*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, *A ^{chem}*. 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,

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.

**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.

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

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.

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 *T _{c}* for an associating component will be significantly higher than a nonassociating component of similar structure (e.g., H

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 temperature^{4} 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.

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, –OCH_{3}, –N(CH_{3})_{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.

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 *k _{ij}*. The

In a generalized binary solution,^{7} the *i ^{th}* complex can be represented by the general form

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 *n _{A}* and

The concentrations and 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, , . Note that *x _{A}* (the apparent mole fraction) is not the same as (the

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 *n _{A}* moles of

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

*n _{A}* = 1

*n _{B}* = 0

The chemical potentials of the true species are designated by μ* _{AM}*, μ

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

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

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

By comparing 19.6 and 19.7 we conclude

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.

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

and for the monomer,

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,

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

which leads to

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

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

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

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

Defining an activity coefficient, α* _{AM},* of the true monomer species, the chemical potential is

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

where the exponential term is a constant at a given temperature. The symmetrical convention of apparent activity requires . For a nonassociating species, the exponential term of Eqn. 19.17 is unity by Eqn. 19.12, and thus , . For an associating species 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 . As such, we write

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

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 α* _{AM}*=1=α

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:

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

Dividing numerator and denominator by the true number of moles, *n _{T},*

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

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

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.

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 *x _{AM}*<

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

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

Once the *K _{i}* are known, then

For ideal chemical theory applied to the vapor phase, the *x _{i}* are replaced with

Eqns. 19.20 and 19.22 then become

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 *K _{i}* values are known,

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 *n _{o}* apparent moles. However, experimentally, in the same total volume, there would be a smaller number of true moles

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

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, , and ; therefore,

Dividing numerator and denominator by *n _{T,}*

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,

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

At *P* = 0.01 bar, for acetic acid, *y _{AM}* = 0.4,

Since the system is pure, *y _{A}* = 1 and the fugacity coefficient on the left-hand side will be for a pure species. Since the model uses ideal chemical theory, . Therefore

The same procedure can be repeated for propionic acid, however it will be even more nonideal. The answers are: y* _{AM}* = 0.333,

**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, C_{4}H_{8}O_{2}, 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 A_{2}B. McGlashan and Rastogi^{a} have studied this system and report the liquid phase can be modeled with ideal chemical theory using *K*_{AB} = 1.11, *K _{A2B}* = 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 A_{2}B compound, *a _{i}* = 2,

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

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

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

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 *x _{AM}* and

For chemical-physical theory applied to the vapor phase,

Eqns. 19.20 and 19.22 then become

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 Harris^{8} 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 Elliott^{12} 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.

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, *A ^{chem}* 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, *A ^{chem},* and consequently

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 *x _{AM}* 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

This “fraction of bonding sites not bonded” is closely related to the fraction monomer, *x _{M}*. 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.

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 *n*_{0} and total true moles *n _{T}* to the fraction of unbonded sites

Dividing Eqn. 19.38 by *n _{0}* gives 1 –

Noting that the solution must satisfy the mass balance, Eqn. 19.20, *x _{M}* +

Substituting Eqn. 19.40 gives 1 – 2*X*/(1 + *X*) = 4*X*^{2}*PK _{a}*(φ

Defining

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

where *g*(σ) is the radial distribution function at contact distance σ, ε* ^{C}* is the bond energy of the complex, and

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 *b _{D}* = 2

Simplifying gives,

Note that the vdW EOS corresponds roughly to *g*(σ) = 1/(1 – η_{P}). Also, the equilibrium constant, *K _{a}*, can be referenced to the critical temperature and written as,

The “best” expression for Δ*Cp/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 *K _{a}*. From a theoretical perspective, Wertheim’s analysis is the most sophisticated. Suresh and Elliott

Substitution shows the resulting relation between *K _{a}* and a portion of Eqn. 19.44,

where *(K _{ac}RT_{c})/(P°)* =

The next objective is to evaluate the impact on Helmholtz energy, *A ^{chem},* the change in Helmholtz energy due to bonding. As discussed in the introduction,

Or, on a apparent molar basis, (dividing by *n _{0}*),

In rearranging, note that *G ^{chem}* = Δμ

Further noting that

Substituting Eqns. 19.39 and 19.53 into Eqn. 19.51, and recalling *X* = *n _{M}/n_{0}*, we have

This turns out to be a very powerful equation.

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,

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,

We can express the fugacity of the fluid in two ways, noting that *f _{M}* =

By Eqns. 19.55 to 19.56,

Equating 19.58 to 19.59, we can immediately cancel terms of ln(1 – η* _{P}*) and ρ

noting that *x _{M}·n_{T}/n_{0}* =

Recall that *Z*/η* _{P}* =

We can evaluate ∂*X*/∂η* _{P}* through Eqn. 19.42 by differentiating implicitly.

