In the previous sections we alluded to equations of state as empirical equations that may have appeared by magic. In this section and the next two, we attempt to de-mystify the origins behind equations of state by systematically describing the current outlook on equation of state development. It may seem like overkill to develop so much theory to justify such simple equations. As empirical equations go, equations of state are not much more difficult to accept than, say, Newton’s laws of motion. Nevertheless, our general purpose is for readers to learn to develop their own engineering model equations and to refute models that are not sensible. By establishing the connection between the nanoscopic potential function and macroscopic properties, molecular modeling lays the foundation for design at the nanoscale.
It is feasible to develop equations of state based solely on fitting experimental data. If the fit is insufficiently precise for a given application, simply add more parameters. We see evidence of this approach in the Peng-Robinson equation, where temperature and density dependencies are added to the parameter “a” in order to fit vapor pressure and density better. A more extensive example of this approach is evident in the 32 parameter Benedict-Webb-Rubin equation that forms the basis of the Lee-Kesler model. The IAPWS model of H2O is representative of the current state of this approach. It is the basis of the steam tables in Appendix E.
The shortcoming of this approach is that we lose the connection between the parameters in the equation of state and their physical meaning. For example, the Peng-Robinson “a” parameter must be related to attractive interactions, like the square-well parameter ε. But ε cannot be a function of temperature and density, so what part of the Peng-Robinson model is due to ε and what part is due to something else? If we could recover that physical connection, then all our efforts to fit data would result in systematically refined characterizations of the molecular interactions. With reliable characterizations of the molecular interactions, we could design molecules to assemble into a myriad of nanostructures: membranes for water purification, nanocomposites, polymer wrappers that block oxygen, artificial kidneys small enough to implant. The possibilities are infinite.
Since about 1960, computers have made it feasible to simulate macroscopic properties based on a specified intermolecular potential. With this tool, the procedure is clear: (1) Specify a potential model for a given molecule, (2) simulate the macroscopic properties, (3) evaluate the deviations between the simulated and experimental properties, (4) repeat until the deviations are minimized. This procedure is straightforward but tedious. Each simulation of Z(T,ρ) can take an hour or so.
Corresponding States in Molecular Dimensions
As engineers, we would like to get results faster. One idea is to leverage the principle of corresponding states. We know that ε has dimensions of J/molecule, so NAε has dimensions of J/mol. Therefore, RT/(NAε) would be dimensionless and serve in similar fashion to the usual reduced temperature, T/Tc. Similarly, the molecular volume, vmol, has dimensions of cm3/molecule and NAvmol has dimensions of cm3/mol. Therefore, NAvmolρ would be dimensionless and serve in very similar fashion to the usual reduced density, ρ/ρc. Another idea would be to tabulate the dimensionless properties from the simulation at many state points, then interpolate, similar to the steam tables. The interpolating equations might even resemble traditional equations of state in form and speed. The difference would be that they retain the connection between the nanoscopic potential model and macroscopic properties. In other words, we can engineer our equations of state to be consistent with specific potential models by expressing our “reduced” temperature and density using molecular dimensions. Then the principle of corresponding states can be applied to match the ε and σ for a particular molecule in the same way that we match a and b parameters in the van der Waals model.
Perhaps the most difficult part of understanding the molecular perspective is making the transformations from the macroscopic scale to the nanoscopic. For example, the “b” parameter has dimensions of cm3/mol. What does that imply about the diameter of the molecule in nm? As another example, the “a” parameter has dimensions of J-cm3/mol2. How does that relate to the molecular properties? Answering these questions leads to the introduction of a few “conversion shortcuts” to facilitate the scale transformations. One valuable conversion shortcut is to note that the transformation from cm3 to nm3 involves a factor of (107)3 or 1021. This transformation from cm3 to nm3 usually goes hand-in-hand with a transformation from moles to molecules, involving a factor of NA. If we write NA = 602(1021) instead of 6.02(1023), then factors of 1021 cancel conveniently. This convenience motivates us to work in cm3 at the macroscopic level. Another shortcut is to write the volume of a sphere in terms of diameter instead of radius. Finally, we note that the molecular volume on a molar basis is equivalent to the “b” parameter in cm3/mol. Altogether,
Example 7.8. Estimating molecular size
Example 7.4 shows that b = 19.9cm3/mol for argon. Estimate the diameter (nm) of argon according to the Peng-Robinson model.
Solution
NAπσ3/6 = 19.9 cm3/mol; σ3 = 6(19.9cm3/mol)(1mol/602(1021)molecules)(1021 nm3/cm3)/π Thus, σ3 = 6(19.9)/(602π) = 0.06313 nm3; σ = (0.06313)⅓ = 0.398 nm.
