model for a single water molecule. These peculiarities may seem esoteric, but they are key obstacles which prevent us from revealing many of the mysteries of biomolecular solutions. Other areas of application such as polymer solutions involving association, as in nylon, can also be imagined. These are the areas which remain to be explored. The methods for engaging in this exploration predominantly involve mathematically formalizing our treatment of the radial distribution function through applications of statistical mechanics. At this point, we leave this engagement to the “satisfaction and good fortune” of the reader.

19.13. Practice Problems

P19.1.

a. A gas-phase A+B system solvates A + B Image AB with Ka = 0.5 at 298.15 K. Calculate the compressibility factor, apparent fugacity coefficients, and the true vapor phase mole fractions in a mixture at 298.15 K and 2 bar when the apparent concentration is yA = 0.45 using ideal chemical theory.

b. A liquid-phase A+B system solvates A + B Image AB with Ka = 0.7 at 298.15 K. Calculate the true liquid-phase mole fractions in a mixture at 298.15 K and 1 bar when the apparent concentration is xA = 0.45 using ideal chemical theory.

c. A gas-phase A+B system associates 2A Image A2 with Ka = 0.5 at 298.15 K. Calculate the compressibility factor, apparent fugacity coefficients, and the true vapor phase mole fractions in a mixture at 298.15 K and 2 bar when the apparent concentration is yA = 0.45 using ideal chemical theory.

19.14. Homework Problems

19.1. Consider a dilute isothermal mixing process of acetic acid(1) in benzene(2). For the dilute region (say, up to 5 mol% acid), draw schematically curves for the following:

Image versus x1; Image versus x1; Image versus x1.

Briefly justify your schematic graphs with suitable explanations. Take standard states as the pure substances.

19.2. Acetic acid dimerizes in the vapor phase. Show that the fugacity of the dimer is proportional to the square of the fugacity of the monomer.

19.3. By assuming that the equilibrium constant for each successive hydrogen bond is equal in the generalized association approach developed in this chapter, what assumptions are being made about the Gibbs energy, enthalpy, and entropy for each successive hydrogen bond?

19.4. The value of the excess Gibbs energy at 298 K for an equimolar chloroform(1) + triethylamine(2) system is GE = –0.91 kJ/mol. Assuming only a 1-1 compound is formed, model the excess Gibbs energy with ideal chemical theory, and plot the P-x-y diagram.

19.5. Suppose that, due to hydrogen bonding, the system A + B forms a 1-1 complex in the vapor phase when mixed. Neither pure species self-associates in the vapor phase. The equilibrium constant for the solvation is KAB = 0.8 bar–1 at 80°C. At 80°C, a mixture with a apparent (bulk) mole fraction of yA = 0.5 is all vapor at 0.78 bar. Calculate the fugacity coefficient of A in the vapor phase using ideal chemical theory at this composition, temperature, and pressure. Use hand calculations.

19.6. At 143.5°C, the vapor pressure of acetic acid is 2.026 bar. The dimerization constant for acetic acid vapor at this temperature is 1.028 bar–1. The molar liquid volume of acetic acid at this temperature is 57.2 cm3/mol. Calculate the fugacity of pure acetic acid at 143.5°C and 10 bar. Use hand calculations.

19.7. An A + B mixture exhibits solvation in the liquid phase, which is to be represented using ideal chemical theory. Because of a Lewis acid/base interaction, the system is expected to form a 1-1 compound.

a. Which one of the following sets of true mole fractions are correct for the system using an equilibrium constant of 3.2 to represent the complex formation at an apparent composition xA = 0.4?

Image

b. Based on your answer for part (a), what are the apparent activity coefficients of A and B?

19.8. Water and acetic acid do not form an azeotrope at 760 mmHg. The normal boiling point of acetic acid is 118.5°C. Therefore, at 118.5°C and 760 mmHg, the mixture will exhibit only vapor behavior across the composition range. The following equilibrium constants have been fitted to represent the vapor-phase behavior:34

Image

a. Let compound A be acetic acid and B be water. Calculate the true mole fractions of all the species from yA = 0.05 to yA = 0.95. At what apparent mole fraction does each specie show a maximum true mole fraction? What is the relation of this apparent mole fraction with the compound’s stoichiometry?

b. Plot the fugacity coefficient of acetic acid and water as a function of acetic acid mole fraction. What is the physical interpretation of the rapid change of the acetic acid fugacity coefficient in the dilute region, if the water fugacity coefficient doesn’t show such a dramatic trend in its dilute region?

19.9.

a. The molar Gibbs energy of mixing (per mole of superficial solution) for a liquid binary system

Image

expressed extensively, this becomes

Image

Introduce the concepts of chemical theory into Eqn. 19.131 to prove that the Gibbs energy of mixing is equivalently given by the sums over true species,

Image

where Ki is unity for the monomers. Hint: nA = ∑aini.

b. Show that on a molar basis for an ideal chemical theory solution that has only solvation, per mole of true solution, the equation reduces to

Image

and provide a physical interpretation relating the Gibbs energy of formation to K.

c. Considering a system where A associates, show that the Gibbs energy of mixing by ideal chemical theory is per mole of true solution given by

Image

Below are tabulated calculations for ideal chemical theory for an A + B system where A forms dimers with K=140. Use Eqns. 19.134 and 19.130 to tabulate the respective Gibbs energies of mixing over RT. Then tabulate nT/n0 (the number of true moles divided by the number of apparent moles) and multiply Eqn. 19.134 by this number and compare with Eqn. 19.130.