Multiplying and dividing the left side by *X* and replacing *X*^{2}Δ with 1 – *X*, then multiplying and dividing the right side by Δ and replacing *X*^{2}Δ with 1 – *X*, we obtain,

Recalling the definition of Δ from Eqn. 19.44,

Substituting *Z ^{chem}* = (–1 + 1/

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 + *Z ^{rep}* +

Example 19.4. The chemical contribution to the equation of state

Assuming Δ is about 1000 at 300 K and ρ = 1.04 g/cm^{3}, estimate *A ^{chem}*/

Solution

Referring to Eqn. 19.66, *A ^{chem}*/

Referring to Eqn. 19.39, *x _{M}* = 2

Comparing *x _{M}* to

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 *n ^{A}* (the mole number of acceptors

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

Note that *n _{o}* refers to the same apparent number of moles discussed previously. Therefore,

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, *x ^{AD},* and the fraction of monomers not bonded,

*x ^{AD}* 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

That transition can be represented by

where *X ^{D}* is the fraction of unbonded donors and Δ

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,

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 Δ* ^{AD},* then Eqn. 19.73 gives

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 *i ^{th}* 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:

where *x _{i}* is the apparent mole fraction of component

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 ε* _{ij}^{AD},* it is implied that the acceptor is on the

The extension of *A ^{chem}* to any number of bonding sites or components becomes,

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 *A ^{chem}* implicitly through Δ, because a larger energy gives a larger value of Δ and a smaller value of

Eqns. 19.75–19.77 provide a powerful and versatile complement to our treatment of phase equilibria. In Chapters 10–12, 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 *aρ*/*RT*.

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, *x _{M}*. 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,

*n _{o}* = 100 cm

But how many moles of H_{2}O 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*·*n _{i}*. Note also that the true mole fractions,

We begin by noting that *K _{i}* =

Substituting Eqn. 19.41–19.44,

*n _{0}* =

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

Since the mole fractions must sum to unity, we can write a second balance, for *x _{i}*,

and again recognizing the series,

Substituting *x _{M}* for (1 –

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

Example 19.5. Molecules of H_{2}O in a 100 ml beaker

Assuming Δ is about 100 at room temperature and ρ = 1 g/cm^{3}, estimate the moles of H_{2}O monomer in a 100 ml beaker of liquid water.

Solution

Note that the problem statement requests moles of H_{2}O, not (H_{2}O)_{2} or (H_{2}O)_{3}, and so on, so we are interested in the true number of H_{2}O monomer moles. We know *n _{0}* = 5.556 by applying the monomer molecular weight, but the number of monomer moles

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

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. 19.75–19.77 completely characterize the temperature, density, and composition dependence of the chemical contribution to Helmholtz energy. The *Z ^{chem}* contribution is implied, but requires differentiation as in

The only way to fully determine all ∂*X _{i}*/∂

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 *A ^{chem}* can be rewritten as the stationary point of a generalized function

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

from *A ^{chem}*. The term

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

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

Note that the summation over 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,

Differentiating Eqn. 19.88,

The derivative of Δ* _{ij}^{AD}* 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. 19.75–19.77 can be further reduced in the special case where , which we refer to as the square root combining rule (SRCR). In general, Eqns. 19.75–19.77 require an iterative solution, as illustrated in Example 19.7. An initial guess for the iterations is

where for all *i,B*. Note: . 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. 19.75–19.77 and 19.91–19.93 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.

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 *A ^{chem}* must approach zero and the log term would then be undefined. This issue can be resolved by substituting, say, 1 –

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 *K*_{12}* ^{AD}* =

**a.** Derive *Z ^{chem}* and ln(φ

**b.** Evaluate the expressions for trimethylamine(1) + methanol(2) at *x*_{1} = 0.5, ρ = 0.0141mol/cm^{3}, and *T* = 300K.

Solution

**a.** Δ* _{ij}^{AD}* = ρ

Eqn. 19.65 shows that η*P*∂Δ/∂η*P* = Δ/(1 – η* _{P}*)

Substituting gives *Z ^{chem}* = –0.5

Similarly, *n*∂(Δ* _{ij}^{DA}*/

**b.** Evaluating these expressions, Δ_{11}* ^{DA}* = Δ

To solve Eqns. 19.93 for *X*_{1}* ^{A}*,

This gives three equations. Rearranging (below) shows that *X*_{1}* ^{A}* =

This shows that *X*_{2}* ^{D}* is almost completely bonded.