Speaking of the “b” parameter, it is useful to note that the combination of bρ appears in the equations quite often. This combined variable is very important. In addition to being dimensionless, and a convenient reduced density, its meaning is quite significant. It represents the volume occupied by molecules divided by the total volume. It makes sense intuitively that the density cannot be higher than when the total volume is completely filled. So this explains why the van der Waals equation includes (1–bρ) in the denominator, forcing divergence as this limit is approached. The prevalence of this combined variable suggests that we give it a special symbol and name, ηP = bρ = b/V, the packing efficiency (aka. packing fraction).10
Finally, we should consider the square-well energy parameter, ε, and the van der Waals parameter, a. Applying Eqn. 7.13 indicates that the dimensions of the “a” parameter are J-cm3/mol2. We can rewrite the van der Waals equation as Z = 1/(1–ηP) – (a/bRT)ηP. In this format, it is clear that the combination of variables “a/b” represents an attractive energy in J/mol. In other words, a/b ~ NAε. Another shortcut for quickly transforming from the macro scale to the nano scale is to recognize that ε/k = NAε/R and both have dimensions of absolute temperature, K. In this context, the combination of variables ε/kT = βε is an especially convenient characterization of dimensionless reciprocal temperature, where β=1/kT.
Distinguishing Repulsive and Attractive Effects
One of the advantages of molecular modeling is that the potential model can be dissected into various parts: the repulsive core, attractive wells, dipole moments, hydrogen bonding, and so forth. The total potential function is the sum of all of these interactions, but simulations can be done separately with one, two, or all interactions. Then we can understand which parts of the equation of state come from each part of the potential model. Fig. 7.7 illustrates Z versus reciprocal temperature. This shows a specific y-intercept at infinite temperature. Analyzing the van der Waals equation shows that this y-intercept corresponds to Z0 = 1/(1–bρ), where the subscript “0” designates the point where reciprocal temperature reaches zero. This contribution represents positive deviations from ideality, and therefore we can call it repulsive. The temperature-dependent part of the van der Waals equation is negative and represents attractive contributions. The reason that the attractive contribution becomes negligible at high temperature is that such a large molecular kinetic energy overwhelms the relatively small “stickiness” of the molecular attractions. Only the repulsive interaction is large enough to contribute at high temperature.
If the y-intercept is so important then what does the x-intercept mean? We have seen Z ~ 0 before, in Example 7.4, where Z = 0.016 for the liquid root. On the scale of Fig. 7.7, Z = 0.016 is practically zero. The pressure of the saturated liquid is low despite having a high density (and high repulsion) because the attractions are comparable to the repulsions when the temperature is low enough. Slow-moving molecules show a greater tendency to “stick together.”
Ultimately, it is necessary to characterize the parameters that relate the intermolecular potential to experimental data. With sufficient data for the compressed liquid density, the problem of characterizing ε and σ becomes a simple matter of matching the slope and intercept of a plot like Fig. 7.7. The procedure is illustrated in Example 7.9(b). This is the most straightforward approach because it relates experimental PVT data directly to PVT data from a molecular simulation.
Fig. 7.7 also compares to experimental compressed liquid data for argon. These data transition quickly to supercritical temperatures and pressures, but the trend is smooth (almost linear) because the density is constant (i.e., isochoric). We can fit the van der Waals equation at one density by tuning the a and b parameters, as shown in Example 7.9(a). Deviations are large, however, when we apply the van der Waals model to a different density using the same a and b. This reflects deficiencies in the physics of the van der Waals model.
We can improve the characterization of argon by using the square-well potential with λ = 1.7. Once again, the parameters (ε and σ this time) are tuned to the Z versus 1/T data at the high density as shown in Example 7.9(b). The predictions (using the same ε and σ) are much better, as shown by the solid lines in Fig. 7.7, reflecting the improved physics underlying the square-well model coupled with molecular simulation. Systematically studying the square-well model, and dissecting the repulsive and attractive contributions, leads to a better understanding of the molecular interactions, and this leads to better predictions.
Other approaches exist to infer potential parameters from experimental data, but they are too complicated for our introductory treatment. One alternative is to apply experimental data for saturated vapor pressure and density. This approach accounts better for vapor pressure being a very important property in chemical engineering, and more data are available. As another alternative, you may be wondering about fitting the critical point, as done by van der Waals. Unfortunately, the behavior at the critical point does not conform to the rules of normal calculus and even simulations are challenging. If you read the fine print on the theorems of calculus, you find the stipulation that functions must be analytic for the theorems to apply. Phenomena in the critical region are non-analytic. The non-analytic behavior is universal for all compounds, so the principle of corresponding states is still valid. On the other hand, fitting a simple analytic function to data outside the critical region leads to inconsistencies inside the critical region, and vice versa. Cubic equations exhibit this inconsistency by predicting a PV phase envelope that is not flat enough on the top. Dealing further with these inconsistencies is a topic of current research. Methods that avoid the critical region are gaining favor at present.