Image

19.10. Furnish a proof that the concentration of true species i is maximum at composition xA* = ai/(ai + bi), xB* = bi/(ai + bi) where ai and bi are given in Eqn. 15.1. [Hint: The Gibbs-Duhem equation is useful for relating derivatives of activity.]

19.11. Show that the result for Zassoc is obtained by taking the appropriate derivative of Aassoc.

19.12. Use the ESD equation to model the monomer, dimer, and trimer in the vapor and liquid phases of saturated water at 373 K, 473 K, and 573 K. How does the monomer fraction of saturated vapor change with respect to temperature? How does monomer fractions of saturated liquid change?

19.13. Derive the equations for determining the critical point of the ESD equation35 based on εHB and KAD being zero by noting that dF/dZ = 0 and d2F/dZ2 = 0, where F = Z3 + a2Z2 + a1Z + a0 when hydrogen bonding is negligible.

19.14. Plot P against V at 647.3 K for water with the ESD equation using the characterization analogous to Eqns. 15.73–15.76. Apply the equal area rule and determine the vapor pressure at that temperature. Raise the temperature until the areas equal zero and compare this temperature to the true value of 647.3 K.

19.15. Apply the ESD equation to the methanol + benzene system and compare to the data in Perry’s Handbook based on matching the bubble pressure at the azeotropic point. Prepare a T-x-y diagram and determine whether the ESD equation indicates a liquid-liquid phase split for any temperatures above 250 K. Perform the same analysis for the Peng-Robinson EOS. Do you see any differences? Compare to Fig. 15.3 on page 599.

19.16. Use the ESD equation to estimate the mutual LLE solubilities of methanol and n-hexane at 285.15 K, 295.15 K, and 310.15 K. Use the value of kij = 0.03 as fitted to a similar system in Fig. 19.12 on page 804.

19.17. The hydrogen halides are unusual. For example, here are the critical properties of various hydrogen halides:

Image

Experimental data for the vapor pressure and the apparent molecular weight of HF vapor are as follows:

Image

These apparent molecular weights have been found by measuring the mass density of the vapor and comparing with an ideal gas of molecular weight 20. Assuming that HF forms only monomers and hexamers, use the ESD EOS with c = q = 1 for both monomer and hexamer to match this value of Zc, and fit the vapor density data as accurately as possible in the least squares sense.

19.18.

a. Compute the values of Ka´, a/bRTc, xMc, and bρc for methanol and ethanol according to the van der Waals hydrogen bonding equation of state.

b. Assuming an enthalpy of hydrogen bonding of 24 kJ/mole and ΔCP = –R, calculate the acentric factors for methanol and ethanol according to the vdw-HB EOS.

19.19. Derive the association model for the Peng-Robinson model, using the van’t Hoff formula with ΔCP/R = –1. Extend the homomorph concept by applying ωPR = ωhomo, where ωhomo is the acentric factor for the nonassociating homomorph and ωPR is the acentric factor substituting for the associating compound into the Peng-Robinson expression for a.

a. For methanol, determine the values of Ka′, b, a/bRTc, xMc that match the critical point.

b. Determine the vapor pressure at Tr = 0.7 for methanol assuming a hydrogen bonding energy of 15 kJ/mole, and compare to the experimental value. Infer the acentric factor and compare to the experimental value.

c. Plot log Prsat versus Tr–1 for the Peng-Robinson EOS and the Peng-Robinson hydrogen bonding EOS, and experiment.

19.20. Acetic acid has a much stronger tendency to dimerize than any alcohol. Therefore, it is not reasonable to assume that Ka2 = Ka3 = ... for acetic acid. The assumption is reasonable for Ka3 = Ka4 = ..., however. We can supplement the theory by adding a single additional equation for the dimerization reaction with an effective equilibrium constant equal to the ratio of the true Ka2 divided by the linear association value. Assume that the linear association is negligible for the saturated vapor at ~300–350 K.

a. Determine the value of Ka2 that matches the saturated vapor compressibility factor in that range. Let NAεHB/R = 4000 K for the dimerization.

b. Determine the values of KAD, b, αc, εHB that match the critical point.

c. Determine the values of KAD, b, αc, εHB that match the vapor pressure at Tr = 0.7 for acetic acid.

19.21. Extend the ESD equation to compounds with more than one bonding segment.

a. Consider ethylene glycol as a compound with both an associating head and tail. Extend the mixture analysis to treat this case with two bonding segments (Nd = 2).

b. Treat water by the same model noting that water is merely a “very short glycol.” Determine the acentric factor of the Peng-Robinson hydrogen bonding EOS with ΔH = 15 kJ/mole.

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