By Eqn. 19.88,

*h* = *x*_{1}^{2}*X*_{1}^{A}*X*_{1}^{D}Δ_{11}^{AD} + 2*x*_{1}*x*_{2}*X*_{1}^{A}*X*_{2}^{D}Δ_{12}^{AD} + *x*_{2}^{2}*X*_{2}^{A}*X*_{2}^{D}Δ_{22}^{AD} + *x*_{1}^{2}*X*_{1}^{D}*X*_{1}^{A}Δ_{11}^{DA} + 2*x*_{1}*x*_{2}*X*1^{D}*X*_{2}^{A}Δ_{12}^{DA} + *x*_{2}^{2}*X*_{2}^{D}*X*_{2}^{A}Δ_{22}^{DA}

Substituting *x*_{1}, *x*_{2}, and *X* gives

*h* = 0 + 2·*x*_{1}*x*_{2}·*X*_{1}^{A}*X*_{2}* ^{D·}*46.4 +

By Eqn. 19.92,

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 *X _{i}^{A}*< 0.5, and, in fact,

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

Evaluate the expressions for *Z ^{chem}* and ln(φ

Solution

The difference between this example and the previous is that ε_{12}* ^{AD}* ≠ ε

*Z ^{chem}* = –0.5

For ln(φ* _{k}^{chem}*): –∑

Substituting the mole fractions and solving for Δ’s,

*b* = 0.4·27.5 + 0.6·20.4 = 23.4; η* _{P}* = 0.0141·23.4 = 0.328. This is slightly less than Eqn 19.99.

Δ_{22}* ^{AD}* = ρ

1 – *X*_{1}^{A} = 0.5*X*_{1}^{A}*X*_{2}^{D}Δ_{12}^{AD}; 1 – *X*_{2}^{D} = 0.5*X*_{1}^{A}*X*_{2}^{D}Δ_{12}^{AD} + 0.5*X*_{2}^{A}*X*_{2}^{D}Δ_{22}^{AD};

1 – *X*_{2}^{A} = 0.5*X*_{2}^{A}*X*_{2}^{D}Δ_{22}^{AD};

*X*_{1}^{A} = 1/(1 + 0.4*X*_{2}^{D}Δ_{12}^{AD}); *X*_{2}^{A} = 1/(1 + 0.6*X*_{2}^{D}Δ_{22}^{AD});

*X*_{2}^{D} = 1/(1 + 0.4*X*_{1}^{A}Δ_{12}^{AD} + **0.6***X*_{2}^{A}Δ_{22}^{AD});

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 *X*_{2}* ^{D}*, then compute

Applying Eqn. 19.94,

(–1 + 1/*X*_{2}* ^{D}*) ≈ Σ

Five more iterations give, *X*_{1}^{A} = 0.678; *X*_{2}* ^{A}* = 0.297;

*X*_{1}* ^{A}* = 1/(1 + 0.4·0.0857·9.21)= 0.678

*X*_{2}* ^{A}* = 1/(1 + 0.6·0.0857·45.8)= 0.297

*X*_{2}* ^{D}* = 1/(1 + 0.4·0.678·9.21 + 0.6·0.297·45.8) = 0.0855

Substituting into Eqn. 19.88

*h* = *x*_{1}^{2}*X*_{1}^{A}*X*_{1}^{D}Δ_{11}^{AD} + 2*x*_{1}*x*_{2}*X*_{1}^{A}*X*_{2}^{D}Δ_{12}^{AD} + *x*_{2}^{2}*X*_{2}^{A}*X*_{2}^{D}Δ_{22}^{AD}

+ *x*_{1}^{2}*X*_{1}^{D}*X*_{1}^{A}Δ_{11}^{DA} + 2*x*_{1}*x*_{2}*X*_{1}^{D}*X*_{2}^{A}Δ_{12}^{DA} + *x*_{2}^{2}*X*_{2}^{D}*X*_{2}^{A}Δ_{22}^{DA}

*h* = 0 + 2·*x*_{1}*x*_{2}·*X*_{1}^{A}*X*_{2}^{D}·9.21 + *x*_{2}^{2}*X*_{2}^{A}*X*_{2}^{D}·45.8 + 0 + 0 + *x*_{2}^{2}*X*_{2}^{D}*X*_{2}^{A}·45.8 = 1.097