We have barely scratched the surface of what is necessary to characterize the intermolecular forces between all the molecules that we can imagine. For example, we have only considered the square-well potential, but the Lennard-Jones model would be more realistic, and those are just two of the possibilities. As another example, the presentation here is limited to spherical molecules. The molecular perspective is not extended to non-spherical molecules until Chapter 19, and then only briefly.11 You might say that the Peng-Robinson equation can be applied to non-spherical molecules, but only because of clever fitting. The physics behind the Peng-Robinson is simply the same as that of van der Waals: spherical. Simple physics and educated empirical fitting are cornerstones of engineering models. The Peng-Robinson model is a prime example of what can be accomplished with that approach. But recognize that we are always learning more about physics and those new insights are the cornerstones of new technology. Accurately characterizing intermolecular forces involves characterizing many small molecules that share common fragments. When those fragments are characterized, they can be assembled to predict the properties of large molecules. Ultimately, we can imagine a time when nanostructures can be designed and constructed the way civil engineers build bridges today. These structures occur naturally in everything from sea shells to proteins. Learning how to do it is a basis for modern research.
Example 7.9. Characterizing molecular interactions
Based on Fig. 7.7, trend lines indicate y-intercept values of, roughly, 5.7 and 4.7 when fit to the isochoric PVT data for argon at 1.38g/cm3 and 1.25 g/cm3, respectively. Similarly, the x-intercepts are roughly 11.2 and 9.5, respectively. Use these values to estimate the EOS parameters.
a. Estimate the values of a and b at 1.38 g/cm3 according to the van der Waals model.
b. Predict the values of x– and y– intercepts at 1.25 g/cm3 using the a and b from part (a).
c. Suppose the square-well simulation data can be represented by:
Z = 1+4 ηP/(1–1.9 ηP)–15.7 ηP βε/(1–0.16 ηP)
Estimate the values of σ and ε/k at 1.38g/cm3 and predict the x– and y-intercepts at 1.25 g/cm3.
Solution
a. At 1.38g/cm3, y-intercept, Z0 = 1/(1 – ηP) = 5.7 => ηP = 1 – 1/5.7 = 0.825 = bρ.
b = 0.825·39.9(g/mol)/1.38(g/cm3) = 23.9 cm3/mol.
At the x-intercept, 0 = 5.7 – (a/bRT)·ηP = 5.7 – (a/bRT)·0.825 => a/bRT = 5.7/0.825 = 6.91. Using the x-intercept to determine temperature, 1000/T = 11.2 => T = 1000/11.2 = 89.3K => a = 23.9(8.314)89.3(6.91) = 123 kJ-cm3/mol2.
b. At 1.25 g/cm3, ηP = 23.9(1.25)/39.9 = 0.7487 => Z0 = 1/(1 – ηP) = 4.0 = y-intercept.
At the x-intercept, 0 = 4.0 – 123000/(23.9RT)·0.7487 = 4.0 – 463/T => T = 463/4 = 116.
Therefore, the x-intercept is 1000/T = 1000/116 = 8.6. These x– and y– intercepts form the basis for the dashed line in Fig. 7.7 at 1.25 g/cm3. The prediction of the van der Waals model is poor.
c. The procedure for finding σ and ε/k is similar. At the 1.38g/cm3,
Z0 = 1 + 4 ηP/(1 – 1.9 ηP) = 5.7 => ηP (4 + 4.7·1.9) = 4.7 => ηP = 0.363 = bρ
b = 0.363·39.9/1.38 = 10.5 cm3/mol = NAπσ3/6 => σ = 0.322 nm
At the x-intercept, 0 = 5.7 – 15.7(0.363)βε/(1 – 0.16·0.363) => βε = 0.942;
1000/T = 11.2 => T = 1000/11.2 = 89.3K => ε/k = (0.942)89.3= 84.1 K
At 1.25 g/cm3, following the same procedure:
ηP = 10.5(1.25)/39.9 = 0.329 => Z0 = 1 + 4ηP/(1 – ηP) = 4.5 = y-intercept.
At the x-intercept, 0 = 4.5 – 15.7(0.329)βε/(1 – 0.16·0.329) = 4.5 – (5.452)βε => βε = 0.827 => T = 84.1/0.827 = 102. Therefore, the x-intercept is 1000/T = 1000/102 = 9.8. These x– and y-intercepts form the basis for the solid line in Fig. 7.7 at 1.25 g/cm3, and the prediction is quite good.
To put the significance of this analysis in perspective, imagine you were designing a material with pores just the right size to capture argon from air using the van der Waals model. The diameters would indicate the appropriate pore size. For the van der Waals model σ = (6·23.9/602π)1/3 =0.423nm compared to 0.322nm by the square-well estimate. This means that your pores could be over sized by more than 30%. Improved physical insight can suggest more successful experiments.
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