*Z ^{chem}* = –0.5

ln(φ_{1}^{chem}) = ln(*X*_{1}^{A}) – [*x*_{1}(0 + 0) + *x*_{2}(*X*_{1}^{A}*X*_{2}^{D}Δ_{12}^{AD} + 0)] – 0.5*hb*_{k}ρ/(1 – η_{P}) = –2.228

ln(φ_{2}^{chem}) = ln(*X*_{2}^{A}) + ln(*X*_{2}^{D}) – [*x*_{1}(0 + *X*_{2}^{D}*X*_{1}^{A}Δ_{21}^{DA})

+ *x*_{2}(*X*_{2}^{A}*X*_{2}^{D}Δ_{22}^{AD} + *X*_{2}^{D}*X*_{2}^{A}Δ_{22}^{DA})] – 0.5*hb*_{kρ}/(1 – η_{P}) = –5.855

These results show that a 20% change in ε_{12}^{AD} compared to Example 19.6 gives a 500% change in Δ_{12}^{AD}. That is fairly sensitive. This change in Δ_{12}^{AD} is primarily responsible for the increase in *X*_{1}^{A} from 0.520 to 0.679 and the decrease of *X*_{2}^{A} 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,

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

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

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}

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

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

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, *Z** ^{att}*. The original version suggested using second order perturbation contributions of the Lennard-Jones fluid for

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 *A ^{att}* given by:

*A ^{att}/RT* =

*A*_{1} = –11.61η_{P} – 8.28η_{P}^{2} – 5.24η_{P}^{3} + 34.21η_{P}^{4}

*A*_{2} = –25.76η* _{P}* + 181.87η

Express this model as an equation of state for alcohols, including *Z ^{chain};* that is

Solution

The CS equation is given by *Z _{hs}*–1 = 4η

Substituting,

*Z ^{chain}* =

*Z ^{chem}* = –0.5

Putting it all together, *Z*(*m*,η* _{P}*,

where *X ^{A}* = [–1 + (1 + 4Δ)

**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.

The PC-SAFT model has the same form as Example 15.8 except for *Z _{att}^{seg}*.

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 *Z _{hs}* from another, while ignoring

In this form, we recognize that *g*(σ)=1/(1–1.9η* _{P}*) provides consistency as a SAFT model. Then,

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–Z _{c}*)

where *a* = 1.9(9.5*q* – *k*_{1}) + 4*ck*_{1} and *k*_{1} = 1.7745

An interesting implication of this result is that *Z _{c}* → ⅓ in the infinite chain limit. Lue et al. showed that this is a general result for all SAFT models, despite the experimental observation that

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 Δ*H ^{vap}* and ρ

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}

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

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.

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.

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.

**a.** A gas-phase A+B system solvates *A* + *B* *AB* with *K _{a}* = 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

**b.** A liquid-phase A+B system solvates *A* + *B* *AB* with *K _{a}* = 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

**c.** A gas-phase A+B system associates 2*A* *A*_{2} with *K _{a}* = 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

**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:

versus *x*_{1}; versus *x*_{1}; versus *x*_{1}.

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 *G ^{E}* = –0.91 kJ/mol. Assuming only a 1-1 compound is formed, model the excess Gibbs energy with ideal chemical theory, and plot the

**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 *K _{AB}* = 0.8 bar

**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 cm^{3}/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 *x*_{A} = 0.4?

**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}

**a.** Let compound *A* be acetic acid and *B* be water. Calculate the true mole fractions of all the species from *y _{A}* = 0.05 to

**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?

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

expressed extensively, this becomes

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,

where *K _{i}* is unity for the monomers. Hint:

**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

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

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 *n _{T}*/

**19.10.** Furnish a proof that the concentration of true species *i* is maximum at composition *x _{A}*

**19.11.** Show that the result for *Z ^{assoc}* is obtained by taking the appropriate derivative of

**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 equation^{35} based on ε* _{HB}* and

**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 *k _{ij}* = 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:

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

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 *Z _{c}*, and fit the vapor density data as accurately as possible in the least squares sense.

**a.** Compute the values of *K _{a}*´,

**b.** Assuming an enthalpy of hydrogen bonding of 24 kJ/mole and Δ*C _{P}* = –

**19.19.** Derive the association model for the Peng-Robinson model, using the van’t Hoff formula with Δ*C _{P}*/

**a.** For methanol, determine the values of *K _{a}*′,

**b.** Determine the vapor pressure at *T _{r}* = 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 *P _{r}^{sat}* versus

**19.20.** Acetic acid has a much stronger tendency to dimerize than any alcohol. Therefore, it is not reasonable to assume that *K _{a}*

**a.** Determine the value of *K _{a}*

**b.** Determine the values of *K _{AD}*,

**c.** Determine the values of *K _{AD}*,

**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 (*N _{d}* = 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 Δ*H* = 15 kJ/mole.

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